A Perturbation of the Dunkl Harmonic Oscillator on the Line
Let Jσ be the Dunkl harmonic oscillator on R (σ>−1/2. For 0<u<1 and ξ>0, it is proved that, if σ>u−1/2, then the operator U=Jσ+ξ|x|⁻²u, with appropriate domain, is essentially self-adjoint in L²(R,|x|²σdx), the Schwartz space S is a core of Ū¹/², and Ū has a discrete spectrum, which i...
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Álvarez López, J.A. Calaza, M. Franco, C. 2019-02-13T17:16:09Z 2019-02-13T17:16:09Z 2015 A Perturbation of the Dunkl Harmonic Oscillator on the Line / J.A. Álvarez López, M. Calaza, C. Franco // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 28 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 47A55; 47B25; 33C45 DOI:10.3842/SIGMA.2015.059 https://nasplib.isofts.kiev.ua/handle/123456789/147130 Let Jσ be the Dunkl harmonic oscillator on R (σ>−1/2. For 0<u<1 and ξ>0, it is proved that, if σ>u−1/2, then the operator U=Jσ+ξ|x|⁻²u, with appropriate domain, is essentially self-adjoint in L²(R,|x|²σdx), the Schwartz space S is a core of Ū¹/², and Ū has a discrete spectrum, which is estimated in terms of the spectrum of Ĵσ. A generalization Jσ,τ of Jσ is also considered by taking different parameters σ and τ on even and odd functions. Then extensions of the above result are proved for Jσ,τ, where the perturbation has an additional term involving, either the factor x⁻¹ on odd functions, or the factor x on even functions. Versions of these results on R+ are derived. The first author was partially supported by MICINN, Grants MTM2011-25656 and MTM2014- 56950-P, and by Xunta de Galicia, Grant Consolidaci´on e estructuraci´on 2015 GPC GI-1574. The third author has received financial support from the Xunta de Galicia and the European Union (European Social Fund - ESF). en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications A Perturbation of the Dunkl Harmonic Oscillator on the Line Article published earlier |
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A Perturbation of the Dunkl Harmonic Oscillator on the Line |
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A Perturbation of the Dunkl Harmonic Oscillator on the Line Álvarez López, J.A. Calaza, M. Franco, C. |
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A Perturbation of the Dunkl Harmonic Oscillator on the Line |
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A Perturbation of the Dunkl Harmonic Oscillator on the Line |
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A Perturbation of the Dunkl Harmonic Oscillator on the Line |
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A Perturbation of the Dunkl Harmonic Oscillator on the Line |
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perturbation of the dunkl harmonic oscillator on the line |
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Álvarez López, J.A. Calaza, M. Franco, C. |
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Álvarez López, J.A. Calaza, M. Franco, C. |
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Symmetry, Integrability and Geometry: Methods and Applications |
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Інститут математики НАН України |
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Let Jσ be the Dunkl harmonic oscillator on R (σ>−1/2. For 0<u<1 and ξ>0, it is proved that, if σ>u−1/2, then the operator U=Jσ+ξ|x|⁻²u, with appropriate domain, is essentially self-adjoint in L²(R,|x|²σdx), the Schwartz space S is a core of Ū¹/², and Ū has a discrete spectrum, which is estimated in terms of the spectrum of Ĵσ. A generalization Jσ,τ of Jσ is also considered by taking different parameters σ and τ on even and odd functions. Then extensions of the above result are proved for Jσ,τ, where the perturbation has an additional term involving, either the factor x⁻¹ on odd functions, or the factor x on even functions. Versions of these results on R+ are derived.
|
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1815-0659 |
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https://nasplib.isofts.kiev.ua/handle/123456789/147130 |
| citation_txt |
A Perturbation of the Dunkl Harmonic Oscillator on the Line / J.A. Álvarez López, M. Calaza, C. Franco // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 28 назв. — англ. |
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2025-11-25T22:54:44Z |
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2025-11-25T22:54:44Z |
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1850576321134985216 |
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 11 (2015), 059, 47 pages
A Perturbation of the Dunkl Harmonic Oscillator
on the Line
Jesús A. ÁLVAREZ LÓPEZ †, Manuel CALAZA ‡ and Carlos FRANCO †
† Departamento de Xeometŕıa e Topolox́ıa, Facultade de Matemáticas,
Universidade de Santiago de Compostela, 15782 Santiago de Compostela, Spain
E-mail: jesus.alvarez@usc.es, carlosluis.franco@usc.es
‡ Laboratorio de Investigación 2 and Rheumatology Unit,
Hospital Clinico Universitario de Santiago, Santiago de Compostela, Spain
E-mail: manuel.calaza@usc.es
Received February 19, 2015, in final form July 20, 2015; Published online July 25, 2015;
Corrected June 28, 2017
https://doi.org/10.3842/SIGMA.2015.059
Abstract. Let Jσ be the Dunkl harmonic oscillator on R (σ > − 1
2 ). For 0 < u < 1
and ξ > 0, it is proved that, if σ > u − 1
2 , then the operator U = Jσ + ξ|x|−2u, with
appropriate domain, is essentially self-adjoint in L2(R, |x|2σdx), the Schwartz space S is
a core of U
1/2
, and U has a discrete spectrum, which is estimated in terms of the spectrum
of Jσ. A generalization Jσ,τ of Jσ is also considered by taking different parameters σ and τ
on even and odd functions. Then extensions of the above result are proved for Jσ,τ , where
the perturbation has an additional term involving, either the factor x−1 on odd functions,
or the factor x on even functions. Versions of these results on R+ are derived.
Key words: Dunkl harmonic oscillator; perturbation theory
2010 Mathematics Subject Classification: 47A55; 47B25; 33C45
1 Introduction
The Dunkl operators on Rn were introduced by Dunkl [8, 9, 10], and gave rise to what is now
called the Dunkl theory [25]. They play an important role in physics and stochastic processes
(see, e.g., [13, 24, 27]). In particular, the Dunkl harmonic oscillators on Rn were studied in
[11, 19, 20, 23]. We will consider only this operator on R, where it is uniquely determined by
one parameter. In this case, a conjugation of the Dunkl operator was previously introduced by
Yang [28] (see also [21]).
Let us fix some notation that is used in the whole paper. Let S = S(R) be the Schwartz
space on R, with its Fréchet topology. It decomposes as direct sum of subspaces of even and
odd functions, S = Sev ⊕ Sodd. The even/odd component of a function in S is denoted with
the subindex ev/odd. Since Sodd = xSev, where x is the standard coordinate of R, x−1φ ∈ Sev
is defined for φ ∈ Sodd. Let L2
σ = L2(R, |x|2σdx) (σ ∈ R), whose scalar product and norm
are denoted by 〈 , 〉σ and ‖ ‖σ. The above decomposition of S extends to an orthogonal
decomposition, L2
σ = L2
σ,ev ⊕ L2
σ,odd, because the function |x|2σ is even. S is a dense subspace
of L2
σ if σ > −1
2 , and Sodd is a dense subspace of L2
τ,odd if τ > −3
2 . Unless otherwise stated, we
assume σ > −1
2 and τ > −3
2 . The domain of a (densely defined) operator P in a Hilbert space is
denoted by D(P ). If P is closable, its closure is denoted by P . The domain of a (densely defined)
sesquilinear form p in a Hilbert space is denoted by D(p). The quadratic form of p is also denoted
by p. If p is closable, its closure is denoted by p̄. For an operator in L2
σ preserving the above
decomposition, its restrictions to L2
σ,ev/odd will be indicated with the subindex ev/odd. The
jesus.alvarez@usc.es
carlosluis.franco@usc.es
manuel.calaza@usc.es
https://doi.org/10.3842/SIGMA.2015.059
2 J.A. Álvarez López, M. Calaza and C. Franco
operator of multiplication by a continuous function h in L2
σ is also denoted by h. The harmonic
oscillator is the operator H = − d2
dx2 + s2x2 (s > 0) in L2
0 with D(H) = S.
The Dunkl operator on R is the operator T in L2
σ, with D(T ) = S, determined by T = d
dx
on Sev and T = d
dx + 2σx−1 on Sodd, and the Dunkl harmonic oscillator on R is the operator
J = −T 2 + s2x2 in L2
σ with D(J) = S. Thus J preserves the above decomposition of S, being
Jev = H − 2σx−1 d
dx and Jodd = H − 2σ d
dxx
−1. The subindex σ is added to J if needed. This J
is essentially self-adjoint, and the spectrum of J is well known [23]; in particular, J > 0. In fact,
even for τ > −3
2 , the operator Jτ,odd is defined in L2
τ,odd with D(Jτ,odd) = Sodd because it is
a conjugation of Jτ+1,ev by a unitary operator (Section 2). Some operators of the form J + ξx−2
(ξ ∈ R) are conjugates of J by powers |x|a (a ∈ R), and therefore their study can be reduced to
the case of J [3]. Our first theorem analyzes a different perturbation of J .
Theorem 1.1. Let 0 < u < 1 and ξ > 0. If σ > u − 1
2 , then there is a positive self-adjoint
operator U in L2
σ satisfying the following:
(i) S is a core of U1/2, and, for all φ, ψ ∈ S,〈
U1/2φ,U1/2ψ
〉
σ
= 〈Jφ, ψ〉σ + ξ
〈
|x|−uφ, |x|−uψ
〉
σ
. (1.1)
(ii) U has a discrete spectrum. Let λ0 ≤ λ1 ≤ · · · be its eigenvalues, repeated according to
their multiplicity. There is some D = D(σ, u) > 0, and, for each ε > 0, there is some
C = C(ε, σ, u) > 0 so that, for all k ∈ N,
(2k + 1 + 2σ)s+ ξDsu(k + 1)−u ≤ λk ≤ (2k + 1 + 2σ)(s+ ξεsu) + ξCsu. (1.2)
Remark 1.2. In Theorem 1.1, observe the following:
(i) The second term of the right hand side of (1.1) makes sense because |x|−uS ⊂ L2
σ since
σ > u− 1
2 .
(ii) U = U , where U := J + ξ|x|−2u with D(U) =
⋂∞
m=0 D(Um) (see [14, Chapter VI, Sec-
tion 2.5]). The more explicit notation Uσ will be also used if necessary.
(iii) The restrictions Uev/odd are self-adjoint in L2
σ,ev/odd and satisfy (1.1) with φ, ψ ∈ Sev/odd
and (1.2) with k even/odd. In fact, by the comments before the statement, Uτ,odd is
defined and satisfies these properties if τ > u− 3
2 .
To prove Theorem 1.1, we consider the positive definite symmetric sesquilinear form u defined
by the right hand side of (1.1). Perturbation theory [14] is used to show that u is closable and ū
induces a self-adjoint operator U , and to relate the spectra of U and J . Most of the work
is devoted to check the conditions to apply this theory so that (1.2) follows; indeed, (1.2) is
stronger than a general eigenvalue estimate given by that theory (Remark 3.22).
The following generalizations of Theorem 1.1 follow with a simple adaptation of the proof.
If ξ < 0, we only have to reverse the inequalities of (1.2). In (1.1), we may use a finite
sum
∑
i ξi〈|x|−uiφ, |x|−uiψ〉σ, where 0 < ui < 1, σ > ui − 1
2 and ξi > 0; then (1.2) would
be modified by using maxi ui and mini ξi in the left hand side, and maxi ξi in the right hand
side. In turn, this can be extended by taking Rp-valued functions (p ∈ Z+), and a finite sum∑
i〈|x|−uiΞiφ, |x|−uiψ〉σ in (1.1), where each Ξi is a positive definite self-adjoint endomorphism
of Rp; then the minimum and maximum eigenvalues of all Ξi would be used in (1.2).
As an open problem, we may ask for a version of Theorem 1.1 using Dunkl operators on Rn,
but we are interested in the following different type of extension. For σ > −1
2 and τ > −3
2 ,
let L2
σ,τ = L2
σ,ev ⊕ L2
τ,odd, whose scalar product and norm are denoted by 〈 , 〉σ,τ and ‖ ‖σ,τ .
Matrix expressions of operators refer to this decomposition. Let Jσ,τ = Jσ,ev ⊕ Jτ,odd in L2
σ,τ ,
with D(Jσ,τ ) = S. The hypotheses of the generalization of Theorem 1.1 are rather involved to
cover enough cases of certain application that will be indicated.
A Perturbation of the Dunkl Harmonic Oscillator on the Line 3
Theorem 1.3. Let ξ > 0 and η ∈ R, let
0 < u < 1, σ > u− 1
2 , τ > u− 3
2 , θ > −1
2 , (1.3)
and set v = σ + τ − 2θ. Suppose that the following conditions hold:
(a) If σ = θ 6= τ and τ − σ 6∈ −N, then
σ − 1 < τ < σ + 1, 2σ + 1
2 . (1.4)
(b) If σ 6= θ = τ and σ − τ 6∈ −N, then
−τ, τ − 1 < σ < 3τ + 1, 11τ + 2, τ + 1. (1.5)
(c) If σ 6= θ = τ + 1 and σ − τ − 1 6∈ −N, then
τ + 1 < σ < τ + 3, 2τ + 7
2 . (1.6)
(d) If σ 6= θ 6= τ and σ − θ, τ − θ 6∈ −N, then
σ−τ
2 − 1, τ−σ2 , σ+τ−1
4 , σ+3τ−2
14 , 3σ+τ−4
14 , σ+τ−1
2 < θ < σ+τ+1
2 ,
τ − 1 < σ < τ + 3.
(1.7)
Then there is a positive self-adjoint operator V in L2
σ,τ satisfying the following:
(i) S is a core of V1/2, and, for all φ, ψ ∈ S,〈
V1/2φ,V1/2ψ
〉
σ,τ
= 〈Jσ,τφ, ψ〉σ,τ + ξ
〈
|x|−uφ, |x|−uψ
〉
σ,τ
+ η
(〈
x−1φodd, ψev
〉
θ
+
〈
φev, x
−1ψodd
〉
θ
)
. (1.8)
(ii) Let ςk = σ if k is even, and ςk = τ if k is odd. V has a discrete spectrum. Its eigenvalues
form two groups, λ0 ≤ λ2 ≤ · · · and λ1 ≤ λ3 ≤ · · · , repeated according to their multiplicity,
such that there is some D = D(σ, τ, u) > 0 and, for every ε > 0, there are some C =
C(ε, σ, τ, u) > 0 and E = E(ε, σ, τ, θ) > 0 so that, for all k ∈ N,
λk ≥ (2k + 1 + 2ςk)
(
s− 2|η|εs
v+1
2
)
+ ξDsu(k + 1)−u − 2|η|Es
v+1
2 , (1.9)
λk ≤ (2k + 1 + 2ςk)
(
s+ ε
(
ξsu + 2|η|s
v+1
2
))
+ ξCsu + 2|η|Es
v+1
2 . (1.10)
(iii) Let ũ ∈ R such that
0, v, τ − 2θ + 1
2 , σ − 2θ − 1
2 < ũ < 1, v + 1, σ + 1
2 , τ + 3
2 , (1.11)
and let û = max{ũ, v+ 1− ũ}. There is some D = D(σ, τ, u) > 0 and, for any ε > 0, there
is some C̃ = C̃(ε, σ, τ, u) > 0 so that, for all k ∈ N,
λk ≥ (2k + 1 + 2ςk)
(
s− |η|εsû
)
+ ξDsu(k + 1)−u − |η|C̃sû. (1.12)
(iv) If u = v+1
2 and ξ ≥ |η|, then there is some D̃ = D̃(σ, τ, u) > 0 so that, for all k ∈ N,
λk ≥ (2k + 1 + 2ςk)s+ (ξ − |η|)D̃su(k + 1)−u. (1.13)
4 J.A. Álvarez López, M. Calaza and C. Franco
(v) If we add the term ξ′〈φev, ψev〉σ + ξ′′〈φodd, ψodd〉τ to the right hand side of (1.8), for some
ξ′, ξ′′ ∈ R, then the result holds as well with the additional term max{ξ′, ξ′′} in the right
hand side of (1.10), and the additional term, ξ′ for k ∈ 2N and ξ′′ for k ∈ 2N + 1, in the
right hand sides of (1.9), (1.12) and (1.13).
Remark 1.4. Note the following in Theorem 1.3:
(i) Like in Remark 1.2(ii), we have V = V , where
V =
(
Uσ,ev η|x|2(θ−σ)x−1
η|x|2(θ−τ)x−1 Uτ,odd
)
,
with D(V ) =
⋂∞
m=0 D(Vm). Note that the adjoint of |x|2(θ−σ)x−1 : Sodd → |x|2(θ−σ)Sev, as
a densely defined operator of L2
τ,odd to L2
σ,ev, is given by |x|2(θ−τ)x−1, with the appropriate
domain.
(ii) Taking θ′ = θ − 1 > −3
2 , since
〈xφ, ψ〉θ′ =
〈
φ, x−1ψ
〉
θ
for all φ ∈ Sev and ψ ∈ Sodd, we can write (1.8) as〈
V1/2φ,V1/2ψ
〉
σ,τ
= 〈Jσ,τφ, ψ〉σ,τ + ξ
〈
|x|−uφ, |x|−uψ
〉
σ,τ
+ η
(
〈φodd, xψev〉θ′ + 〈xφev, ψodd〉θ′
)
for all φ, ψ ∈ S, and, correspondingly,
V =
(
Uσ,ev η|x|2(θ′−σ)x
η|x|2(θ′−τ)x Uτ,odd
)
.
(iii) The conditions (1.4), (1.5) and (1.6) describe three convex open subsets of R2 (Fig. 1).
The condition (1.7) describes a convex open subset of R3 (Fig. 2), which is symmetric with
respect to the plane defined by σ = τ + 1. It is a “semi-infinite bar” with 4 lateral faces,
and 5 faces at the “bounded end.”
(iv) In Theorem 1.3(iii), the condition (1.11) means that (1.3) also holds with ũ and v+ 1− ũ
instead of u. There exists ũ satisfying (1.11) just when
0, v, τ − 2θ + 1
2 , σ − 2θ − 1
2 < 1, v + 1, σ + 1
2 , τ + 3
2 . (1.14)
This property is satisfied in the cases (b) and (d) by (1.3), (1.5) and (1.7); in particular,
we can take ũ = v+1
2 . In the case (a), if τ < 3σ, then (1.14) holds by (1.3) and (1.4). In
the case (c), if σ < 3τ + 4, then (1.14) holds by (1.3) and (1.6).
The main arguments of the proofs of Theorems 1.1 and 1.3 are given in Sections 3–5. But
some needed estimates are postponed to Sections 6 and 7 because they are of rather independent
nature, and with rather long and tedious proofs.
Versions of these results on R+ are also derived in Section 8 (Corollaries 8.1, 8.2 and 8.3).
In [4], these corollaries are used to study a version of the Witten’s perturbation ∆s of the
Laplacian on strata with the general adapted metrics of [6, 17, 18]. This gives rise to an analytic
proof of Morse inequalities in strata involving intersection homology of arbitrary perversity,
which was our original motivation. The simplest case of adapted metrics, corresponding to
the lower middle perversity, was treated in [2] using an operator induced by J on R+. The
perturbations of J studied here show up in the local models of ∆s when general adapted metrics
are considered. Some details of this application are given in Section 9.
A Perturbation of the Dunkl Harmonic Oscillator on the Line 5
(a) Set defined by (1.4). (b) Set defined by (1.5). (c) Set defined by (1.6).
Figure 1. Sets in Theorem 1.3(a),(b),(c).
Figure 2. Set defined by (1.7) in Theorem 1.3(d).
2 Preliminaries
The Dunkl annihilation and creation operators are B = sx+T and B′ = sx−T (s > 0). Like J ,
the operators B and B′ are considered in L2
σ with domain S. They are perturbations of the
usual annihilation and creation operators. The operators T , B, B′ and J are continuous on S.
The following properties hold [3, 23]:
• B′ is adjoint of B, and J is essentially self-adjoint.
• The spectrum of J consists of the eigenvalues1 (2k + 1 + 2σ)s, k ∈ N, of multiplicity one.
• The corresponding normalized eigenfunctions φk are inductively defined by
φ0 = s(2σ+1)/4Γ
(
σ + 1
2
)− 1
2 e−sx
2/2, (2.1)
φk =
{
(2ks)−
1
2B′φk−1 if k is even,
(2(k + 2σ)s)−
1
2B′φk−1 if k is odd,
k ≥ 1. (2.2)
• The eigenfunctions φk also satisfy
Bφ0 = 0, (2.3)
Bφk =
{
(2ks)
1
2φk−1 if k is even,
(2(k + 2σ)s)
1
2φk−1 if k is odd,
k ≥ 1. (2.4)
•
⋂∞
m=0 D
(
J
m)
= S.
1It is assumed that 0 ∈ N.
6 J.A. Álvarez López, M. Calaza and C. Franco
By (2.1) and (2.2), we get φk = pke
−sx2/2, where pk is the sequence of polynomials inductively
given by p0 = s(2σ+1)/4Γ(σ + 1
2)−
1
2 and
pk =
{
(2ks)−
1
2 (2sxpk−1 − Tpk−1) if k is even,
(2(k + 2σ)s)−
1
2 (2sxpk−1 − Tpk−1) if k is odd,
k ≥ 1.
Up to normalization, pk is the sequence of generalized Hermite polynomials [26, p. 380, Prob-
lem 25], and φk is the sequence of generalized Hermite functions. Each pk is of degree k, even/odd
if k is even/odd, and with positive leading coefficient. They satisfy the recursion formula [3,
equation (13)]
pk =
{
k−
1
2
(
(2s)
1
2xpk−1 − (k − 1 + 2σ)
1
2 pk−2
)
if k is even,
(k + 2σ)−
1
2
(
(2s)
1
2xpk−1 − (k − 1)
1
2 pk−2
)
if k is odd.
(2.5)
When k = 2m+ 1 (m ∈ N), we have [3, equation (14)]
x−1pk =
m∑
i=0
(−1)m−i
√
m!Γ(i+ 1
2 + σ)s
i!Γ(m+ 3
2 + σ)
p2i. (2.6)
The Pochhammer symbol could be used to simplify this expression, as well as many other
expressions in Sections 3 and 4. However there are quotients of gamma functions in Sections 4
and 5 that can not be simplified in this way (see e.g. Proposition 4.7). Thus, for the sake of
uniformity, we use gamma functions in all quotients of this type.
Let j be the positive definite symmetric sesquilinear form in L2
σ, with D(j) = S, given by
j(φ, ψ) = 〈Jφ, ψ〉σ. Like in the case of J , the subindex σ will be added to the notation T , B, B′
and φk and j if necessary. Observe that
Bσ =
{
Bτ on Sev,
Bτ + 2(σ − τ)x−1 on Sodd,
(2.7)
B′σ =
{
B′τ on Sev,
B′τ + 2(τ − σ)x−1 on Sodd.
(2.8)
The operator x : Sev → Sodd is a homeomorphism [3], which extends to a unitary operator
x : L2
σ,ev → L2
σ−1,odd. We get xJσ,evx
−1 = Jσ−1,odd because x
[
d2
dx2 , x
−1
]
= −2 d
dxx
−1. Thus, even
for any τ > −3
2 , the operator Jτ,odd is densely defined in L2
τ,odd, with D(Jτ,odd) = Sodd, and has
the same spectral properties as Jτ+1,ev; in particular, the eigenvalues of Jτ,odd are (2k+ 1 + 2τ)s
(k ∈ 2N + 1), and φτ,k = xφτ+1,k−1.
To prove the results of the paper, alternative arguments could be given by using the expression
of the generalized Hermite polynomials in terms of the Laguerre ones (see, e.g., [24, p. 525] or
[25, p. 23]). In particular, certain asymptotic estimates of Laguerre functions [12, 15] (see
also [5, 16]), yield the following asymptotic estimates of the generalized Hermite functions [1,
Section 2.4]: there are some C, c > 0, depending only on σ, such that
|φk(x)xσ| ≤
Cs
σ̄
2
+ 1
4xσ̄ν
σ̄
2
− 1
4 if 0 < x ≤
√
1
sν ,
Cs
1
4 ν−
1
4 if
√
1
sν < x ≤
√
ν
2s ,
Cs
1
4 (ν
1
3 + |sx2 − ν|)−
1
4 if
√
ν
2s < x ≤
√
3ν
2s ,
C(sx)
1
2 e−csx
2
if
√
3ν
2s < x,
(2.9)
where σ̄ = σ̄k = σ + 1−(−1)k
2 and ν = νk = 2k + 1 + 2σ, with the proviso that we must take
ν = 2 if k = 0 and σ < 1
2 .
A Perturbation of the Dunkl Harmonic Oscillator on the Line 7
3 The sesquilinear form t
Let 0 < u < 1 such that σ > u − 1
2 . Then |x|−uS ⊂ L2
σ, and therefore a positive definite
symmetric sesquilinear form t in L2
σ, with D(t) = S, is defined by
t(φ, ψ) =
〈
|x|−uφ, |x|−uψ
〉
σ
= 〈φ, ψ〉σ−u.
The notation tσ may be also used. The goal of this section is to study t and apply it to prove
Theorem 1.1. Precisely, an estimation of the values t(φk, φ`) is needed.
Lemma 3.1. For all φ ∈ Sodd and ψ ∈ Sev,
t(B′φ, ψ)− t(φ,Bψ) = t(φ,B′ψ)− t(Bφ,ψ) = −2ut
(
x−1φ, ψ
)
.
Proof. By (2.7) and (2.8), for all φ ∈ Sodd and ψ ∈ Sev,
t(B′σφ, ψ)− t(φ,Bσψ) = 〈B′σ−uφ, ψ〉σ−u − 2u
〈
x−1φ, ψ
〉
σ−u − 〈φ,Bσ−uψ〉σ−u
= −2ut
(
x−1φ, ψ
)
,
t(φ,B′σψ)− t(Bσφ, ψ) = 〈φ,B′σ−uψ〉σ−u − 〈Bσ−uφ, ψ〉σ−u − 2u
〈
x−1φ, ψ
〉
σ−u
= −2ut
(
x−1φ, ψ
)
. �
In the whole of this section, k, `, m, n, i, j, p and q will be natural numbers. Let ck,` =
t(φk, φ`) and dk,` = ck,`/c0,0. Thus dk,` = d`,k, and dk,` = 0 when k + ` is odd. Since∫ ∞
−∞
e−sx
2 |x|2κdx = s−
2κ+1
2 Γ
(
κ+ 1
2
)
(3.1)
for κ > −1
2 , we get
c0,0 = Γ
(
σ − u+ 1
2
)
Γ
(
σ + 1
2
)−1
su. (3.2)
Lemma 3.2. If k = 2m > 0, then
dk,0 =
u√
m
m−1∑
j=0
(−1)m−j
√
(m− 1)!Γ(j + 1
2 + σ)
j!Γ(m+ 1
2 + σ)
d2j,0.
Proof. By (2.2), (2.3), (2.6) and Lemma 3.1,
ck,0 =
1√
2sk
t(B′φk−1, φ0) =
1√
2sk
t(φk−1, Bφ0)− 2u√
2sk
t
(
x−1φk−1, φ0
)
= − 2u√
2sk
t
(
x−1φk−1, φ0
)
=
u√
m
m−1∑
j=0
(−1)m−j
√
(m− 1)!Γ(j + 1
2 + σ)
j!Γ(m+ 1
2 + σ)
c2j,0. �
Lemma 3.3. If k = 2m > 0 and ` = 2n > 0, then
dk,` =
√
m
n
dk−1,`−1 +
u√
n
n−1∑
j=0
(−1)n−j
√
(n− 1)!Γ(j + 1
2 + σ)
j!Γ(n+ 1
2 + σ)
dk,2j .
Proof. By (2.2), (2.4), (2.6) and Lemma 3.1,
ck,` =
1√
2s`
t(φk, B
′φ`−1) =
1√
2`s
t(Bφk, φ`−1)− 2u√
2`s
t
(
φk, x
−1φ`−1
)
=
√
m
n
ck−1,`−1 +
u√
n
n−1∑
j=0
(−1)n−j
√
(n− 1)!Γ(j + 1
2 + σ)
j!Γ(n+ 1
2 + σ)
ck,2j . �
8 J.A. Álvarez López, M. Calaza and C. Franco
Lemma 3.4. If k = 2m+ 1 and ` = 2n+ 1, then
dk,` =
√
n+ 1
2 + σ
m+ 1
2 + σ
dk−1,`−1 −
u√
m+ 1
2 + σ
n∑
j=0
(−1)n−j
√
n!Γ(j + 1
2 + σ)
j!Γ(n+ 3
2 + σ)
dk−1,2j .
Proof. By (2.2), (2.4), (2.6) and Lemma 3.1,
ck,` =
1√
2(k + 2σ)s
t(B′φk−1, φ`)
=
1√
2(k + 2σ)s
t(φk−1, Bφ`)−
2u√
2(k + 2σ)s
t
(
φk−1, x
−1φ`
)
=
√
n+ 1
2 + σ
m+ 1
2 + σ
ck−1,`−1 −
u√
m+ 1
2 + σ
n∑
j=0
(−1)n−j
√
n!Γ(j + 1
2 + σ)
j!Γ(n+ 3
2 + σ)
ck−1,2j . �
The following definitions are given for k ≥ ` with k + ` even. Let
Πk,` =
√
m!Γ(n+ 1
2 + σ)
n!Γ(m+ 1
2 + σ)
(3.3)
if k = 2m ≥ ` = 2n, and
Πk,` =
√
m!Γ(n+ 3
2 + σ)
n!Γ(m+ 3
2 + σ)
(3.4)
if k = 2m+ 1 ≥ ` = 2n+ 1. Let Σk,` be inductively defined as follows2:
Σk,0 =
m∏
i=1
(
1− 1− u
i
)
=
Γ(m+ u)
m!Γ(u)
(3.5)
if k = 2m;
Σk,` = Σk−1,`−1 + u
n−1∑
j=0
(n− 1)!Γ(j + 1
2 + σ)
j!Γ(n+ 1
2 + σ)
Σk,2j (3.6)
if k = 2m ≥ ` = 2n > 0; and
Σk,` = Σk−1,`−1 − u
n∑
j=0
n!Γ(j + 1
2 + σ)
j!Γ(n+ 3
2 + σ)
Σk−1,2j (3.7)
=
(
1− u
n+ 1
2 + σ
)
Σk−1,`−1 −
nu
n+ 1
2 + σ
n−1∑
j=0
(n− 1)!Γ(j + 1
2 + σ)
j!Γ(n+ 1
2 + σ)
Σk−1,2j (3.8)
if k = 2m+ 1 ≥ ` = 2n+ 1. Thus Σ0,0 = 1, Σ2,0 = u, Σ4,0 = 1
2u(1 + u), and
Σk,1 =
(
1− u
1
2 + σ
)
Σk−1,0 (3.9)
2We use the convention that a product of an empty set of factors is 1. Such empty products are possible in (3.5)
(when m = 0), in Lemma 3.10 and its proof, and in the proofs of Lemma 3.11 and Remark 3.19. Consistently,
the sum of an empty set of terms is 0. Such empty sums are possible in Lemma 4.4 and its proof, and in the
proof of Proposition 4.7.
A Perturbation of the Dunkl Harmonic Oscillator on the Line 9
if k is odd. From (3.5) and using induction on m, it easily follows that
Σk,0 =
u
m
m−1∑
j=0
Σ2j,0 (3.10)
for k = 2m > 0. Combining (3.6) with (3.7), and (3.8) with (3.6), we get
Σk,` = Σk−2,`−2 − u
n−1∑
j=0
(n− 1)!Γ(j + 1
2 + σ)
j!Γ(n+ 1
2 + σ)
(Σk−2,2j − Σk,2j) (3.11)
if k = 2m ≥ ` = 2n > 0; and
Σk,` =
(
1− u
n+ 1
2 + σ
)
Σk−2,`−2
+
(
1− u+ n
n+ 1
2 + σ
)
u
n−1∑
j=0
(n− 1)!Γ(j + 1
2 + σ)
j!Γ(n+ 1
2 + σ)
Σk−1,2j (3.12)
if k = 2m+ 1 ≥ ` = 2n+ 1 > 1.
Proposition 3.5. dk,` = (−1)m+nΠk,`Σk,` if k = 2m ≥ ` = 2n, or if k = 2m+ 1 ≥ ` = 2n+ 1.
Proof. We proceed by induction on k and l. The statement is obvious for k = ` = 0 because
d0,0 = Π0,0 = Σ0,0 = 1.
Let k = 2m > 0, and assume that the result is true for all d2j,0 with j < m. Then, by
Lemma 3.2, (3.3) and (3.10),
dk,0 =
u√
m
m−1∑
j=0
(−1)m−j
√
(m− 1)!Γ(j + 1
2 + σ)
j!Γ(m+ 1
2 + σ)
(−1)jΠ2j,0Σ2j,0
= (−1)m
u√
m
m−1∑
j=0
√
(m− 1)!Γ(j + 1
2 + σ)
j!Γ(m+ 1
2 + σ)
√
j!Γ(1
2 + σ)
Γ(j + 1
2 + σ)
Σ2j,0
= (−1)mΠk,0
u
m
m−1∑
j=0
Σ2j,0 = (−1)mΠk,0Σk,0.
Now, take k = 2m ≥ ` = 2n > 0 so that the equality of the statement holds for dk−1,`−1 and
all dk,2j with j < n. Then, by Lemma 3.3,
dk,` =
√
m
n
(−1)m+nΠk−1,`−1Σk−1,`−1
+
u√
n
n−1∑
j=0
(−1)n−j
√
(n− 1)!Γ(j + 1
2 + σ)
j!Γ(n+ 1
2 + σ)
(−1)m+jΠk,2jΣk,2j .
Here, by (3.3) and (3.4),
√
m/nΠk−1,`−1 = Πk,`, and
1√
n
√
(n− 1)!Γ(j + 1
2 + σ)
j!Γ(n+ 1
2 + σ)
Πk,2j =
1√
n
√
m!Γ(n+ 1
2 + σ)
(n− 1)!Γ(m+ 1
2 + σ)
(n− 1)!Γ(j + 1
2 + σ)
j!Γ(n+ 1
2 + σ)
= Πk,`
(n− 1)!Γ(j + 1
2 + σ)
j!Γ(n+ 1
2 + σ)
.
Thus, by (3.6), dk,` = (−1)m+nΠk,`Σk,`.
10 J.A. Álvarez López, M. Calaza and C. Franco
Finally, take k = 2m + 1 ≥ ` = 2n + 1 such that the equality of the statement holds for all
dk−1,2j with j ≤ n. Then, by Lemma 3.4,
dk,` =
√
n+ 1
2 + σ
m+ 1
2 + σ
(−1)m+nΠk−1,`−1Σk−1,`−1
− u√
m+ 1
2 + σ
n∑
j=0
(−1)n−j
√
n!Γ(j + 1
2 + σ)
j!Γ(n+ 3
2 + σ)
(−1)m+jΠk−1,2jΣk−1,2j .
Here, by (3.3) and (3.4),√
n+ 1
2 + σ
m+ 1
2 + σ
Πk−1,`−1 = Πk,`,
and
1√
m+ 1
2 + σ
√
n!Γ(j + 1
2 + σ)
j!Γ(n+ 3
2 + σ)
Πk−1,2j
=
1√
m+ 1
2 + σ
√
m!Γ(n+ 3
2 + σ)
n!Γ(m+ 1
2 + σ)
n!Γ(j + 1
2 + σ)
j!Γ(n+ 3
2 + σ)
= Πk,`
n!Γ(j + 1
2 + σ)
j!Γ(n+ 3
2 + σ)
.
Thus, by (3.7), dk,` = (−1)m+nΠk,`Σk,`. �
Lemma 3.6. Σk,` > 0 for all k and `.
Proof. We proceed by induction on `. For ` ∈ {0, 1}, this is true by (3.5) and (3.9) because
σ > u− 1
2 . If ` > 1 and the results holds for Σk′,`′ with `′ < `, then Σk,` > 0 by (3.6) and (3.12)
since σ > u− 1
2 . �
Lemma 3.7. If k = 2m > ` = 2n or k = 2m+ 1 > ` = 2n+ 1, then
Σk,` ≤
(
1− 1− u
m
)
Σk−2,`.
Proof. We proceed by induction on `. This is true for ` ∈ {0, 1} by (3.5) and (3.9).
Now, suppose that the result is satisfied by Σk′,`′ with `′ < `. If k = 2m > ` = 2n > 0, then,
by (3.6) and Lemma 3.6,
Σk,` ≤
(
1− 1− u
m− 1
)
Σk−3,`−1 + u
n−1∑
j=0
(n− 1)!Γ(j + 1
2 + σ)
j!Γ(n+ 1
2 + σ)
(
1− 1− u
m
)
Σk−2,2j
≤
(
1− 1− u
m
)
Σk−2,`.
If k = 2m+ 1 > ` = 2n+ 1 > 1, then, by (3.12) and Lemma 3.6, and since σ > u− 1
2 ,
Σk,` ≤
(
1− u
n+ 1
2 + σ
)(
1− 1− u
m− 1
)
Σk−4,`−2
+
(
1− u+ n
n+ 1
2 + σ
)
u
n−1∑
j=0
(n− 1)!Γ(j + 1
2 + σ)
j!Γ(n+ 1
2 + σ)
(
1− 1− u
m
)
Σk−3,2j
<
(
1− 1− u
m
)
Σk−2,`. �
A Perturbation of the Dunkl Harmonic Oscillator on the Line 11
Corollary 3.8. If k = 2m ≥ ` = 2n > 0, then
Σk−1,`−1 < Σk,` ≤
(
1− u(1− u)
m
)
Σk−2,`−2.
Proof. The first inequality is a direct consequence of (3.6), and Lemma 3.6. On the other
hand, by (3.11), and Lemmas 3.6 and 3.7,
Σk,` ≤ Σk−2,`−2 −
u(1− u)
m
n−1∑
j=0
(n− 1)!Γ(j + 1
2 + σ)
j!Γ(n+ 1
2 + σ)
Σk−2,2j
=
(
1− u(1− u)
m
)
Σk−2,`−2 −
u(1− u)
m
n−2∑
j=0
(n− 1)!Γ(j + 1
2 + σ)
j!Γ(n+ 1
2 + σ)
Σk−2,2j
≤
(
1− u(1− u)
m
)
Σk−2,`−2. �
Corollary 3.9. If k = 2m+ 1 ≥ ` = 2n+ 1, then(
1− u
n+ 1
2 + σ
)
Σk−2,`−2 < Σk,` ≤
(
1− u
n+ 1
2 + σ
)
Σk−1,`−1.
Proof. This follows from (3.8), (3.12) and Lemma 3.6 because σ > u− 1
2 . �
Lemma 3.10. For 0 < t < 1, there is some C0 = C0(t) ≥ 1 such that, for all p,
C−1
0 (p+ 1)−t ≤
p∏
i=1
(
1− t
i
)
≤ C0(p+ 1)−t.
Proof. For each t > 0, by the Weierstrass definition of the gamma function,
Γ(t) =
e−γt
t
∞∏
i=1
(
1 +
t
i
)−1
et/i,
where γ = lim
j→∞
( j∑
i=1
1
i − ln j
)
(the Euler–Mascheroni constant), there is some K0 ≥ 1 such that,
for all p ∈ Z+,
K−1
0
p∏
i=1
e−t/i ≤
p∏
i=1
(
1 +
t
i
)−1
≤ K0
p∏
i=1
e−t/i. (3.13)
Now, assume that 0 < t < 1, and observe that
p∏
i=1
(
1− t
i
)
=
p∏
i=1
(
1 +
t
i− t
)−1
.
By the second inequality of (3.13), for p ≥ 1,
p∏
i=1
(
1 +
t
i− t
)−1
<
p∏
i=1
(
1 +
t
i
)−1
≤ K0
p∏
i=1
e−t/i = K0 exp
(
−t
p∑
i=1
1
i
)
≤ K0 exp
(
−t
∫ p+1
1
dx
x
)
= K0(p+ 1)−t.
12 J.A. Álvarez López, M. Calaza and C. Franco
On the other hand, by the first inequality of (3.13), for p ≥ 2,
p∏
i=1
(
1 +
t
i− t
)−1
≥ (1− t)
p−1∏
i=1
(
1 +
t
i
)−1
≥ (1− t)K−1
0
p−1∏
i=1
e−t/i
= (1− t)K−1
0 exp
(
−t
p−1∑
i=1
1
i
)
≥ (1− t)K−1
0 exp
(
−t
(
1 +
∫ p−1
1
dx
x
))
= (1− t)K−1
0 e−t(p− 1)−t > (1− t)K−1
0 e−t(p+ 1)−t. �
Lemma 3.11. There is some C ′ = C ′(u) > 0 such that
Σk,` ≤ C ′(m+ 1)−u(1−u)(m− n+ 1)−(1−u)2
for k = 2m ≥ ` = 2n or k = 2m+ 1 ≥ ` = 2n+ 1.
Proof. Suppose first that k = 2m ≥ ` = 2n. By Lemma 3.7 and Corollary 3.8, we get
Σk,` ≤
m∏
i=m−n+1
(
1− u(1− u)
i
)m−n∏
i=1
(
1− 1− u
i
)
=
m∏
i=1
(
1− u(1− u)
i
)m−n∏
i=1
(
1− u(1− u)
i
)−1 m−n∏
i=1
(
1− 1− u
i
)
.
Then the result follows in this case from Lemma 3.10.
When k = 2m+ 1 ≥ ` = 2n+ 1, the result follows from the above case and Corollary 3.9. �
Lemma 3.12. For each t ∈ R \ (−N), there is some C1 = C1(t) ≥ 1 such that, for all p,
C−1
1 (p+ 1)1−t ≤ Γ(p+ 1)
|Γ(p+ t)|
≤ C1(p+ 1)1−t.
Proof. We can assume that p ≥ 1. Write t = q + r, where q = btc. If q = 0, then 0 < r < 1
and the result follows from the Gautschi’s inequality, stating that
x1−r ≤ Γ(x+ 1)
Γ(x+ r)
≤ (x+ 1)1−r (3.14)
for 0 < r < 1 and x > 0, because x1−r ≥ 2r−1(x+ 1)1−r for x ≥ 1.
If q ≥ 1 and r = 0, then
Γ(p+ 1)
Γ(p+ t)
=
p!
(p+ q − 1)!
≤ 1
(p+ 1)q−1
= (p+ 1)1−t,
Γ(p+ 1)
Γ(p+ t)
=
p!
(p+ q − 1)!
≥ 1
(p+ q − 1)q−1
≥ 1
(qp)q−1
≥ t1−t(p+ 1)1−t.
If q ≥ 1 and r > 0, then, by (3.14),
Γ(p+ 1)
Γ(p+ t)
≤ Γ(p+ 1)
(p+ 1)q−1(p+ r)Γ(p+ r)
≤ (p+ 1)2−q−r
p+ r
≤ 2(p+ 1)1−t,
Γ(p+ 1)
Γ(p+ t)
≥ Γ(p+ 1)
(p+ t− 1)qΓ(p+ r)
≥ p1−r
(p+ t− 1)q
≥ (p+ 1)1−r
21−r(p+ t− 1)q
≥ min{1, (t− 1)−q}2r−1(p+ 1)1−t,
A Perturbation of the Dunkl Harmonic Oscillator on the Line 13
because
(p+ t− 1)−q ≥
{
(p+ 1)−q if 0 < t ≤ 2,
(t− 1)−q(p+ 1)−q if t > 2.
In the case q < 0 (t < 0), apply reverse induction on q: with C1 = C1(t+ 1), we get
Γ(p+ 1)
|Γ(p+ t)|
=
|p+ t|Γ(p+ 1)
|Γ(p+ t+ 1)|
≤ |p+ t|C1(p+ 1)−t ≤ C1|q|(p+ 1)1−t,
Γ(p+ 1)
|Γ(p+ t)|
=
|p+ t|Γ(p+ 1)
|Γ(p+ t+ 1)|
≥ |p+ t|C−1
1 (p+ 1)−t =
|p+ t|
p+ 1
C−1
1 (p+ 1)1−t,
where |p+ t|/(p+ 1) is bounded uniformly on p. �
Corollary 3.13. There is some C ′′ = C ′′(σ) > 0 such that
Πk,` ≤
C ′′
(
n+ 1
m+ 1
)σ
2
− 1
4
if k = 2m ≥ ` = 2n,
C ′′
(
n+ 1
m+ 1
)σ
2
+ 1
4
if k = 2m+ 1 ≥ ` = 2n+ 1.
Proof. This follows from (3.3), (3.4) and Lemma 3.12. �
For the sake of simplicity, let us use the following notation. For real valued functions f and g
of (m,n), for (m,n) in some subset of N × N, write f 4 g if there is some C > 0 such that
f(m,n) ≤ Cg(m,n) for all (m,n). The same notation is used for functions depending also on
other variables, s, σ, u, . . . , taking C independent of m, n and s, but possibly depending on the
rest of variables.
Lemma 3.14. For α, β, γ ∈ R, if α + β, α + γ, α + β + γ < 0, then there is some ω > 0 such
that, for all naturals m ≥ n,
(m+ 1)α(n+ 1)β(m− n+ 1)γ 4 (m+ 1)−ω(n+ 1)−ω.
Proof. We consider the following cases:
1. If α, β, γ < 0, then
(m+ 1)α(n+ 1)β(m− n+ 1)γ ≤ (m+ 1)α(n+ 1)β.
2. If β ≥ 0 and γ < 0, then
(m+ 1)α(n+ 1)β(m− n+ 1)γ ≤ (m+ 1)α+β ≤ (m+ 1)
α+β
2 (n+ 1)
α+β
2 .
3. If α ≥ 0 and m+ 1 ≤ 2(n+ 1), then β, γ < 0 and
(m+ 1)α(n+ 1)β(m− n+ 1)γ ≤ 2−β(m+ 1)α+β ≤ 2−β(m+ 1)
α+β
2 (n+ 1)
α+β
2 .
4. If α ≥ 0 and m+ 1 > 2(n+ 1), then β, γ < 0 and m− n+ 1 > (m+ 1)/2, and therefore
(m+ 1)α(n+ 1)β(m− n+ 1)γ ≤ 2−γ(m+ 1)α+γ(n+ 1)β.
5. If β < 0 and γ ≥ 0, then
(m+ 1)α(n+ 1)β(m− n+ 1)γ ≤ (m+ 1)α+γ(n+ 1)β.
6. If β ≥ 0 and γ ≥ 0, then
(m+ 1)α(n+ 1)β(m− n+ 1)γ ≤ (m+ 1)α+β+γ ≤ (m+ 1)
α+β+γ
2 (n+ 1)
α+β+γ
2 . �
14 J.A. Álvarez López, M. Calaza and C. Franco
Proposition 3.15. There is some ω = ω(σ, u) > 0 such that
|dk,`| 4 (m+ 1)−ω(n+ 1)−ω
for k = 2m and ` = 2n, or for k = 2m+ 1 and ` = 2n+ 1.
Proof. We can assume k ≥ ` because dk,` = d`,k.
If k = 2m+1 ≥ ` = 2n+1, then, according to Proposition 3.5, Lemma 3.11 and Corollary 3.13,
|dk,`| 4 (m+ 1)−
σ
2
− 1
4
−u(1−u)(n+ 1)
σ
2
+ 1
4 (m− n+ 1)−(1−u)2
.
Thus the result follows by Lemma 3.14 since
−σ
2 −
1
4 − u(1− u)− (1− u)2 = −σ
2 + u− 5
4 <
u
2 − 1 < 0.
If k = 2m ≥ ` = 2n, then, according to Proposition 3.5, Lemma 3.11 and Corollary 3.13,
|dk,`| 4 (m+ 1)−
σ
2
+ 1
4
−u(1−u)(n+ 1)
σ
2
− 1
4 (m− n+ 1)−(1−u)2
.
Thus the result follows by Lemma 3.14 since
−σ
2 + 1
4 − u(1− u)− (1− u)2 = −σ
2 + u− 3
4 <
u
2 −
1
2 < 0. �
Corollary 3.16. There is some ω = ω(σ, u) > 0 such that, for k = 2m and ` = 2n, or for
k = 2m+ 1 and ` = 2n+ 1,
|ck,`| 4 su(m+ 1)−ω(n+ 1)−ω.
Proof. This follows from Proposition 3.15 and (3.2). �
Proposition 3.17. For any ε > 0, there is some C = C(ε, σ, u) > 0 such that, for all φ ∈ S,
t(φ) ≤ εsu−1j(φ) + Csu‖φ‖2σ.
Proof. For each k, let νk = 2k + 1 + 2σ. By Corollary 3.16, there are K0 = K0(σ, u) > 0 and
ω = ω(σ, u) > 0 such that
|ck,`| ≤ K0s
uν−ωk ν−ω` (3.15)
for all k and `. Since S = S(σ, u) :=
∑
k ν
−1−2ω
k <∞, given ε > 0, there is some k0 = k0(ε, σ, u)
so that
S0 = S0(ε, σ, u) :=
∑
k>k0
ν−1−2ω
k <
ε2
4K2
0S
.
Let S1 = S1(ε, σ, u) =
∑
k≤k0
ν−ωk . For φ =
∑
k tkφk ∈ S, by (3.15) and the Schwartz inequality,
we have
t(φ) =
∑
k,`
tkt`ck,` ≤
∑
k,`
|tk||t`||ck,`|
≤ K0s
u− 1
2
∑
k≤k0
|tk|
νωk
∑
`
|t`|(ν`s)
1
2
ν
1
2
+ω
`
+K0s
u−1
∑
k>k0
|tk|(νks)
1
2
ν
1
2
+ω
k
∑
`
|t`|(ν`s)
1
2
ν
1
2
+ω
`
≤ K0S1S
1
2 su−
1
2 ‖φ‖σj(φ)
1
2 +K0S
1
2
0 S
1
2 su−1j(φ)
≤ K0S1S
1
2 su−
1
2 ‖φ‖σj(φ)
1
2 +
εsu−1
2
j(φ) ≤ K2
0S
2
1Ss
u
2ε
‖φ‖2σ + εsu−1j(φ). �
A Perturbation of the Dunkl Harmonic Oscillator on the Line 15
Proposition 3.18. There is some D = D(σ, u) > 0 such that, for all k ∈ N and φ in the linear
span of φ0, . . . , φk,
t(φ) ≥ Dsu(k + 1)−u‖φ‖2σ.
Proof. Let φ =
k∑
i=0
tiφi (ti ∈ C) and ν = νk = 2k + 1 + 2σ. Let K ≥ 3, which will be fixed
later. By (2.9),∫
|x|≥
√
Kν
2s
|φ(x)|2|x|2σdx =
k∑
i,j=0
titj
∫
|x|≥
√
Kν
2s
φi(x)φj(x)|x|2σdx
≤ 2
k∑
i,j=0
|ti||tj |
∫ ∞√
Kν
2s
|φi(x)||φj(x)|x2σdx
≤
k∑
i,j=0
(
|ti|2 + |tj |2
)
C2s
∫ ∞√
Kν
2s
xe−2csx2
dx
= 2(k + 1)‖φ‖2σC2s
∫ ∞√
Kν
2s
xe−2csx2
dx
=
C2(k + 1)
2c
e−Kcν‖φ‖2σ,
where C, c > 0 depend only on σ. We can choose K = K(σ) ≥ 3 and D = D(σ, u) > 0 such that(
2s
Kν
)u(
1− C2(k + 1)
2c
e−Kcν
)
≥ Dsu(k + 1)−u
for all s > 0 and k ∈ N, obtaining
t(φ) ≥
∫
|x|≤
√
Kν
2s
|φ(x)|2|x|2σ−2udx ≥
(
2s
Kν
)u ∫
|x|≤
√
Kν
2s
|φ(x)|2|x|2σdx
≥
(
2s
Kν
)u
‖φ‖2σ
(
1− C2(k + 1)
2c
e−Kcν
)
≥ Dsu(k + 1)−u‖φ‖2σ. �
Remark 3.19. For φ = φk, we can also use the following argument. By Proposition 3.5
and (3.2), and since Πk,k = 1, it is enough to prove that there is some D0 = D0(σ, u) > 0 so
that Σk,k ≥ D0(k + 1)−u. Moreover we can assume that k = 2m+ 1 by Corollary 3.8. We have
p0 := b1
2 + σc ≥ 0 because 1
2 + σ > u. According to Corollary 3.9, Lemma 3.10 and (3.9), there
is some C0 = C0(u) ≥ 1 such that
Σk,k ≥
m∏
i=0
(
1− u
i+ 1
2 + σ
)
≥
(
1− u
1
2 + σ
)
m+p0∏
p=1+p0
(
1− u
p
)
=
(
1− u
1
2 + σ
)
m+p0∏
p=1
(
1− u
p
) p0∏
p=1
(
1− u
p
)−1
≥
(
1− u
1
2 + σ
)
C−2
0 (m+ p0 + 1)−u(p0 + 1)u ≥
(
1− u
1
2 + σ
)
C−2
0 (k + 1)−u.
Remark 3.20. If 0 < u < 1
2 , then lim
m
t(φ2m+1) = 0. To check it, we use that there is some
K = K(σ, s) > 0 so that |x|2σφ2
k(x) ≤ Kk−
1
6 for all x ∈ R and all odd k ∈ N [1, Theorem 1.1(ii)]
16 J.A. Álvarez López, M. Calaza and C. Franco
(this also follows from (2.9)). For any ε > 0, take some x0 > 0 and k0 ∈ N such that x−2u
0 < ε/2
and Kk
− 1
6
0 x1−2u
0 < ε(1− 2u)/4. Then, for all odd natural k ≥ k0,
t(φk) = 2
∫ x0
0
φ2
k(x)x2(σ−u)dx+ 2
∫ ∞
x0
φ2
k(x)x2(σ−u)dx
≤ 2Kk−
1
6
∫ x0
0
x−2udx+ 2x−2u
0
∫ ∞
x0
φ2
k(x)x2σdx ≤ 2Kk−
1
6
x1−2u
0
1− 2u
+ x−2u
0 < ε,
because 1− 2u > 0 and ‖φk‖σ = 1. In the case where σ ≥ 0, this argument is also valid when k
is even. We do not know if infk t(φk) > 0 when 1
2 ≤ u < 1.
Proof of Theorem 1.1. The positive definite sesquilinear form j of Section 2 is closable by
[14, Theorems VI-2.1 and VI-2.7]. Then, taking ε > 0 so that ξεsu−1 < 1, it follows from [14,
Theorem VI-1.33] and Proposition 3.17 that the positive definite sesquilinear form u := j+ ξt is
also closable, and D(ū) = D(j). By [14, Theorems VI-2.1, VI-2.6 and VI-2.7], there is a unique
positive definite self-adjoint operator U such that D(U) is a core of D(ū), which consists of the
elements φ ∈ D(ū) so that, for some χ ∈ L2
σ, we have ū(φ, ψ) = 〈χ, ψ〉σ for all ψ in some core
of ū (in this case, U(φ) = χ). By [14, Theorem VI-2.23], we have D(U1/2) = D(ū), S is a core
of U1/2 (since it is a core of u), and (1.1) is satisfied. By Proposition 3.18, there is some D(σ, u)
so that, for all s > 0 and k ∈ N, and every φ is in the linear span of φ0, . . . , φk, we have t(φ) ≥
Dsu(k+ 1)−u‖φ‖2σ. Moreover we can assume that the sequence (2k+ 1 + 2σ)s+ ξDsu(k+ 1)−u
is strictly increasing after reducing D if necessary. So
u(φ) ≥
(
(2k + 1 + 2σ)s+ ξDsu(k + 1)−u
)
‖φ‖2σ
if φ ∈ S is orthogonal in L2
σ to the linear span of φ0, . . . , φk−1 (assuming that this span is 0
when k = 0). Therefore U has a discrete spectrum satisfying the first inequality of (1.2) by
the form version of the min-max principle [22, Theorem XIII.2]. The second inequality of (1.2)
holds because
ū(φ) ≤
(
1 + ξεsu−1
)̄
j(φ) + ξCsu‖φ‖2σ
for all φ ∈ D(ū) by Proposition 3.17 and [14, Theorem VI-1.18], since S is a core of ū and j̄. �
Remark 3.21. In the above proof, note that ū = j̄+ ξt̄ and D(̄j) = D
(
J
1/2)
. Thus (1.1) can be
extended to φ, ψ ∈ D
(
U1/2
)
using
〈
J
1/2
φ, J
1/2
ψ
〉
σ
instead of 〈Jφ, ψ〉σ.
Remark 3.22. Extend the definition of the above forms and operators to the case of ξ ∈ C.
Then |̄t(φ)| ≤ εsu−1 <̄j(φ)+Csu‖φ‖2σ for all φ ∈ D(̄j), like in the proof of Theorem 1.1. Thus the
family ū = ū(ξ) becomes holomorphic of type (a) by Remark 3.21 and [14, Theorem VII-4.8], and
therefore U = U(ξ) is a self-adjoint holomorphic family of type (B). So the functions λk = λk(ξ)
(ξ ∈ R) are continuous and piecewise holomorphic [14, Remark VII-4.22, Theorem VII-3.9,
and VII-§ 3.4], with λk(0) = (2k+1+2σ)s. Moreover [14, Theorem VII-4.21] gives an exponential
estimate of |λk(ξ)− λk(0)| in terms of ξ. But (1.2) is a better estimate.
4 Scalar products of mixed generalized Hermite functions
Let σ, τ, θ > −1
2 , and write v = σ+τ−2θ. This section is devoted to describe the scalar products
ĉk,` = ĉσ,τ,θ,k,` = 〈φσ,k, φτ,`〉θ,
A Perturbation of the Dunkl Harmonic Oscillator on the Line 17
which will be needed to prove Theorem 1.3. Note that ĉk,` = 0 if k + ` is odd, and
ĉσ,τ,θ,k,` = ĉτ,σ,θ,`,k (4.1)
for all k and `. Of course, ĉk,` = δk,` if σ = τ = θ.
According to Section 2, if k and ` are odd, then ĉσ,τ,θ,k,` is also defined when σ, τ, θ > −3
2 ,
and we have
ĉσ,τ,θ,k,` = 〈xφσ+1,k−1, xφτ+1,`−1〉θ = ĉσ+1,τ+1,θ+1,k−1,`−1. (4.2)
4.1 Case where σ = θ 6= τ and τ − σ 6∈ −N
In this case, we have v = τ − σ. By (2.1) and (3.1),
ĉ0,0 = s
v
2 Γ
(
σ + 1
2
) 1
2 Γ
(
τ + 1
2
)− 1
2 . (4.3)
Lemma 4.1. If k > 0 is even, then ĉk,0 = 0.
Proof. By (2.2), (2.3) and (2.7),
ĉk,0 =
1√
2ks
〈B′σφσ,k−1, φτ,0〉σ =
1√
2ks
〈φσ,k−1, Bτφτ,0〉σ = 0. �
Lemma 4.2. If ` = 2n > 0, then
ĉ0,` =
v√
n
n−1∑
j=0
(−1)n−j
√
(n− 1)!Γ(j + 1
2 + τ)
j!Γ(n+ 1
2 + τ)
ĉ0,2j .
Proof. By (2.2), (2.3), (2.6) and (2.8),
ĉ0,` =
1√
2`s
〈φσ,0, B′τφτ,`−1〉σ =
1√
2`s
〈φσ,0, (B′σ − 2vx−1)φτ,`−1〉σ
=
1√
2`s
〈Bσφσ,0, φτ,`−1〉σ −
2v√
2`
n−1∑
j=0
(−1)n−1−j
√
(n− 1)!Γ(j + 1
2 + τ)
j!Γ(n+ 1
2 + τ)
ĉ0,2j
=
v√
n
n−1∑
j=0
(−1)n−j
√
(n− 1)!Γ(j + 1
2 + τ)
j!Γ(n+ 1
2 + τ)
ĉ0,2j . �
Lemma 4.3. If k = 2m > 0 and ` = 2n > 0, then ĉk,` =
√
n/mĉk−1,`−1.
Proof. By (2.2), (2.4) and (2.7),
ĉk,` =
1√
2ks
〈B′σφσ,k−1, φτ,`〉σ =
1√
2ks
〈φσ,k−1, Bτφτ,`〉σ =
√
n
m
ĉk−1,`−1. �
Lemma 4.4. If k = 2m+ 1 and ` = 2n+ 1, then
ĉk,` =
n+ 1
2 + σ√
(m+ 1
2 + σ)(n+ 1
2 + τ)
ĉk−1,`−1
− v√
m+ 1
2 + σ
n−1∑
j=0
(−1)n−j
√
n!Γ(j + 1
2 + τ)
j!Γ(n+ 3
2 + τ)
ĉk−1,2j .
18 J.A. Álvarez López, M. Calaza and C. Franco
Proof. By (2.2), (2.4), (2.6) and (2.7),
ĉk,` =
1√
2(k + 2σ)s
〈B′σφσ,k−1, φτ,`〉σ =
1√
2(k + 2σ)s
〈
φσ,k−1,
(
Bτ − 2vx−1
)
φτ,`
〉
σ
=
√
n+ 1
2 + τ
m+ 1
2 + σ
ĉk−1,`−1 −
v√
m+ 1
2 + σ
n∑
j=0
(−1)n−j
√
n!Γ(j + 1
2 + τ)
j!Γ(n+ 3
2 + τ)
ĉk−1,2j
=
n+ 1
2 + σ√
(m+ 1
2 + σ)(n+ 1
2 + τ)
ĉk−1,`−1
− v√
m+ 1
2 + σ
n−1∑
j=0
(−1)n−j
√
n!Γ(j + 1
2 + τ)
j!Γ(n+ 3
2 + τ)
ĉk−1,2j . �
Corollary 4.5. If k > `, then ĉk,` = 0.
Proof. This follows by induction on ` using Lemmas 4.1, 4.3 and 4.4. �
Remark 4.6. By Corollary 4.5, in Lemma 4.4, it is enough to consider the sum with j running
from m to n− 1.
Proposition 4.7. If k = 2m ≤ ` = 2n, then
ĉk,` = (−1)m+ns
v
2
√
n!Γ(m+ 1
2 + σ)
m!Γ(n+ 1
2 + τ)
Γ(n−m+ v)
(n−m)!Γ(v)
,
and, if k = 2m+ 1 ≤ ` = 2n+ 1, then
ĉk,` = (−1)m+ns
v
2
√
n!Γ(m+ 3
2 + σ)
m!Γ(n+ 3
2 + τ)
Γ(n−m+ v)
(n−m)!Γ(v)
.
Proof. This is proved by induction on k. In turn, the case k = 0,
ĉ0,` = (−1)ns
v
2
√
Γ(1
2 + σ)
n!Γ(n+ 1
2 + τ)
Γ(n+ v)
Γ(v)
, (4.4)
is proved by induction on `. If k = ` = 0, (4.4) is (4.3). Given ` = 2n > 0, assume that the
result holds for k = 0 and all `′ = 2n′ < `. Then, by Lemma 4.2,
ĉ0,` =
v√
n
n−1∑
j=0
(−1)n−j
√
(n− 1)!Γ(j + 1
2 + τ)
j!Γ(n+ 1
2 + τ)
(−1)js
v
2
√
Γ(1
2 + σ)
j!Γ(j + 1
2 + τ)
Γ(j + v)
Γ(v)
= (−1)ns
v
2
√
(n− 1)!Γ(1
2 + σ)
nΓ(n+ 1
2 + τ)
v
Γ(v)
n−1∑
j=0
Γ(j + v)
j!
,
obtaining (4.4) because
Γ(p+ 1 + t)
p!
= t
p∑
i=0
Γ(i+ t)
i!
(4.5)
for all p ∈ N and t ∈ R \ (−N), as can be easily checked by induction on p.
A Perturbation of the Dunkl Harmonic Oscillator on the Line 19
Given k > 0, assume that the result holds for all k′ < k. If k is even, the statement follows
directly from Lemma 4.3. If k is odd, by Lemma 4.4, Remark 4.6 and (4.5),
ĉk,` =
n+ 1
2 + σ√
(m+ 1
2 + σ)(n+ 1
2 + τ)
(−1)m+ns
v
2
√
n!Γ(m+ 1
2 + σ)
m!Γ(n+ 1
2 + τ)
Γ(n−m+ v)
(n−m)!Γ(v)
− v√
m+ 1
2 + σ
n−1∑
j=m
(−1)n−j
√
n!Γ(j + 1
2 + τ)
j!Γ(n+ 3
2 + τ)
× (−1)m+js
v
2
√
j!Γ(m+ 1
2 + σ)
m!Γ(j + 1
2 + τ)
Γ(j −m+ v)
(j −m)!Γ(v)
= (−1)m+ns
v
2
√
n!Γ(m+ 1
2 + σ)
(m+ 1
2 + σ)m!Γ(n+ 3
2 + τ)
1
Γ(v)
×
(
Γ(n−m+ v)(n+ 1
2 + σ)
(n−m)!
− v
n−m−1∑
i=0
Γ(i+ v)
i!
)
= (−1)m+ns
v
2
√
n!Γ(m+ 3
2 + σ)
m!Γ(n+ 3
2 + τ)
Γ(n−m+ v)
(n−m)!Γ(v)
. �
Remark 4.8. By (4.2), if k and ` are odd, then Corollary 4.5 and Proposition 4.7 also hold
when σ, τ > −3
2 .
4.2 Case where σ 6= θ 6= τ and σ − θ, τ − θ 6∈ −N
By (2.1) and (3.1),
ĉ0,0 = s
v
2 Γ
(
σ + 1
2
)− 1
2 Γ
(
τ + 1
2
)− 1
2 Γ(θ + 1
2). (4.6)
Lemma 4.9. If k = 2m > 0, then
ĉk,0 =
σ − θ√
m
m−1∑
i=0
(−1)m−i
√
(m− 1)!Γ(i+ 1
2 + σ)
i!Γ(m+ 1
2 + σ)
ĉ2i,0.
Proof. By (2.2) and (2.8),
ĉk,0 =
1√
2ks
〈B′σφσ,k−1, φτ,0〉θ =
1
2
√
ms
〈B′θφσ,k−1, φτ,0〉θ +
θ − σ√
ms
〈
x−1φσ,k−1, φτ,0
〉
θ
.
Here, by (2.3), (2.6) and (2.7),
〈B′θφσ,k−1, φτ,0〉θ = 〈φσ,k−1, Bθφτ,0〉θ = 〈φσ,k−1, Bτφτ,0〉θ = 0,
〈x−1φσ,k−1, φτ,0〉θ = −
m−1∑
i=0
(−1)m−i
√
(m− 1)!Γ(i+ 1
2 + σ)s
i!Γ(m+ 1
2 + σ)
ĉ2i,0. �
Lemma 4.10. If k = 2m > 0 and ` = 2n > 0, then
ĉk,` =
√
n
m
ĉk−1,`−1 +
σ − θ
m
m−1∑
i=0
(−1)m−i
√
m!Γ(i+ 1
2 + σ)
i!Γ(m+ 1
2 + σ)
ĉ2i,`.
20 J.A. Álvarez López, M. Calaza and C. Franco
Proof. Like in the proof of Lemma 4.9,
ĉk,` =
1
2
√
ms
〈B′θφσ,k−1, φτ,`〉θ +
θ − σ√
ms
〈
x−1φσ,k−1, φτ,`
〉
θ
.
Now, by (2.4), (2.6) and (2.7),
〈B′θφσ,k−1, φτ,`〉θ = 〈φσ,k−1, Bθφτ,`〉θ = 〈φσ,k−1, Bτφτ,`〉θ = 2
√
nsĉk−1,`−1,〈
x−1φσ,k−1, φτ,`
〉
θ
= −
m−1∑
i=0
(−1)m−i
√
(m− 1)!Γ(i+ 1
2 + σ)s
i!Γ(m+ 1
2 + σ)
ĉ2i,`. �
Lemma 4.11. If k = 2m+ 1 and ` = 2n+ 1, then
ĉk,` =
m+ 1
2 + θ√
(m+ 1
2 + σ)(n+ 1
2 + τ)
ĉk−1,`−1
− σ − θ√
n+ 1
2 + τ
m−1∑
i=0
(−1)m−i
√
m!Γ(i+ 1
2 + σ)
i!Γ(m+ 3
2 + σ)
ĉ2i,`−1.
Proof. By (2.2),
ĉk,` =
1
2
√
(n+ 1
2 + τ)s
〈φσ,k, B′τφτ,`−1〉θ,
where, by (2.8),
〈φσ,k, B′τφτ,`−1〉θ = 〈φσ,k, B′θφτ,`−1〉θ = 〈Bθφσ,k, φτ,`−1〉θ
= 〈Bσφσ,k, φτ,`−1〉θ + 2(θ − σ)〈x−1φσ,k, φτ,`−1〉θ.
Hence, by (2.4) and (2.6),
ĉk,` =
√
m+ 1
2 + σ
n+ 1
2 + τ
ĉk−1,`−1 −
σ − θ√
n+ 1
2 + τ
m∑
i=0
(−1)m−i
√
m!Γ(i+ 1
2 + σ)
i!Γ(m+ 3
2 + σ)
ĉ2i,`−1
=
m+ 1
2 + θ√
(n+ 1
2 + τ)(m+ 1
2 + σ)
ĉk−1,`−1
− σ − θ√
n+ 1
2 + τ
m−1∑
i=0
(−1)m−i
√
m!Γ(i+ 1
2 + σ)
i!Γ(m+ 3
2 + σ)
ĉ2i,`−1. �
Proposition 4.12. If k = 2m and ` = 2n, then
ĉk,` = (−1)m+ns
v
2
√
m!n!
Γ(m+ 1
2 + σ)Γ(n+ 1
2 + τ)
×
min{m,n}∑
p=0
Γ(p+ 1
2 + θ)Γ(m− p+ σ − θ)Γ(n− p+ τ − θ)
p!(m− p)!(n− p)!Γ(σ − θ)Γ(τ − θ)
,
and, if k = 2m+ 1 and ` = 2n+ 1, then
ĉk,` = (−1)m+ns
v
2
√
m!n!
Γ(m+ 3
2 + σ)Γ(n+ 3
2 + τ)
×
min{m,n}∑
p=0
Γ(p+ 3
2 + θ)Γ(m− p+ σ − θ)Γ(n− p+ τ − θ)
p!(m− p)!(n− p)!Γ(σ − θ)Γ(τ − θ)
.
A Perturbation of the Dunkl Harmonic Oscillator on the Line 21
Proof. The result is proved by induction on k and `. First, consider the case ` = 0. When
k = ` = 0, the result is given by (4.6). Now, take any k = 2m > 0, and assume that the result
holds for all ĉk′,0 with k′ = 2m′ < k. Then, by Lemma 4.9 and (4.5),
ĉk,0 =
σ − θ√
m
m−1∑
i=0
(−1)m−i
√
(m− 1)!Γ(i+ 1
2 + σ)
i!Γ(m+ 1
2 + σ)
× (−1)is
v
2
√
1
i!Γ(i+ 1
2 + σ)Γ(1
2 + τ)
Γ(1
2 + θ)Γ(i+ σ − θ)
Γ(σ − θ)
= (−1)ms
v
2
√
m!
Γ(m+ 1
2 + σ)Γ(1
2 + τ)
Γ(1
2 + θ)(σ − θ)
m
m−1∑
i=0
Γ(i+ σ − θ)
i!Γ(σ − θ)
= (−1)ms
v
2
√
1
m!Γ(m+ 1
2 + σ)Γ(1
2 + τ)
Γ(1
2 + θ)Γ(m+ σ − θ)
Γ(σ − θ)
.
From the case ` = 0, the result also follows for the case k = 0 by (4.1).
Now, take k = 2m > 0 and ` = 2n > 0, and assume that the result holds for all ĉk′,`′ with
k′ < k and `′ ≤ `. By Lemma 4.10,
ĉk,` =
√
n
m
(−1)m+n−2s
v
2
√
(m− 1)!(n− 1)!
Γ(m+ 1
2 + σ)Γ(n+ 1
2 + τ)
×
min{m−1,n−1}∑
q=0
Γ(q + 3
2 + θ)Γ(m− 1− q + σ − θ)Γ(n− 1− q + τ − θ)
q!(m− 1− q)!(n− 1− q)!Γ(σ − θ)Γ(τ − θ)
+
σ − θ
m
m−1∑
i=0
(−1)m−i
√
m!Γ(i+ 1
2 + σ)
i!Γ(m+ 1
2 + σ)
(−1)i+ns
v
2
√
i!n!
Γ(i+ 1
2 + σ)Γ(n+ 1
2 + τ)
×
min{i,n}∑
p=0
Γ(p+ 1
2 + θ)Γ(i− p+ σ − θ)Γ(n− p+ τ − θ)
p!(i− p)!(n− p)!Γ(σ − θ)Γ(τ − θ)
= (−1)m+ns
v
2
1
m
√
m!n!
Γ(m+ 1
2 + σ)Γ(n+ 1
2 + τ)
×
(
min{m−1,n−1}∑
q=0
Γ(q + 3
2 + θ)Γ(m− 1− q + σ − θ)Γ(n− 1− q + τ − θ)
q!(m− 1− q)!(n− 1− q)!Γ(σ − θ)Γ(τ − θ)
+ (σ − θ)
m−1∑
i=0
min{i,n}∑
p=0
Γ(p+ 1
2 + θ)Γ(i− p+ σ − θ)Γ(n− p+ τ − θ)
p!(i− p)!(n− p)!Γ(σ − θ)Γ(τ − θ)
)
.
Then the desired expression for ĉk,` follows because
min{m−1,n−1}∑
q=0
Γ(q + 3
2 + θ)Γ(m− 1− q + σ − θ)Γ(n− 1− q + τ − θ)
q!(m− 1− q)!(n− 1− q)!Γ(σ − θ)Γ(τ − θ)
=
min{m,n}∑
p=0
pΓ(p+ 1
2 + θ)Γ(m− p+ σ − θ)Γ(n− p+ τ − θ)
p!(m− p)!(n− p)!Γ(σ − θ)Γ(τ − θ)
,
and, by (4.5),
(σ − θ)
m−1∑
i=0
min{i,n}∑
p=0
Γ(p+ 1
2 + θ)Γ(i− p+ σ − θ)Γ(n− p+ τ − θ)
p!(i− p)!(n− p)!Γ(σ − θ)Γ(τ − θ)
22 J.A. Álvarez López, M. Calaza and C. Franco
= (σ − θ)
min{m−1,n}∑
p=0
m−1−p∑
j=0
Γ(p+ 1
2 + θ)Γ(j + σ − θ)Γ(n− p+ τ − θ)
p!j!(n− p)!Γ(σ − θ)Γ(τ − θ)
=
min{m,n}∑
p=0
Γ(p+ 1
2 + θ)(m− p)Γ(m− p+ σ − θ)Γ(n− p+ τ − θ)
p!(m− p)!(n− p)!Γ(σ − θ)Γ(τ − θ)
. (4.7)
Finally, take k = 2m+ 1 and ` = 2n+ 1, and assume that the result holds for all ĉk′,`′ with
k′ < k and `′ < `. By Lemma 4.11,
ĉk,` =
(m+ 1
2 + θ)(−1)m+ns
v
2√
(m+ 1
2 + σ)(n+ 1
2 + τ)
√
m!n!
Γ(m+ 1
2 + σ)Γ(n+ 1
2 + τ)
×
min{m,n}∑
p=0
Γ(p+ 1
2 + θ)Γ(m− p+ σ − θ)Γ(n− p+ τ − θ)
p!(m− p)!(n− p)!Γ(σ − θ)Γ(τ − θ)
− σ − θ√
n+ 1
2 + τ
m−1∑
i=0
(−1)m−i
√
m!Γ(i+ 1
2 + σ)
i!Γ(m+ 3
2 + σ)
× (−1)i+ns
v
2
√
i!n!
Γ(i+ 1
2 + σ)Γ(n+ 1
2 + τ)
×
min{i,n}∑
p=0
Γ(p+ 1
2 + θ)Γ(i− p+ σ − θ)Γ(n− p+ τ − θ)
p!(i− p)!(n− p)!Γ(σ − θ)Γ(τ − θ)
= (−1)m+ns
v
2
√
m!n!
Γ(m+ 3
2 + σ)Γ(n+ 3
2 + τ)
×
(
min{m,n}∑
p=0
(m+ 1
2 + θ)Γ(p+ 1
2 + θ)Γ(m− p+ σ − θ)Γ(n− p+ τ − θ)
p!(m− p)!(n− p)!Γ(σ − θ)Γ(τ − θ)
− (σ − θ)
m−1∑
i=0
min{i,n}∑
p=0
Γ(p+ 1
2 + θ)Γ(i− p+ σ − θ)Γ(n− p+ τ − θ)
p!(i− p)!(n− p)!Γ(σ − θ)Γ(τ − θ)
)
.
Then we get the stated expression for ĉk,` using (4.7) again. �
Remark 4.13. By (4.2), if k and ` are odd, then Proposition 4.12 also holds when σ, τ > −3
2 .
5 The sesquilinear form t′
Consider the notation of Section 4. Since x−1Sodd = Sev, a sesquilinear form t′ in L2
σ,τ , with
D(t′) = S, is defined by
t′(φ, ψ) =
〈
φev, x
−1ψodd
〉
θ
= 〈xφev, ψodd〉θ−1.
Note that t′ is neither symmetric nor bounded from the left. The goal of this section is to
study t′, and use it to prove Theorem 1.3.
Let c′k,` = t′(φσ,k, φτ,`). Clearly, c′k,` = 0 if k is odd or ` is even.
A Perturbation of the Dunkl Harmonic Oscillator on the Line 23
5.1 Case where σ = θ = τ
In this case, we have v = 0.
Proposition 5.1. For k = 2m and ` = 2n + 1, if k > ` (m > n), then c′k,` = 0, and, if k < `
(m ≤ n), then
c′k,` = (−1)n−ms
1
2
√
n!Γ(m+ 1
2 + σ)
m!Γ(n+ 3
2 + σ)
.
Proof. This follows from (2.6) since ĉk,` = δk,` in this case. �
Proposition 5.2. There is some ω = ω(σ) > 0 so that, for k = 2m and ` = 2n+ 1,
|c′k,`| 4 s
1
2 (m+ 1)−ω(n+ 1)−ω.
Proof. We can assume that m ≤ n according to Proposition 5.1. Moreover
|c′k,`| 4 s
1
2 (m+ 1)
σ
2
− 1
4 (n+ 1)−
σ
2
− 1
4
for allm ≤ n by Proposition 5.1 and Lemma 3.12. Therefore the result follows using Lemma 3.14,
reversing the roles of m and n, because −σ
2 −
1
4 < −
u
2 < 0. �
5.2 Case where σ = θ 6= τ and τ − σ 6∈ −N
Recall that v = τ − σ in this case. Moreover c′k,` = 0 if k > ` by (2.6) and Corollary 4.5.
Proposition 5.3. For k = 2m < ` = 2n+ 1 (m ≤ n),
c′k,` = (−1)m+ns
1+v
2
√
n!Γ(m+ 1
2 + σ)
m!Γ(n+ 3
2 + τ)
Γ(n−m+ 1 + v)
(n−m)!Γ(1 + v)
.
Proof. By (2.6), Corollary 4.5, Proposition 4.7 and (4.5),
c′k,` = s
1
2
n∑
j=m
(−1)n−j
√
n!Γ(j + 1
2 + τ)
j!Γ(n+ 3
2 + τ)
(−1)m+js
v
2
√
j!Γ(m+ 1
2 + σ)
m!Γ(j + 1
2 + τ)
Γ(j −m+ v)
(j −m)!Γ(v)
= (−1)m+ns
1+v
2
√
n!Γ(m+ 1
2 + σ)
m!Γ(n+ 3
2 + τ)
1
Γ(v)
n−m∑
i=0
Γ(i+ v)
i!
= (−1)m+ns
1+v
2
√
n!Γ(m+ 1
2 + σ)
m!Γ(n+ 3
2 + τ)
Γ(n−m+ 1 + v)
(n−m)!Γ(1 + v)
. �
Proposition 5.4. If (σ, τ) satisfies (1.4), then there is some ω = ω(σ, τ) > 0 so that, for
k = 2m < ` = 2n+ 1,
|c′k,`| 4 s
1+v
2 (m+ 1)−ω(n+ 1)−ω.
Proof. By Proposition 5.3 and Lemma 3.12,
|c′k,`| 4 s
1+v
2 (m+ 1)
σ
2
− 1
4 (n+ 1)−
τ
2
− 1
4 (n−m+ 1)v.
Then the result follows by Lemma 3.14, interchanging the roles of m and n, using the condition
of Theorem 1.3(a). �
24 J.A. Álvarez López, M. Calaza and C. Franco
5.3 Case where σ 6= θ = τ and σ − θ 6∈ −N
Recall that v = σ − τ in this case.
Proposition 5.5. For k = 2m and ` = 2n+ 1,
c′k,` = (−1)m+ns
1+v
2
√
m!n!
Γ(m+ 1
2 + σ)Γ(n+ 3
2 + τ)
min{m,n}∑
j=0
Γ(j + 1
2 + τ)Γ(m− j + v)
j!(m− j)!Γ(v)
.
Proof. Let j run from 0 to min{m,n}. By (2.6), Corollary 4.5, Proposition 4.7 and (4.1),
c′k,` = s
1
2
∑
j
(−1)n−j
√
n!Γ(j + 1
2 + τ)
j!Γ(n+ 3
2 + τ)
(−1)j+ms
v
2
√
m!Γ(j + 1
2 + τ)
j!Γ(m+ 1
2 + σ)
Γ(m− j + v)
(m− j)!Γ(v)
= (−1)m+ns
1+v
2
√
m!n!
Γ(m+ 1
2 + σ)Γ(n+ 3
2 + τ)
∑
j
Γ(j + 1
2 + τ)Γ(m− j + v)
j!(m− j)!Γ(v)
. �
Proposition 5.6. If (σ, τ) satisfies (1.5), then there is some ω = ω(σ, τ) > 0 so that, for
k = 2m and ` = 2n+ 1,
|c′k,`| 4 s
1+v
2 (m+ 1)−ω(n+ 1)−ω.
Proof. By Proposition 5.5 and Lemma 3.12,
|c′k,`| 4 s
1+v
2 (m+ 1)
1
4
−σ
2 (n+ 1)−
1
4
− τ
2
min{m,n}∑
j=0
(m− j + 1)v−1(j + 1)τ−
1
2 .
Then the result follows by Corollary 7.4, proved in Section 7, since (σ, τ) satisfies (1.5). �
5.4 Case where σ 6= θ = τ + 1 and σ − τ − 1 6∈ −N
Note that v = σ − τ − 2 in this case. Moreover
c′k,` =
〈
φσ,k, x
−1φτ,`
〉
τ+1
= 〈xφσ,k, φτ,`〉τ = 〈φτ,`, xφσ,k〉τ (5.1)
for k = 2m and ` = 2n+ 1 (Remark 1.4(ii)).
Proposition 5.7. Let k = 2m and ` = 2n+ 1. If k+ 1 < ` (m < n), then c′k,` = 0. If k+ 1 ≥ `
(m ≥ n), then
c′k,` = (−1)m+ns
v+1
2
√
m!Γ(n+ 3
2 + τ)
n!Γ(m+ 1
2 + σ)
Γ(m− n+ v + 1)
(m− n)!Γ(v + 1)
.
Proof. By (2.5) and (5.1),
c′k,` =
√
m+ 1
2 + σ
s
ĉτ,σ,τ,`,k+1 +
√
m
s
ĉτ,σ,τ,`,k−1. (5.2)
So c′k,` = 0 if k+ 1 < ` by Corollary 4.5. When k+ 1 = ` (m = n), by (5.2) and Proposition 4.7,
c′k,` =
√
m+ 1
2 + σ
s
s
v+2
2
√
Γ(n+ 3
2 + τ)
Γ(m+ 3
2 + σ)
= s
v+1
2
√
Γ(n+ 3
2 + τ)
Γ(m+ 1
2 + σ)
.
A Perturbation of the Dunkl Harmonic Oscillator on the Line 25
When k − 1 ≥ ` (m > n), by (5.2) and Proposition 4.7,
c′k,` =
√
m+ 1
2 + σ
s
(−1)m+ns
v+2
2
√
m!Γ(n+ 3
2 + τ)
n!Γ(m+ 3
2 + σ)
Γ(m− n+ v + 2)
(m− n)!Γ(v + 2)
+
√
m
s
(−1)m+n−1s
v+2
2
√
(m− 1)!Γ(n+ 3
2 + τ)
n!Γ(m+ 1
2 + σ)
Γ(m− n+ v + 1)
(m− 1− n)!Γ(v + 2)
= (−1)m+ns
v+1
2
√
m!Γ(n+ 3
2 + τ)
n!Γ(m+ 1
2 + σ)
Γ(m− n+ v + 1)
(m− 1− n)!Γ(v + 2)
(
m− n+ v + 1
m− n
− 1
)
= (−1)m+ns
v+1
2
√
m!Γ(n+ 3
2 + τ)
n!Γ(m+ 1
2 + σ)
Γ(m− n+ v + 1)
(m− n)!Γ(v + 1)
. �
Proposition 5.8. If (σ, τ) satisfies (1.6), then there is some ω = ω(σ, τ) > 0 so that, for
k = 2m and ` = 2n+ 1,
|c′k,`| 4 s
v+1
2 (m+ 1)−ω(n+ 1)−ω.
Proof. By Proposition 5.7, we can assume that k+ 1 ≥ ` (m ≥ n), and, in this case, using also
Lemma 3.12, we get
|c′k,`| 4 s
v+1
2 (m+ 1)
1
4
−σ
2 (n+ 1)
1
4
+ τ
2 (m− n+ 1)v.
Then the result follows using Lemma 3.14. �
5.5 Case where σ 6= θ 6= τ and σ − θ, τ − θ 6∈ −N
Proposition 5.9. For k = 2m and ` = 2n+ 1,
c′k,` = (−1)m+ns
1+v
2
√
m!n!
Γ(m+ 1
2 + σ)Γ(n+ 3
2 + τ)
×
min{m,n}∑
p=0
Γ(p+ 1
2 + θ)Γ(m− p+ σ − θ)Γ(n− p+ 1 + τ − θ)
p!(m− p)!(n− p)!Γ(σ − θ)Γ(1 + τ − θ)
.
Proof. By (2.6) and Proposition 4.12,
c′k,` = s
1
2
n∑
j=0
(−1)n−j
√
n!Γ(j + 1
2 + τ)
j!Γ(n+ 3
2 + τ)
(−1)m+js
v
2
√
m!j!
Γ(m+ 1
2 + σ)Γ(j + 1
2 + τ)
×
min{m,j}∑
p=0
Γ(p+ 1
2 + θ)Γ(m− p+ σ − θ)Γ(j − p+ τ − θ)
p!(m− p)!(j − p)!Γ(σ − θ)Γ(τ − θ)
= (−1)m+ns
1+v
2
√
m!n!
Γ(m+ 1
2 + σ)Γ(n+ 3
2 + τ)
×
n∑
j=0
min{m,j}∑
p=0
Γ(p+ 1
2 + θ)Γ(m− p+ σ − θ)Γ(j − p+ τ − θ)
p!(m− p)!(j − p)!Γ(σ − θ)Γ(τ − θ)
.
But, by (4.5),
n∑
j=0
min{m,j}∑
p=0
Γ(m− p+ σ − θ)Γ(j − p+ τ − θ)
(m− p)!(j − p)!Γ(σ − θ)Γ(τ − θ)
26 J.A. Álvarez López, M. Calaza and C. Franco
=
min{m,n}∑
p=0
n∑
j=p
Γ(m− p+ σ − θ)Γ(j − p+ τ − θ)
(m− p)!(j − p)!Γ(σ − θ)Γ(τ − θ)
=
min{m,n}∑
p=0
n−p∑
i=0
Γ(m− p+ σ − θ)Γ(i+ τ − θ)
(m− p)!i!Γ(σ − θ)Γ(τ − θ)
=
min{m,n}∑
p=0
Γ(m− p+ σ − θ)Γ(n− p+ 1 + τ − θ)
(m− p)!(n− p)!Γ(σ − θ)Γ(1 + τ − θ)
. �
Proposition 5.10. If (σ, τ, θ) satisfies (1.7), then there is some ω = ω(σ, τ, θ) > 0 so that, for
k = 2m and ` = 2n+ 1,
|c′k,`| 4 s
1+v
2 (m+ 1)−ω(n+ 1)−ω.
Proof. Let p run from 0 to min{m,n}. By Proposition 5.9 and Lemma 3.12,
|c′k,`| 4 s
1+v
2 (m+ 1)
1
4
−σ
2 (n+ 1)−
1
4
− τ
2
∑
p
(m− p+ 1)σ−θ−1(n− p+ 1)τ−θ(p+ 1)θ−
1
2 .
Then the result follows by Corollary 7.2, proved in Section 7, since (σ, τ, θ) satisfies (1.7). �
5.6 Proof of Theorem 1.3
Assume the conditions of Theorem 1.3. Let jσ,τ be the positive definite symmetric sesquilinear
form in L2
σ,τ , with domain S, defined by jσ,τ (φ, ψ) = 〈Jσ,τφ, ψ〉σ,τ .
Proposition 5.11. For any ε > 0, there is some E = E(ε, σ, τ, θ) > 0 such that, for all φ ∈ S,
|t′(φ)| ≤ εs
v−1
2 jσ,τ (φ) + Es
1+v
2 ‖φ‖2σ,τ .
Proof. This follows from Propositions 5.2, 5.4, 5.6, 5.8 and 5.10 using the arguments of the
proof of Proposition 3.17. �
Proof of Theorem 1.3. This is analogous to the proof of Theorem 1.1. Thus some details
and the bibliographic references are omitted.
Let tσ,τ be the positive definite symmetric sesquilinear form in L2
σ,τ , with D(tσ,τ ) = S, defined
by tσ on Sev and tτ on Sodd, and vanishing on Sev × Sodd. Let s be the symmetric sesquilinear
form in L2
σ,τ , with D(s) = S, defined by s(φ, ψ) = t′(φ, ψ) + t′(ψ, φ). Then the symmetric
sesquilinear form v = jσ,τ + ξtσ,τ + ηs in L2
σ,τ , with D(v) = S, is given by the right hand side
of (1.8). Using Propositions 3.17 and 5.11, for any ε > 0, there are some C = C(ε, σ, τ, u) > 0
and E = E(ε, σ, τ, θ) > 0 such that, for all φ ∈ S,
|(ξtσ,τ + ηs(φ)| ≤ ε
(
ξsu−1 + 2|η|s
v−1
2
)
jσ,τ (φ) +
(
ξCsu + 2|η|Es
1+v
2
)
‖φ‖2σ,τ . (5.3)
Then, taking ε so that ε
(
ξsu−1 + 2|η|s
v−1
2
)
< 1, since jσ,τ is closable and positive definite, it
follows that v is sectorial and closable, and D(v̄) = D(jσ,τ ); in particular, v is bounded from
below because it is also symmetric. So v̄ is induced by a self-adjoint operator V in L2
σ,τ with
D(V1/2) = D(v̄). Thus S is a core of v̄ and V1/2.
For all φ ∈ S,
v(φ) ≥ jσ,τ (φ) + ξtσ,τ (φ)− |η|s(φ) ≥ jσ,τ (φ) + ξtσ,τ (φ)− 2|η||t′(φ)|. (5.4)
A Perturbation of the Dunkl Harmonic Oscillator on the Line 27
Since S is a core of v̄ and jσ,τ , using Propositions 3.18 and 5.11 like in the proof of Theorem 1.1,
it follows from (5.4) that V has a discrete spectrum, which consists of two groups of eigenvalues,
λ0 ≤ λ2 ≤ · · · and λ1 ≤ λ3 ≤ · · · , repeated according to their multiplicity, satisfying (1.9). On
the other hand, by (5.3), for all φ ∈ S,
v(φ) ≤
(
1 + ε
(
ξsu−1 + 2|η|s
v−1
2
))
jσ,τ (φ) +
(
ξCsu + 2|η|Es
1+v
2
)
‖φ‖2σ,τ , (5.5)
obtaining (1.10) because S is a core of v̄ and jσ,τ .
With the notation of (iii), let t̃σ (respectively, t̃τ ) be the symmetric sesquilinear form in L2
σ
(respectively, L2
τ ), with D(̃tσ) = S (respectively, D(̃tτ ) = S), defined like tσ (respectively, tτ ),
using ũ (respectively, v − ũ + 1) instead of u. Let t̃σ,τ be the positive definite symmetric
sesquilinear form in L2
σ,τ , with D(̃tσ,τ ) = S, defined by t̃σ on Sev and t̃τ on Sodd, and vanishing
on Sev × Sodd. By the Schwartz inequality,
2|t′(φ)| = 2
∣∣〈φev|x|−ũ+σ−θ, x−1φodd|x|ũ−σ+θ
〉
θ
∣∣
≤ 2
∥∥φev|x|−ũ+σ−θ∥∥
θ
·
∥∥φodd|x|ũ−σ+θ−1
∥∥
θ
= 2
∥∥φev|x|−ũ
∥∥
σ
·
∥∥φodd|x|ũ−v−1
∥∥
τ
≤
∥∥φev|x|−ũ
∥∥2
σ
+
∥∥φodd|x|ũ−v−1
∥∥2
τ
= t̃σ,τ (φ). (5.6)
Hence (1.12) follows like in the proof of Theorem 1.1, using Propositions 3.17 and 3.18.
If u = v+1
2 , then we can take ũ = u = v − ũ+ 1 in (iii), yielding
v(φ) ≥ jσ,τ (φ) + (ξ − |η|)tσ,τ (φ)
by (5.4) and (5.6). Thus (1.13) follows if moreover |η| ≤ ξ, like in the proof of Theorem 1.1,
using Proposition 3.18.
If we add the term ξ′〈φev, ψev〉σ + ξ′′〈φodd, ψodd〉τ to the right hand side of (1.8), for some
ξ′, ξ′′ ∈ R, then the same argument can be used by adding the term ξ′‖φev‖2σ + ξ′′‖φodd‖2τ to jσ,τ ,
obtaining (v). �
6 A preliminary estimate
6.1 Statement
The standard coordinates of R5 are denoted by (α, β, γ, δ,κ). Consider the partition of R into the
following intervals: I1 = (−∞,−1], I2 = (−1,−1
2 ], I3 = (−1
2 ,−
1
3 ], I4 = (−1
3 , 0) and I5 = [0,∞).
Let Qijk = Ii × Ij × Ik, and consider the following subsets of R5:
S515: This is the subset of R2 ×Q515 defined by
α+ γ, α+ β + γ + κ < 0. (6.1)
S522: This is the subset of R2 ×Q522 defined by
α+ γ, α+ β + γ < 0. (6.2)
S252: This is the subset of R2 ×Q252 defined by
0 ≤ γ + δ ⇒ α+ γ, α+ β + γ + δ + κ + 1 < 0, (6.3)
γ + δ < 0⇒
α+ γ + δ, α+ β + κ + 1 < 0, or
α+ γ + 1
2 , α+ β + δ < 0, or
α+ γ + 1
3 , α+ β + δ + 1
3 < 0, or
α+ γ, α+ β + δ + κ + 1 < 0.
(6.4)
28 J.A. Álvarez López, M. Calaza and C. Franco
S155: This is the subset of R2 ×Q155 defined by (6.3) and
γ + δ < 0⇒
α+ γ + δ, α+ β + κ + 1 < 0, or
α+ γ + 1, α+ β + δ + κ < 0, or
α+ γ + 1
2 , α+ β + δ + κ + 1
2 < 0, or
α+ γ + 1
3 , α+ β + δ + κ + 2
3 < 0, or
α+ γ, α+ β + δ + κ + 1 < 0.
(6.5)
S212: This is the subset of R2 ×Q212 defined by
γ ≤ κ ⇒ α+ γ, α+ β + κ < 0, (6.6)
κ ≤ γ ⇒ α+ γ, α+ β + γ < 0. (6.7)
Let Š = S515 ∪S522 ∪S252 ∪S155 ∪S212. On the other hand, consider the linear isomorphism
of R5 defined by
(α, β, γ, δ,κ) 7→ (β, α, δ, γ,κ). (6.8)
This is the reflection with respect to the linear subspace defined by α = β and γ = δ. The
image of any subset X ⊂ R5 by the mapping (6.8) is denoted by X ′, and let Xconv be the convex
hull of X. Thus X ′conv := (X ′)conv = (Xconv)′.
Lemma 6.1. If (α, β, γ, δ,κ) ∈ Šconv ∩ Š′conv, then there is some ω > 0 such that, for all
m,n ∈ N,
(m+ 1)α(n+ 1)β
min{m,n}∑
p=0
(m− p+ 1)γ(n− p+ 1)δ(p+ 1)κ 4 (m+ 1)−ω(n+ 1)−ω. (6.9)
6.2 Proof of Lemma 6.1
Since the roles of m and n in Lemma 6.1 are interchanged by the mapping (6.8), we can assume
that m ≥ n. Then Lemma 3.14 gives (6.9) once
n∑
p=0
(m− p+ 1)γ(n− p+ 1)δ(p+ 1)κ
is appropriately estimated. Estimates of this expression are achieved with several strategies
explained in Sections 6.2.1–6.2.7, giving rise to several lists of conditions that guarantee (6.9)
when m ≥ n. Then, for the chosen subindices ijk equal to 515, 522, 252, 155, 212, every Sijk
is defined by the most general of those conditions on R2×Qijk. This will show that (6.9) holds
for m ≥ n and (α, β, γ, δ,κ) ∈ Š. In Section 6.2.9, it will be shown that this property can be
extended to the convex hull Šconv, completing the proof of Lemma 6.1.
6.2.1 First list of conditions
For all ε > 0,
n∑
p=0
(p+ 1)κ =
n+1∑
q=1
qκ ≤
∫ n+2
1
xκdx if κ ≥ 0,
1 +
∫ n+1
1
xκdx if κ < 0
A Perturbation of the Dunkl Harmonic Oscillator on the Line 29
4
(n+ 1)κ+1 if κ > −1,
1 + ln(n+ 1) if κ = −1,
1 if κ < −1
4
(n+ 1)κ+1 if κ > −1,
(n+ 1)ε if κ = −1,
1 if κ < −1.
(6.10)
On the other hand, we claim that
(m− p+ 1)γ(n− p+ 1)δ 4
(m+ 1)γ(n+ 1)δ if δ ≥ −γ, 0,
(m− n+ 1)γ+δ and
(m− n+ 1)γ(n+ 1)δ
}
if 0 ≤ δ < −γ,
(m− n+ 1)γ if δ ≤ −γ, 0,
(m− n+ 1)γ or
(m+ 1)γ(n+ 1)δ
}
if −γ < δ < 0,
(6.11)
for all p = 0, . . . , n. Combining (6.10) and (6.11), it follows that
(m+ 1)α(n+ 1)β
n∑
p=0
(m− p+ 1)γ(n− p+ 1)δ(p+ 1)κ 4 A1, (6.12)
where A1 = A1(m,n, α, β, γ, δ,κ) can be taken to be equal to
(m+ 1)α+γ(n+ 1)β+δ+κ+1 if −γ, 0 ≤ δ, −1 < κ,
(m+ 1)α(n+ 1)β+κ+1(m− n+ 1)γ+δ and
(m+ 1)α(n+ 1)β+δ+κ+1(m− n+ 1)γ
}
if 0 ≤ δ < −γ, −1 < κ.
In the definition of A1, the other cases of γ, δ,κ are omitted because they will not be used. We
will continue omitting such cases, often without further comment. Resulting tautologies will be
also removed without further comment. By (6.12), applying Lemma 3.14 to the above list, we
get the first list of conditions that guarantee (6.9) when m ≥ n:
−γ, 0 ≤ δ, −1 < κ ⇒ α+ γ, α+ β + γ + δ + κ + 1 < 0, (6.13)
0 ≤ δ < −γ, −1 < κ ⇒
{
α+ γ + δ, α+ β + κ + 1 < 0, or
α+ γ, α+ β + δ + κ + 1 < 0.
(6.14)
To prove (6.11), it is enough to study the maximum of the C∞ function
f(x) = (m− x+ 1)γ(n− x+ 1)δ
on [0, n] (the natural domain of f contains (−∞, n+ 1)). We have
f ′(x) = (m− x+ 1)γ−1(n− x+ 1)δ−1h(x),
where
h(x) = (γ + δ)x− γ(n+ 1)− δ(m+ 1).
Observe that this expression is valid even when γ = 0 or δ = 0. Since f ′ and h have the same
zero set on [0, n], and they have the same sign on the complement of the zero set in [0, n], it is
enough to analyze h to know where f reaches its maximum on [0, n]. We consider several cases.
Case where γ+δ = 0. Then h ≡ γ(m−n). If m > n and γ 6= 0, then h 6= 0 and signh = sign γ.
If m = n or γ = 0, then h ≡ 0. Hence:
max
0≤x≤n
f(x) =
{
f(n) = (m− n+ 1)γ if γ = −δ ≥ 0,
f(0) = (m+ 1)γ(n+ 1)δ if γ = −δ ≤ 0.
(6.15)
30 J.A. Álvarez López, M. Calaza and C. Franco
Case where γ + δ 6= 0. Then h vanishes just at the point
x0 :=
γ(n+ 1) + δ(m+ 1)
γ + δ
.
Case where γ + δ < 0. We have h > 0 on (−∞, x0) and h < 0 on (x0,∞), yielding
max
0≤x≤n
f(x) =
f(0) = (m+ 1)γ(n+ 1)δ if x0 ≤ 0,
f(x0) if 0 ≤ x0 ≤ n,
f(n) = (m− n+ 1)γ if x0 ≥ n.
(6.16)
Case where γ + δ < 0 and δ ≤ 0. Then x0 ≥ n+ 1, and therefore, by (6.16),
γ + δ < 0, δ ≤ 0⇒ max
0≤x≤n
f(x) = (m− n+ 1)γ . (6.17)
Case where γ + δ < 0 and δ > 0; i.e., 0 < δ < −γ. We may have x0 ≤ 0, 0 ≤ x0 ≤ n or
n ≤ x0. Moreover
f(x0) =
(−γ)γδδ
(−γ − δ)γ+δ
(m− n)γ+δ 4 (m− n+ 1)γ+δ.
Therefore
0 < δ < −γ ⇒ max
0≤x≤n
f(x) 4 (m− n+ 1)γ+δ (6.18)
by (6.16) and since
(m− n+ 1)γ , (m+ 1)γ(n+ 1)δ < (m− n+ 1)γ+δ,
which follows using that γ < γ + δ and
n ≥ m
2
⇒ m− n+ 1
n+ 1
≤ 1⇒ m− n+ 1
m+ 1
<
m− n+ 1
n+ 1
≤
(
m− n+ 1
n+ 1
)− δ
γ
,
n <
m
2
⇒ m− n+ 1
n+ 1
> 1⇒ m− n+ 1
m+ 1
≤ 1 <
(
m− n+ 1
n+ 1
)− δ
γ
,
because 0 < − δ
γ < 1. On the other hand, in this case,
max
0≤x≤n
f(x) ≤ max
0≤x≤n
(m− x+ 1)γ max
0≤y≤n
(n− y + 1)δ
= (m− n+ 1)γ(n+ 1)δ < (m− n+ 1)γ+δ
if n < m
2 . So (6.18) can be improved by
0 < δ < −γ ⇒ max
0≤x≤n
f(x) 4
{
(m− n+ 1)γ+δ and
(m− n+ 1)γ(n+ 1)δ.
(6.19)
Case where γ + δ > 0. We have h < 0 on (−∞, x0) and h > 0 on (x0,∞), yielding
max
0≤x≤n
f(x) =
f(n) = (m− n+ 1)γ if x0 ≤ 0,
max{f(0), f(n)} if 0 ≤ x0 ≤ n,
f(0) = (m+ 1)γ(n+ 1)δ if x0 ≥ n.
(6.20)
Case where γ + δ > 0 and δ ≥ 0. We get x0 ≥ n+ 1, and therefore, by (6.20),
γ + δ > 0, δ ≥ 0⇒ max
0≤x≤n
f(x) = (m+ 1)γ(n+ 1)δ. (6.21)
Gathering together (6.15), (6.17), (6.19) and (6.21), we get the first two cases of (6.11). The
other cases will not be used, and they follow with similar arguments.
A Perturbation of the Dunkl Harmonic Oscillator on the Line 31
6.2.2 Second list of conditions
According to (6.10), for all ε > 0,
n∑
p=0
(n− p+ 1)δ =
n+1∑
q=1
qδ 4
(n+ 1)δ+1 if δ > −1,
(n+ 1)ε if δ = −1,
1 if δ < −1.
(6.22)
On the other hand, we claim that
(m− p+ 1)γ(p+ 1)κ ≤
(m+ 1)γ(n+ 1)κ if γ,κ ≥ 0,
(m− n+ 1)γ(n+ 1)κ if γ ≤ 0 ≤ κ,
(m− n+ 1)γ(n+ 1)κ or
(m+ 1)γ
}
if γ < κ < 0,
(m+ 1)γ if κ ≤ γ, 0,
(6.23)
for all p = 0, . . . , n. Combining (6.22) and (6.23), it follows that, for all ε > 0,
(m+ 1)α(n+ 1)β
n∑
p=0
(m− p+ 1)γ(n− p+ 1)δ(p+ 1)κ 4 A2, (6.24)
where A2 = A2(m,n, α, β, γ, δ,κ, ε) can be taken to be equal to
(m+ 1)α+γ(n+ 1)β+κ+ε if 0 ≤ γ,κ, δ = −1,
(m+ 1)α+γ(n+ 1)β+κ if 0 ≤ γ,κ, δ < −1,
(m+ 1)α(n+ 1)β+κ+ε(m− n+ 1)γ if γ ≤ 0 ≤ κ, δ = −1,
(m+ 1)α(n+ 1)β+κ(m− n+ 1)γ if γ ≤ 0 ≤ κ, δ < −1,
(m+ 1)α(n+ 1)β+κ+ε(m− n+ 1)γ or
(m+ 1)α+γ(n+ 1)β+ε
}
if γ < κ < 0, δ = −1,
(m+ 1)α(n+ 1)β+κ(m− n+ 1)γ or
(m+ 1)α+γ(n+ 1)β
}
if γ < κ < 0, δ < −1,
(m+ 1)α+γ(n+ 1)β+ε if κ ≤ γ, 0, δ = −1,
(m+ 1)α+γ(n+ 1)β if κ ≤ γ, 0, δ < −1.
By (6.24), applying Lemma 3.14 to the above list, we get the second list of conditions that
guarantee (6.9) when m ≥ n:
0 ≤ γ,κ, δ ≤ −1⇒ α+ γ, α+ β + γ + κ < 0, (6.25)
γ ≤ κ, 0, δ ≤ −1⇒ α+ γ, α+ β + κ < 0, (6.26)
κ ≤ γ, 0, δ ≤ −1⇒ α+ γ, α+ β + γ < 0. (6.27)
To prove (6.23), it is enough to study the maximum of the C∞ function
f(x) = (m− x+ 1)γ(x+ 1)κ
on [0, n] (the natural domain of f contains (−1,m+ 1)). We have
f ′(x) = (m− x+ 1)γ−1(x+ 1)κ−1h(x),
32 J.A. Álvarez López, M. Calaza and C. Franco
where
h(x) = −(γ + κ)x+ κ(m+ 1)− γ.
Observe that this expression is valid even when γ = 0 or κ = 0. Since f ′ and h have the same
zero set on [0, n], and they have the same sign on the complement of the zero set in [0, n], it is
enough to analyze h to know where f reaches its maximum on [0, n]. We consider several cases.
Case where γ + κ = 0. Then h ≡ κ(m + 2). If κ 6= 0, then h 6= 0 and signh = signκ. If
κ = 0, then h ≡ 0. Hence:
max
0≤x≤n
f(x) =
{
f(n) = (m− n+ 1)γ(n+ 1)κ if κ = −γ ≥ 0,
f(0) = (m+ 1)γ if κ = −γ ≤ 0.
(6.28)
Case where γ + κ 6= 0. Then h vanishes just at the point
x0 :=
κ(m+ 1)− γ
γ + κ
.
Case where γ + κ < 0. We have h < 0 on (−∞, x0) and h > 0 on (x0,∞), yielding
max
0≤x≤n
f(x) =
f(n) = (m− n+ 1)γ(n+ 1)κ if x0 ≤ 0,
max{f(0), f(n)} if 0 ≤ x0 ≤ n,
f(0) = (m+ 1)γ if x0 ≥ n.
(6.29)
Case where γ + κ < 0 and κ ≥ 0; i.e., 0 ≤ κ < −γ. Then x0 < 0, and therefore, by (6.29),
0 ≤ κ < −γ ⇒ max
0≤x≤n
f(x) = (m− n+ 1)γ(n+ 1)κ. (6.30)
Case where γ + κ < 0 and γ ≥ 0; i.e., 0 ≤ γ < −κ. Then x0 ≥ m + 1, and therefore,
by (6.29),
0 ≤ γ < −κ ⇒ max
0≤x≤n
f(x) = (m+ 1)γ . (6.31)
Case where κ ≤ γ < 0. Then x0 ≥ m
2 , and we may have x0 ≤ n or x0 ≥ n. In any case,
by (6.29),
κ ≤ γ < 0⇒ max
0≤x≤n
f(x) = (m+ 1)γ . (6.32)
Case where γ < κ < 0. Then x0 <
m
2 , and we may have x0 ≤ 0, 0 ≤ x0 ≤ n or x0 ≥ n. In
any case, by (6.29),
γ < κ < 0⇒ max
0≤x≤n
f(x) =
{
(m− n+ 1)γ(n+ 1)κ or
(m+ 1)γ .
(6.33)
Case where γ + κ > 0. We have h > 0 on (−∞, x0) and h < 0 on (x0,∞), yielding
max
0≤x≤n
f(x) =
f(0) = (m+ 1)γ if x0 ≤ 0,
f(x0) if 0 ≤ x0 ≤ n,
f(n) = (m− n+ 1)γ(n+ 1)κ if x0 ≥ n.
(6.34)
Case where γ + κ > 0 and κ ≤ 0; i.e., −γ < κ ≤ 0. Then x0 < 0, and therefore, by (6.34),
−γ < κ ≤ 0⇒ max
0≤x≤n
f(x) = (m+ 1)γ . (6.35)
A Perturbation of the Dunkl Harmonic Oscillator on the Line 33
Case where γ + κ > 0 and γ ≤ 0; i.e., −κ < γ ≤ 0. Then x0 ≥ m + 1, and therefore,
by (6.34),
−κ < γ ≤ 0⇒ max
0≤x≤n
f(x) = (m− n+ 1)γ(n+ 1)κ. (6.36)
Case where γ + κ > 0 and γ,κ ≥ 0. We may have x0 ≤ 0, 0 ≤ x0 ≤ n or n ≤ x0. Moreover
f(x0) =
γγκκ
(γ + κ)γ+κ (m+ 2)γ+κ 4 (m+ 1)γ+κ.
But, in this case,
max
0≤x≤n
f(x) ≤ max
0≤x≤n
(m− x+ 1)γ max
0≤y≤n
(y + 1)κ = (m+ 1)γ(n+ 1)κ ≤ (m+ 1)γ+κ.
Therefore, by (6.34),
γ,κ ≥ 0⇒ max
0≤x≤n
f(x) ≤ (m+ 1)γ(n+ 1)κ. (6.37)
Gathering together (6.28), (6.30)–(6.33) and (6.35)–(6.37), we get (6.23).
6.2.3 Third list of conditions
For all ε > 0,
n∑
p=0
(m− p+ 1)γ =
m+1∑
q=m−n+1
qγ ≤
∫ m+2
m−n+1
xγdx if γ ≥ 0,
(m− n+ 1)γ +
∫ m+1
m−n+1
xγdx if γ < 0
4
(m+ 1)γ+1 if γ > −1,
1 + ln(m+ 1) if γ = −1,
(m− n+ 1)γ+1 if γ < −1
4
(m+ 1)γ+1 if γ > −1,
(m+ 1)ε if γ = −1,
(m− n+ 1)γ+1 if γ < −1.
(6.38)
The following gives a better estimate when γ ≥ 0, and an alternative estimate when γ < 0:
n∑
p=0
(m− p+ 1)γ ≤
{
(m+ 1)γ(n+ 1) if γ ≥ 0,
(m− n+ 1)γ(n+ 1) if γ < 0.
(6.39)
The estimate (6.39) is better than (6.38) when γ ≥ 0 since m ≥ n, and (6.39) may be better or
worse than (6.38) when γ < 0, depending on the values of m and n. Note that estimates of the
type (6.39) for
n∑
p=0
(p+1)κ and
n∑
p=0
(n−p+1)δ are worse than (6.10) and (6.22). Thus it makes no
sense to add this kind of estimate in Sections 6.2.1 and 6.2.2. On the other hand, we claim that
(n− p+ 1)δ(p+ 1)κ ≤
(n+ 1)κ if δ ≤ κ, 0,
(n+ 1)δ+κ if δ,κ ≥ 0,
(n+ 1)δ if κ ≤ δ, 0
(6.40)
for all p = 0, . . . , n. Combining (6.38)–(6.40), it follows that, for all ε > 0,
(m+ 1)α(n+ 1)β
n∑
p=0
(m− p+ 1)γ(n− p+ 1)δ(p+ 1)κ 4 A3, (6.41)
34 J.A. Álvarez López, M. Calaza and C. Franco
where A3 = A3(m,n, α, β, γ, δ,κ, ε) can be taken to be equal to
(m+ 1)α+ε(n+ 1)β+δ+κ and
(m+ 1)α(n+ 1)β+δ+κ+1(m− n+ 1)γ
}
if γ = −1, 0 ≤ δ,κ,
(m+ 1)α(n+ 1)β+δ+κ(m− n+ 1)γ+1 and
(m+ 1)α(n+ 1)β+δ+κ+1(m− n+ 1)γ
}
if γ < −1, 0 ≤ δ,κ.
By (6.41), applying Lemma 3.14 to the above list, we get the third list of conditions that guar-
antee (6.9) when m ≥ n:
γ ≤ −1, 0 ≤ δ,κ ⇒
{
α+ γ + 1, α+ β + δ + κ < 0, or
α+ γ, α+ β + δ + κ + 1 < 0.
(6.42)
To prove (6.40), it is enough to study the maximum of the C∞ function
f(x) = (n− x+ 1)δ(x+ 1)κ
on [0, n] (the natural domain of f contains (−1, n+ 1)). We have
f ′(x) = (n− x+ 1)δ−1(x+ 1)κ−1h(x),
where
h(x) = −(δ + κ)x+ κ(n+ 1)− δ.
Observe that this expression is valid even when δ = 0 or κ = 0. Since f ′ and h have the same
zero set on [0, n], and they have the same sign on the complement of the zero set in [0, n], it is
enough to analyze h to know where f reaches its maximum on [0, n]. We consider several cases.
Case where δ + κ = 0. Then h ≡ κ(n + 2). If κ 6= 0, then h 6= 0 and signh = signκ. If
κ = 0, then h ≡ 0. Hence:
max
0≤x≤n
f(x) =
{
f(n) = (n+ 1)κ if κ = −δ ≥ 0,
f(0) = (n+ 1)δ if κ = −δ ≤ 0.
(6.43)
Case where δ + κ 6= 0. Then h vanishes just at the point
x0 :=
κ(n+ 1)− δ
δ + κ
.
Case where δ + κ > 0. We have h > 0 on (−∞, x0) and h < 0 on (x0,∞), yielding
max
0≤x≤n
f(x) =
f(0) = (n+ 1)δ if x0 ≤ 0,
f(x0) if 0 ≤ x0 ≤ n,
f(n) = (n+ 1)κ if x0 ≥ n.
(6.44)
Case where δ + κ > 0 and δ,κ ≥ 0. We may have x0 ≤ 0, 0 ≤ x0 ≤ n or n ≤ x0. Moreover
f(x0) =
δδκκ
(δ + κ)δ+κ (n+ 2)δ+κ 4 (n+ 1)δ+κ.
But, in this case,
max
0≤x≤n
f(x) ≤ max
0≤x≤n
(n− x+ 1)δ max
0≤y≤n
(y + 1)κ = (n+ 1)δ+κ.
Therefore, by (6.44),
δ,κ ≥ 0⇒ max
0≤x≤n
f(x) ≤ (n+ 1)δ+κ. (6.45)
Gathering together (6.43) and (6.45), we get the second case of (6.40). The other cases will
not be used, and they follow with similar arguments.
A Perturbation of the Dunkl Harmonic Oscillator on the Line 35
6.2.4 Fourth list of conditions
We have
(p+ 1)κ ≤
{
(n+ 1)κ if κ ≥ 0,
1 if κ ≤ 0
(6.46)
for p = 0, . . . , n. Moreover, by (6.10), (6.38) and (6.39), for all ε > 0,
n∑
p=0
(n− p+ 1)2δ =
n+1∑
q=1
q2δ 4
(n+ 1)2δ+1 if δ > −1
2 ,
(n+ 1)ε if δ = −1
2 ,
1 if δ < −1
2 ,
(6.47)
n∑
p=0
(m− p+ 1)2γ 4
(m+ 1)2γ+1 if γ > −1
2 ,
(m+ 1)ε if γ = −1
2 ,
(m− n+ 1)2γ+1 if γ < −1
2 ,
(6.48)
n∑
p=0
(m− p+ 1)2γ ≤
{
(m+ 1)2γ(n+ 1) if γ ≥ 0,
(m− n+ 1)2γ(n+ 1) if γ < 0.
(6.49)
The estimate (6.49) is better than (6.48) when γ ≥ 0, and it may be better or worse than (6.48)
when γ < 0, depending on the values of m and n. By the Cauchy–Schwartz inequality,
n∑
p=0
(m− p+ 1)γ(n− p+ 1)δ ≤
n∑
p=0
(m− p+ 1)2γ
1
2
n∑
p=0
(n− p+ 1)2δ
1
2
.
Therefore, by (6.46)–(6.49),
(m+ 1)α(n+ 1)β
n∑
p=0
(m− p+ 1)γ(n− p+ 1)δ(p+ 1)κ 4 A4, (6.50)
where A4 = A4(m,n, α, β, γ, δ,κ) can be taken to be equal to
(m+ 1)α(n+ 1)β+δ+κ+ 1
2 (m− n+ 1)γ+ 1
2 and
(m+ 1)α(n+ 1)β+δ+κ+1(m− n+ 1)γ
}
if γ < −1
2 < δ, 0 ≤ κ.
By (6.50), applying Lemma 3.14 to the above list, we get the fourth list of conditions that
guarantee (6.9) when m ≥ n:
γ < −1
2 < δ, 0 ≤ κ ⇒
{
α+ γ + 1
2 , α+ β + δ + κ + 1
2 < 0, or
α+ γ, α+ β + δ + κ + 1 < 0.
(6.51)
6.2.5 Fifth list of conditions
This is analogous to the estimates of Section 6.2.4, interchanging the roles of δ and κ. We have
(n− p+ 1)δ ≤
{
(n+ 1)δ if δ ≥ 0,
1 if δ ≤ 0
(6.52)
for p = 0, . . . , n. Moreover, by (6.10), for all ε > 0,
n∑
p=0
(p+ 1)2κ 4
(n+ 1)2κ+1 if κ > −1
2 ,
(n+ 1)ε if κ = −1
2 ,
1 if κ < −1
2 .
(6.53)
36 J.A. Álvarez López, M. Calaza and C. Franco
Applying the Cauchy–Schwartz inequality, we get
n∑
p=0
(m− p+ 1)γ(p+ 1)κ ≤
n∑
p=0
(m− p+ 1)2γ
1
2
n∑
p=0
(p+ 1)2κ
1
2
.
Therefore, by (6.52), (6.53), (6.48) and (6.49), for all ε > 0,
(m+ 1)α(n+ 1)β
n∑
p=0
(m− p+ 1)γ(n− p+ 1)δ(p+ 1)κ 4 A5, (6.54)
where A5 = A5(m,n, α, β, γ, δ,κ, ε) can be taken to be equal to
(m+ 1)α+ε(n+ 1)β+δ+ε and
(m+ 1)α(n+ 1)β+δ+ 1
2
+ε(m− n+ 1)γ
}
if γ = κ = −1
2 , 0 ≤ δ,
(m+ 1)α+ε(n+ 1)β+δ and
(m+ 1)α(n+ 1)β+δ+ 1
2 (m− n+ 1)γ
}
if κ < γ = −1
2 , 0 ≤ δ,
(m+ 1)α(n+ 1)β+δ+ε(m− n+ 1)γ+ 1
2 and
(m+ 1)α(n+ 1)β+δ+ 1
2
+ε(m− n+ 1)γ
}
if γ < κ = −1
2 , 0 ≤ δ,
(m+ 1)α(n+ 1)β+δ(m− n+ 1)γ+ 1
2 and
(m+ 1)α(n+ 1)β+δ+ 1
2 (m− n+ 1)γ
}
if γ,κ < −1
2 , 0 ≤ δ.
By (6.54), applying Lemma 3.14 to the above list, we get the fifth list of conditions that guar-
antee (6.9) when m ≥ n:
γ,κ ≤ −1
2 , 0 ≤ δ ⇒
{
α+ γ + 1
2 , α+ β + δ < 0, or
α+ γ, α+ β + δ + 1
2 < 0.
(6.55)
6.2.6 Sixth list of conditions
We have
(m− p+ 1)γ ≤
{
(m+ 1)γ if γ ≥ 0,
(m− n+ 1)γ if γ ≤ 0
(6.56)
for p = 0, . . . , n. Moreover, by (6.10), for all ε > 0,
n∑
p=0
(n− p+ 1)2δ =
n+1∑
q=1
q2δ 4
(n+ 1)2δ+1 if δ > −1
2 ,
(n+ 1)ε if δ = −1
2 ,
1 if δ < −1
2 .
(6.57)
Applying the Cauchy–Schwartz inequality, we get
n∑
p=0
(n− p+ 1)δ(p+ 1)κ ≤
n∑
p=0
(n− p+ 1)2δ
1
2
n∑
p=0
(p+ 1)2κ
1
2
.
Therefore, by (6.53), (6.56) and (6.57), for all ε > 0,
(m+ 1)α(n+ 1)β
n∑
p=0
(m− p+ 1)γ(n− p+ 1)δ(p+ 1)κ 4 A6, (6.58)
A Perturbation of the Dunkl Harmonic Oscillator on the Line 37
where A6 = A6(m,n, α, β, γ, δ,κ, ε) can be taken to be equal to
(m+ 1)α+γ(n+ 1)β+ε if
{
κ ≤ δ = −1
2 , 0 ≤ γ, or
δ ≤ κ = −1
2 , 0 ≤ γ,
(m+ 1)α+γ(n+ 1)β if δ,κ < −1
2 , 0 ≤ γ.
By (6.58), applying Lemma 3.14 to the above list, we get the sixth list of conditions that
guarantee (6.9) when m ≥ n:
δ,κ ≤ −1
2 , 0 ≤ γ ⇒ α+ γ, α+ β + γ < 0. (6.59)
6.2.7 Seventh list of conditions
By (6.10), (6.38) and (6.39), for all ε > 0,
n∑
p=0
(n− p+ 1)3δ =
n+1∑
q=1
q3δ 4
(n+ 1)3δ+1 if δ > −1
3 ,
(n+ 1)ε if δ = −1
3 ,
1 if δ < −1
3 ,
(6.60)
n∑
p=0
(p+ 1)3κ 4
(n+ 1)3κ+1 if κ > −1
3 ,
(n+ 1)ε if κ = −1
3 ,
1 if κ < −1
3 ,
(6.61)
n∑
p=0
(m− p+ 1)3γ 4
(m+ 1)3γ+1 if γ > −1
3 ,
(m+ 1)ε if γ = −1
3 ,
(m− n+ 1)3γ+1 if γ < −1
3 ,
(6.62)
n∑
p=0
(m− p+ 1)3γ ≤
{
(m+ 1)3γ(n+ 1) if γ ≥ 0,
(m− n+ 1)3γ(n+ 1) if γ < 0.
(6.63)
Note that (6.63) is better than (6.62) for γ ≥ 0, and it is an alternative estimate for γ < 0.
Applying the generalized Hölder inequality [7], we get
n∑
p=0
(m− p+ 1)γ(n− p+ 1)δ(p+ 1)κ
≤
n∑
p=0
(m− p+ 1)3γ
1
3
n∑
p=0
(n− p+ 1)3δ
1
3
n∑
p=0
(p+ 1)3κ
1
3
.
Therefore, by (6.60)–(6.63),
(m+ 1)α(n+ 1)β
n∑
p=0
(m− p+ 1)γ(n− p+ 1)δ(p+ 1)κ 4 A7, (6.64)
where A7 = A7(m,n, α, β, γ, δ,κ) can be taken to be equal to
(m+ 1)α(n+ 1)β+δ+κ+ 2
3 (m− n+ 1)γ+ 1
3 and
(m+ 1)α(n+ 1)β+δ+κ+1(m− n+ 1)γ
}
if γ < −1
3 < δ,κ,
(m+ 1)α(n+ 1)β+δ+ 1
3 (m− n+ 1)γ+ 1
3 and
(m+ 1)α(n+ 1)β+δ+ 2
3 (m− n+ 1)γ
}
if γ,κ < −1
3 < δ.
38 J.A. Álvarez López, M. Calaza and C. Franco
By (6.64), applying Lemma 3.14 to the above list, we get the seventh list of conditions that
guarantee (6.9) when m ≥ n:
γ < −1
3 < δ,κ ⇒
{
α+ γ + 1
3 , α+ β + δ + κ + 2
3 < 0, or
α+ γ, α+ β + δ + κ + 1 < 0,
(6.65)
γ,κ < −1
3 < δ ⇒
{
α+ γ + 1
3 , α+ β + δ + 1
3 < 0, or
α+ γ, α+ β + δ + 2
3 < 0.
(6.66)
6.2.8 Obtaining the sets Sijk from the lists of conditions
The left hand side of the conditions from the lists of Sections 6.2.1–6.2.7 involve only (γ, δ,κ).
Now, we indicate which of them define sets covering every Qijk, for the chosen subindices ijk
equal to 515, 522, 252, 155, 212. Those conditions will produce the definition of Sijk ⊂ R2×Qijk
so that (6.9) holds for m ≥ n.
The set Q515 is given by the left hand side of (6.25), whose right hand side is (6.1), defi-
ning S515.
Any (γ, δ,κ) ∈ Q522 satisfies the left hand side of (6.59), whose right hand side is (6.2),
defining S522.
Any (γ, δ,κ) ∈ Q252 satisfies the left hand side of (6.13) or (6.14), and satisfies the left
hand side of (6.55) and (6.66). So, when m ≥ n, the estimate (6.9) is guaranteed for any
(α, β, γ, δ,κ) ∈ R2 × Q252 satisfying both (6.13) and (6.14), or any of (6.55) or (6.66). On
R2 ×Q252, these conditions mean that (α, β, γ, δ,κ) satisfies (6.3) and (6.4), defining S252.
Any (γ, δ,κ) ∈ Q155 satisfies the left hand side of (6.13) or (6.14), and satisfies the left hand
side of (6.51), (6.42) and (6.65). So, when m ≥ n, the estimate (6.9) is guaranteed for any
(α, β, γ, δ,κ) ∈ R2 × Q155 satisfying both (6.13) and (6.14), or any of (6.51), (6.42) or (6.65).
On R2 ×Q155, these conditions mean that (α, β, γ, δ,κ) satisfies (6.3) and (6.5), defining S155.
Any (γ, δ,κ) ∈ Q212 satisfies the left hand side of (6.26) or (6.27). So, when m ≥ n, the
estimate (6.9) is guaranteed for any (α, β, γ, δ,κ) ∈ R2×Q212 satisfying both (6.26) and (6.27).
On R2 ×Q212, these conditions become (6.6) and (6.7), defining S212.
6.2.9 The preliminary estimate is satisf ied on a convex set
Let us show the convexity of the set of elements x = (α, β, γ, δ,κ) ∈ R5 satisfying (6.9) for
m ≥ n, with ω = ω(x) > 0. For i = 0, 1, suppose that xi = (αi, βi, γi, δi,κi) satisfies (6.9) for
m ≥ n with ωi = ω(xi) > 0. Recall that the case where m ≤ n follows from the case where
m ≥ n by using the mapping (6.8).
For 0 < t < 1, let
xt = (αt, βt, γt, δt,κt) = (1− t)x0 + tx1, ωt = (1− t)ω0 + tω1 > 0.
By Hölder inequality, for all m ≥ n,
n∑
p=0
(m− p+ 1)γt(n− p+ 1)δt(p+ 1)κt
=
n∑
p=0
(
(m− p+ 1)γ0(n− p+ 1)δ0(p+ 1)κ0
)1−t(
(m− p+ 1)γ1(n− p+ 1)δ1(p+ 1)κ1
)t
≤
n∑
p=0
(m− p+ 1)γ0(n− p+ 1)δ0(p+ 1)κ0
1−t
A Perturbation of the Dunkl Harmonic Oscillator on the Line 39
×
n∑
p=0
(m− p+ 1)γ1(n− p+ 1)δ1(p+ 1)κ1
t
.
So
(m+ 1)αt(n+ 1)βt
n∑
p=0
(m− p+ 1)γt(n− p+ 1)δt(p+ 1)κt
4
(
(m+ 1)−ω0(n+ 1)−ω0
)1−t(
(m+ 1)−ω1(n+ 1)−ω1
)t
= (m+ 1)−ωt(n+ 1)−ωt .
Thus xt satisfies (6.9) for m ≥ n with ωt. This completes the proof of Lemma 6.1.
7 The main estimates
Here, we show the estimates used in the proofs of Propositions 5.6 and 5.10. We continue with
the notation of Section 6. Moreover let (σ, τ, θ) denote the standard coordinates of R3. Consider
the affine injection R3 → R5 and the affine isomorphism of R3 defined by
(σ, τ, θ) 7→
(
1
4 −
σ
2 ,−
1
4 −
τ
2 , σ − θ − 1, τ − θ, θ − 1
2
)
, (7.1)
(σ, τ, θ) 7→ (τ + 1, σ − 1, θ). (7.2)
The mapping (7.2) is the reflection with respect to the plane defined by σ = τ + 1, and it
corresponds to the mapping (6.8) via (7.1). Let Ǩ, Ǩ′ ⊂ R3 be the inverse images of Š, Š′
by (7.1), and let Ǩconv, Ǩ′conv be their convex hulls. So Ǩconv, Ǩ′conv are contained in the inverse
images of Šconv, Š′conv by (7.1), and Ǩ′, Ǩ′conv are the images of Ǩ, Ǩconv by (7.2). Thus
Ǩconv ∩ Ǩ′conv is symmetric with respect to the plane σ = τ + 1. We will show the following.
Lemma 7.1. Ǩconv ∩ Ǩ′conv consists of the elements (σ, τ, θ) ∈ R3 that satisfy (1.7).
The following is a direct consequence of Lemmas 6.1 and 7.1.
Corollary 7.2. If (σ, τ, θ) ∈ R3 satisfies (1.7), then there is some ω > 0 such that (6.9) holds
with the image (α, β, γ, δ,κ) of (σ, τ, θ) by (7.1).
Let J̌ ⊂ R2 be the inverse image of Ǩconv ∩ Ǩ′conv by the affine injection R2 → R3, (σ, τ) 7→
(σ, τ, τ). The following is a direct consequence of Lemma 7.1.
Lemma 7.3. J̌ consists of the elements (σ, τ) ∈ R2 that satisfy (1.5).
Lemma 7.3 and Corollary 7.2 have the following direct consequence.
Corollary 7.4. If (σ, τ) ∈ R2 satisfies (1.5), then there is some ω > 0 such that (6.9) holds
with the image (α, β, γ, δ,κ) of (σ, τ, τ) by (7.1).
Let us prove Lemma 7.1. For the subindices ijk equal to 515, 522, 252, 155, 212, let Kijk and
Rijk be the inverse images of Sijk and R2×Qijk by the mapping (7.1). Thus Ǩ = K515 ∪K522 ∪
K252 ∪ K155 ∪ K212. Moreover, for every θ ∈ R, let
I1
1 (θ) = (−∞, θ], I2
1 (θ) = (−∞, θ − 1], I3
1 =
(
−∞,−1
2
]
,
I1
2 (θ) =
(
θ, θ + 1
2
]
, I2
2 (θ) =
(
θ − 1, θ − 1
2
]
, I3
2 =
(
−1
2 , 0
]
,
I1
3 (θ) =
(
θ + 1
2 , θ + 2
3
]
, I2
3 (θ) =
(
θ − 1
2 , θ −
1
3
]
, I3
3 =
(
0, 1
6
]
,
I1
4 (θ) =
(
θ + 2
3 , θ + 1
)
, I2
4 (θ) =
(
θ − 1
3 , θ
)
, I3
4 =
(
1
6 ,
1
2
)
,
40 J.A. Álvarez López, M. Calaza and C. Franco
I1
5 (θ) = [θ + 1,∞), I2
5 (θ) = [θ,∞), I3
5 =
[
1
2 ,∞
)
.
It can be directly checked that
Rijk =
{
(σ, τ, θ) ∈ R3 | (σ, τ) ∈ I1
i (θ)× I2
j (θ), θ ∈ I3
k
}
.
Simple computations show that, via (7.1), the conditions defining the sets Sijk (Section 6.1)
become the following descriptions of the sets Kijk (Fig. 3(a)):
K515: This is the subset of R515 defined by
σ
2 −
3
4 < θ, σ − τ − 3 < 0.
K522: This is the subset of R522 defined by
σ
2 −
3
4 ,
σ−τ
2 − 1 < θ.
K252: This is the subset of R252 defined by
σ+τ−1
2 < θ, (7.3)
σ
2 −
1
4 ,
τ−σ
2 < θ, or
σ
2 −
5
12 ,
τ−σ
2 + 1
3 < θ, or
σ
2 −
3
4 < θ, τ − σ + 1 < 0.
K155: This is the subset of R155 defined by (7.3) and
σ
2 + 1
4 < θ, τ − σ − 1 < 0, or
σ
2 −
1
4 < θ, τ − σ < 0, or
σ
2 −
5
12 < θ, τ − σ + 1
3 < 0, or
σ
2 −
3
4 < θ, τ − σ + 1 < 0.
K212: This is the subset of R212 defined by
σ
2 −
1
4 ≤ θ ⇒ θ < σ+τ+1
2 ,
θ ≤ σ
2 −
1
4 ⇒
σ
2 −
3
4 ,
σ−τ
2 − 1 < θ.
With tedious computations assisted by graphics produced with Mathematica, it follows that
Ǩconv is the open subset of R3 defined by (Fig. 3(b))
σ−τ
2 − 1, τ−σ2 , σ+τ−1
4 , σ+3τ−2
14 , σ+τ−1
2 < θ < σ+τ+1
2 , τ − 1 < σ < τ + 3. (7.4)
This is a “semi-infinite bar” with 4 lateral faces, and 4 faces at the “bounded end”.
Applying the affine transformation (7.2) to this description, we get that Ǩ′conv consists of the
triples (σ, τ, θ) ∈ R3 satisfying the following conditions:
σ−τ
2 − 1, τ−σ2 , σ+τ−1
4 , 3σ+τ−4
14 , σ+τ−1
2 < θ < σ+τ+1
2 , τ − 1 < σ < τ + 3. (7.5)
Combining (7.4) and (7.5), it follows that Ǩconv ∩ Ǩ′conv is given by (1.7) (Fig. 2), completing
the proof of Lemma 7.1.
A Perturbation of the Dunkl Harmonic Oscillator on the Line 41
(a) Ǩ (b) Ǩconv
Figure 3. The sets Ǩ and Ǩconv.
Remark 7.5. In Sections 6.2.1–6.2.7, we have only written the cases that provide the most
general conditions to define S515,S522,S252,S155,S212. But indeed much more hidden work
was needed to produce this shorter proof:
• We have computed all cases in Sections 6.2.1–6.2.7, giving rise to seven long lists of con-
ditions that guarantee (6.9) when m ≥ n.
• We have studied which of those conditions are the most general ones on every R2 ×Qijk,
for all ijk = 1, . . . , 5. This produces 125 sets Sijk, whose inverse images by (7.1) give
125 sets Kijk. The corresponding unions are denoted by S and K, and their convex hulls
by Sconv and Kconv.
• We got that 41 sets Kijk are empty, including the 25 sets of the form Kij1, and the remaining
84 sets Kijk fit together forming a “semi-infinite bar” (Fig. 4).
• With tedious computations, we have shown that Kconv is given by (7.4).
• We have chosen the most simple family, K515, K522, K252, K155, K212, defining the same
convex hull (Ǩconv = Kconv).
• Finally, we have made some attempts to improve the estimates of Section 6.2.7 by using
more general versions of the Hölder inequality [7]. Some better estimates were obtained
in this way, but they produce the same set Kconv after taking the convex hull.
Remark 7.6. The set Sconv may have a simple expression, like Kconv, but its computation
became too involved. This is the reason we have used Kconv, obtaining the conditions of Theo-
rem 1.3, which are general enough for our applications in [4]. But, of course, the inverse image
of Sconv by (7.1) is possibly larger than Kconv. Therefore a simple expression of Sconv would
possibly give a better version of Theorem 1.3. Even a simple expression of Šconv would possibly
give a better version of Theorem 1.3.
8 Operators induced on R+
Let Sev/odd,+ = {φ|R+ |φ ∈ Sev/odd}. For c, d > −1
2 , let L2
c,+ = L2(R+, x
2cdx) and L2
c,d,+ =
L2
c,+ ⊕ L2
d,+, whose scalar products are denoted by 〈 , 〉c and 〈 , 〉c,d, respectively. For
c1, c2, d1, d2 ∈ R, let
P0 = H − 2c1x
−1 d
dx
+ c2x
−2, Q0 = H − 2d1
d
dx
x−1 + d2x
−2.
Morever let ξ > 0 and η, θ ∈ R.
42 J.A. Álvarez López, M. Calaza and C. Franco
(a)
⋃
i,j Kij2 (b)
⋃
i,j Kij3 (c)
⋃
i,j Kij4
(d)
⋃
i,j Kij5 (e) K
Figure 4. Construction of K.
Corollary 8.1. If a2 + (2c1 − 1)a − c2 = 0, 0 < u < 1 and σ := a + c1 > u − 1
2 , then there is
a positive self-adjoint operator P in L2
c1,+ satisfying the following:
(i) xaSev,+ is a core of P1/2 and, for all φ, ψ ∈ xaSev,+,〈
P1/2φ,P1/2ψ
〉
c1
= 〈P0φ, ψ〉c1 + ξ
〈
x−uφ, x−uψ
〉
c1
.
(ii) P has a discrete spectrum. Let λ0 ≤ λ2 ≤ · · · be its eigenvalues, repeated according to
their multiplicity. There is some D = D(σ, u) > 0, and, for each ε > 0, there is some
C = C(ε, σ, u) > 0 so that (1.2) holds for all k ∈ 2N.
Corollary 8.2. If b2 + (2d1 + 1)b − d2 = 0, 0 < u < 1 and τ := b + d1 > u − 3
2 , then there is
a positive self-adjoint operator Q in L2
d1,+
satisfying the following:
(i) xbSodd,+ is a core of Q1/2 and, for all φ, ψ ∈ xbSodd,+,〈
Q1/2φ,Q1/2ψ
〉
d1
= 〈Q0φ, ψ〉d1 + ξ
〈
x−uφ, x−uψ
〉
d1
.
(ii) Q has a discrete spectrum. Let λ1 ≤ λ3 ≤ · · · be its eigenvalues, repeated according to
their multiplicity. There is some D = D(τ, u) > 0, and, for each ε > 0, there is some
C = C(ε, τ, u) > 0 so that (1.2) holds for all k ∈ 2N + 1, with τ instead of σ.
A Perturbation of the Dunkl Harmonic Oscillator on the Line 43
Corollary 8.3. Under the conditions of Corollaries 8.1 and 8.2, if moreover the conditions of
Theorem 1.3 are satisfied with some θ > −1
2 , then there is a positive self-adjoint operator W in
L2
c1,d1,+
satisfying the following:
(i) xaSev,+ ⊕ xbSodd,+ is a core of W1/2, and, for φ = (φ1, φ2) and ψ = (ψ1, ψ2) in xaSev,+ ⊕
xbSodd,+,〈
W1/2φ,W1/2ψ
〉
c1,d1
= 〈(P0 ⊕Q0)φ, ψ〉c1,d1 + ξ
〈
x−uφ, x−uψ
〉
c1,d1
+ η
(〈
x−a−b−1φ2, ψ1
〉
θ
+
〈
φ1, x
−a−b−1ψ2
〉
θ
)
. (8.1)
(ii) W has a discrete spectrum. Its eigenvalues form two groups, λ0 ≤ λ2 ≤ · · · and λ1 ≤ λ3 ≤
· · · , repeated according to their multiplicity, such that there is some D = D(σ, τ, u) > 0
and, for every ε > 0, there are some C = C(ε, σ, τ, u) > 0 and E = E(ε, σ, τ, θ) > 0 so
that (1.9) and (1.10) hold for all k ∈ N.
(iii) If ũ ∈ R satisf ies (1.11), then there is some D = D(σ, τ, u) > 0 and, for any ε > 0, there
is some C̃ = C̃(ε, σ, τ, u) > 0 so that (1.12) holds for all k ∈ N.
(iv) If u = v+1
2 and ξ ≥ |η|, then there is some D̃ = D̃(σ, τ, u) > 0 so that (1.13) holds for all
k ∈ N.
(v) If we add the term ξ′〈φ1, ψ1〉c1 + ξ′′〈φ2, ψ2〉d1 to the right hand side of (8.1), for some
ξ′, ξ′′ ∈ R, then the result holds as well with the additional term max{ξ′, ξ′′} in the right
hand side of (1.10), and the additional term, ξ′ for k ∈ 2N and ξ′′ for k ∈ 2N + 1, in the
right hand sides of (1.9), (1.12) and (1.13).
These corollaries follow directly from Theorems 1.1 and 1.3 because the given conditions on a
and b characterize the cases where P0 and Q0 correspond to |x|aUσ,ev|x|−a and |x|bUτ,odd|x|−b,
respectively, via the isomorphisms |x|aSev → xaSev,+ and |x|bSodd → xbSodd,+ defined by re-
striction [3, Theorem 1.4 and Section 5]. In fact, Corollaries 8.1 and 8.2 are equivalent because,
if c1 = d1 + 1 and c2 = d2, then Q0 = xP0x
−1 and x : L2
c1,+ → L2
d1,+
is a unitary operator.
Remarks 1.2(ii) and 3.21 have obvious versions for these corollaries. In particular, P = P ,
Q = Q and W = W , where P = P0 + ξx−2u, Q = Q0 + ξx−2u and
W =
(
P ηx2(θ−σ)+a−b−1
ηx2(θ−τ)+b−a−1 Q
)
=
(
P ηx2(θ−c1)−a−b−1
ηx2(θ−d1)−a−b−1 Q
)
,
with D(P ) =
⋂∞
m=0 D(Pm), D(Q) =
⋂∞
m=0 D(Qm) and D(W ) =
⋂∞
m=0 D(Wm). According to
Remark 1.4(ii), we can write (8.1) as〈
W1/2φ,W1/2ψ
〉
c1,d1
= 〈(P0 ⊕Q0)φ, ψ〉c1,d1 + ξ
〈
x−uφ, x−uψ
〉
c1,d1
+ η
(〈
x−a−b+1φ2, ψ1
〉
θ′
+
〈
φ1, x
−a−b+1ψ2
〉
θ′
)
,
and we have
W =
(
P ηx2(θ′−c1)−a−b+1
ηx2(θ′−d1)−a−b+1 Q
)
.
9 Application to the Witten’s perturbation on strata
Let M be a Riemannian n-manifold. Let d, δ and ∆ denote the de Rham derivative and
coderivative, and the Laplacian, with domain the graded space Ω0(M) of compactly supported
differential forms, and let L2Ω(M) be the graded Hilbert space of square integrable differential
44 J.A. Álvarez López, M. Calaza and C. Franco
forms. Any closed extension d of d in L2Ω(M), defining a complex (d2 = 0), is called an ideal
boundary condition (i.b.c.) of d, which defines a self-adjoint extension ∆ = d∗d+dd∗ of ∆, called
the Laplacian of d. There always exists a minimum/maximum i.b.c., dmin = d and dmax = δ∗,
whose Laplacians are denoted by ∆min /max. We get corresponding cohomologies Hmin /max(M),
and versions of Betti numbers and Euler characteristic, βimin /max and χmin /max. These are quasi-
isometric invariants; in particular, Hmax(M) is the usual L2 cohomology. If M is complete, then
there is a unique i.b.c., but these concepts become interesting in the non-complete case. For
instance, if M is the interior of a compact manifold with non-empty boundary, then dmin /max
is defined by taking relative/absolute boundary conditions. Given s > 0 and f ∈ C∞(M), the
above ideas can be considered as well for the Witten’s perturbations ds = e−sfdesf = d+ sdf∧,
with formal adjoint δs = esfδe−sf = δ − sdfy and Laplacian ∆s. In fact, this theory can be
considered for any elliptic complex.
On the other hand, let us give a rough idea of the concept of stratified space. It is a Hausdorff,
locally compact and second countable space A with a partition into C∞ manifolds (strata)
satisfying certain conditions. An order on the family of strata is defined so that X ≤ Y means
that X ⊂ Y . With this order relation, the maximum length of chains of strata is called the
depth of A. Then we continue describing A by induction on depthA, as well as its group Aut(A)
of automorphisms. If depthA = 0, then A is just a C∞ manifold, whose automorphisms are its
diffeomorphisms. Now, assume that depthA > 0, and the descriptions are given for lower depth.
Then it is required that each stratum X has an open neighborhood T (a tube) that is a fiber
bundle whose typical fiber is a cone c(L) = (L×[0,∞))/(L×{0}) and structural group c(Aut(L)),
where L is a compact stratification of lower depth (the link of X), and c(Aut(L)) consists of
the homeomorphisms c(φ) of c(L) induced by the maps φ× id on L× [0,∞) (φ ∈ Aut(L)). The
point ∗ = L× {0} ∈ c(L) is called the vertex. An automorphism of A is a homeomorphism that
restricts to diffeomorphisms between the strata, and whose restrictions to their tubes are fiber
bundle homomorphisms. This completes the description because the depth is locally finite by
the local compactness.
The local trivializations of the tubes can be considered as “stratification charts”, giving
a local description of the form Rm × c(L). Via these charts, a stratum M of A corresponds,
either to Rm×{∗} ≡ Rm, or to Rm×N ×R+ for some stratum N of L. The concept of general
adapted metric on M is defined by induction on the depth. It is any Riemannian metric in the
case of depth zero. For positive depth, a Riemannian metric g on M is called a general adapted
metric if, on each local chart as above, g is quasi-isometric, either to the flat Euclidean metric g0
if M corresponds to Rm, or to g0 + x2ug̃ + (dx)2 if M corresponds to Rm × N × R+, where g̃
is a general adapted metric on N , x is the canonical coordinate of R+, and u > 0 depends
on M and each stratum X < M , whose tube is considered to define the chart. This assignment
X 7→ u is called the type of the metric. We omit the term “general” when we take u = 1 for all
strata.
Assuming that A is compact, it is proved in [4] that, for any general adapted metric g on
a stratum M of A with numbers u ≤ 1, the Laplacian ∆min /max has a discrete spectrum, its
eigenvalues satisfy a weak version of the Weyl’s asymptotic formula, and the method of Witten
is extended to get Morse inequalities involving the numbers βimin /max and another numbers
νimin /max defined by the local data around the “critical points” of a version of Morse functions
on M ; here, the “critical points” live in the metric completion of M . This is specially important
in the case of a stratified pseudo-manifold A with regular stratum M , where Hmax(M) is the
intersection homology with perversity depending on the type of the metric [17, 18]. Again, we
proceed by induction on the depth to prove these assertions. In the case of depth zero, these
properties hold because we are in the case of closed manifolds. Now, assume that the depth is
positive, and these properties hold for lower depth. Via a globalization procedure and a version
of the Künneth formula, the computations boil down to the case of the Witten’s perturbation ds
A Perturbation of the Dunkl Harmonic Oscillator on the Line 45
Table 1. Self-adjoint extensions of ∆s,r−1 and ∆s,r+1.
a σ condition b τ condition
0 κ+ u κ > −1
2 0 κ κ > u− 3
2
1− 2(κ+ u) 1− κ− u κ < 3
2 − 2u 1− 2κ −1− κ κ < 1
2 − u
Table 2. Self-adjoint extensions of ∆s,r.
a b σ τ θ Condition
0 0 κ κ+ u κ κ > u− 1
2
1− 2κ −1− 2(κ+ u) 1− κ −1− κ− u −κ− u κ < 1
2 − 2u
0 −1− 2(κ+ u) κ −1− κ− u −1
2 − u Impossible
1− 2κ 0 1− κ κ+ u 1
2 −1− u
2 < κ < 1− u
2
for a stratum M = N × (0,∞) of a cone c(L) with an adapted metric g = x2ug̃ + (dx)2, where
we consider the “Morse function” f = ±x2/2.
Let d̃min /max, δ̃min /max and ∆̃min /max denote the operators defined as above for N with g̃.
Take differential forms 0 6= γ ∈ ker ∆̃min /max, of degree r, and 0 6= α, β ∈ D(∆̃min /max), of
degrees r and r−1, with d̃min /maxβ = µα and δ̃min /maxα = µβ for some µ > 0. Since ∆̃min /max
is assumed to have a discrete spectrum, L2Ω(N) has a complete orthonormal system consisting
of forms of these types. Correspondingly, there is a “direct sum splitting” of ds into the following
two types of subcomplexes:
C∞0 (R+) γ
ds,r−−−−→ C∞0 (R+) γ ∧ dx,
C∞0 (R+)β
ds,r−1−−−−→ C∞0 (R+)α+ C∞0 (R+)β ∧ dx ds,r−−−−→ C∞0 (R+)α ∧ dx.
Forgetting the differential form part, they can be considered as two types of simple elliptic
complexes of lengths one and two,
C∞0 (R+)
ds,r−−−−→ C∞0 (R+),
C∞0 (R+)
ds,r−1−−−−→ C∞0 (R+)⊕ C∞0 (R+)
ds,r−−−−→ C∞0 (R+).
Let κ = (n− 2r − 1)u2 . In the complex of length one, ds,r is a densely defined operator of L2
κ,+
to L2
κ,+, we have
ds,r =
d
dx
± sx, δs,r = − d
dx
− 2κx−1 ± sx,
and the corresponding components of the Laplacian are
∆s,r = H − 2κx−1 d
dx
∓ s(1 + 2κ), ∆s,r+1 = H − 2κ
d
dx
x−1 ∓ s(−1 + 2κ).
Up to the constant terms, these operators are of the form already considered in [3], without the
term with x−2u, and the spectrum of ∆s,min /max,r and ∆s,min /max,r+1 is well known.
In the complex of length two, ds,r−1 is a densely defined operator of L2
κ+u,+ to L2
κ,+⊕L2
κ+u,+,
ds,r is a densely defined operator of L2
κ,+ ⊕ L2
κ+u,+ to L2
κ,+, we have
ds,r−1 =
(
µ
d
dx ± sx
)
, δs,r−1 =
(
µx−2u − d
dx − 2(κ+ u)x−1 ± sx
)
,
46 J.A. Álvarez López, M. Calaza and C. Franco
ds,r =
(
d
dx ± sx −µ
)
, δs,r =
(
− d
dx − 2κx−1 ± sx
−µx−2u
)
,
and the corresponding components of the Laplacian are
∆s,r−1 = H − 2(κ+ u)x−1 d
dx
+ µ2x−2u ∓ s(1 + 2(κ+ u)),
∆s,r+1 = H − 2κ
d
dx
x−1 + µ2x−2u ∓ s(−1 + 2κ),
∆s,r =
(
A −2µux−1
−2µux−2u−1 B
)
,
where
A = H − 2κx−1 d
dx
+ µ2x−2u ∓ s(1 + 2κ),
B = H − 2(κ+ u)
d
dx
x−1 + µ2x−2u ∓ s(−1 + 2(κ+ u)).
Up to the constant terms, ∆s,r−1 and A are of the form of P , and ∆s,r+1 and B are of the form
of Q, in Section 8. In the case u = 1, these operators were studied in [3]. Thus assume that
u < 1. Then, according to Corollaries 8.1–8.3, we get self-adjoint extensions of ∆s,r−1, ∆s,r+1
and ∆s,r as indicated in Tables 1 and 2, where the conditions are determined by the hypotheses;
indeed most possibilities of the hypothesis are needed. With further analysis [4], the maximum
and minimum Laplacians can be given by appropriate choices of these operators, depending on
the values of κ. Moreover the eigenvalue estimates of these corollaries play a key role in this
research.
If A is a stratified pseudo-manifold, our restrictions on u allow to get enough metrics to
represent all intersection cohomologies of A with perversity less or equal than the lower middle
perversity, according to [17, 18].
Acknowledgements
The first author was partially supported by MICINN, Grants MTM2011-25656 and MTM2014-
56950-P, and by Xunta de Galicia, Grant Consolidación e estructuración 2015 GPC GI-1574.
The third author has received financial support from the Xunta de Galicia and the European
Union (European Social Fund - ESF).
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1 Introduction
2 Preliminaries
3 The sesquilinear form t
4 Scalar products of mixed generalized Hermite functions
4.1 Case where == and –N
4.2 Case where == and -,–N
5 The sesquilinear form t'
5.1 Case where ==
5.2 Case where == and –N
5.3 Case where == and –N
5.4 Case where ==+1 and –1-N
5.5 Case where == and -,–N
5.6 Proof of Theorem 1.3
6 A preliminary estimate
6.1 Statement
6.2 Proof of Lemma 6.1
6.2.1 First list of conditions
6.2.2 Second list of conditions
6.2.3 Third list of conditions
6.2.4 Fourth list of conditions
6.2.5 Fifth list of conditions
6.2.6 Sixth list of conditions
6.2.7 Seventh list of conditions
6.2.8 Obtaining the sets Sijk from the lists of conditions
6.2.9 The preliminary estimate is satisfied on a convex set
7 The main estimates
8 Operators induced on R+
9 Application to the Witten's perturbation on strata
References
|