A Sharp Lieb-Thirring Inequality for Functional Difference Operators
We prove sharp Lieb-Thirring type inequalities for the eigenvalues of a class of one-dimensional functional difference operators associated with mirror curves. We furthermore prove that the bottom of the essential spectrum of these operators is a resonance state.
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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| Дата: | 2021 |
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| Мова: | Англійська |
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Інститут математики НАН України
2021
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| Цитувати: | A Sharp Lieb-Thirring Inequality for Functional Difference Operators. Ari Laptev and Lukas Schimmer. SIGMA 17 (2021), 105, 10 pages |
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| author | Laptev, Ari Schimmer, Lukas |
| author_facet | Laptev, Ari Schimmer, Lukas |
| citation_txt | A Sharp Lieb-Thirring Inequality for Functional Difference Operators. Ari Laptev and Lukas Schimmer. SIGMA 17 (2021), 105, 10 pages |
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| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | We prove sharp Lieb-Thirring type inequalities for the eigenvalues of a class of one-dimensional functional difference operators associated with mirror curves. We furthermore prove that the bottom of the essential spectrum of these operators is a resonance state.
|
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 17 (2021), 105, 10 pages
A Sharp Lieb–Thirring Inequality
for Functional Difference Operators
Ari LAPTEV ab and Lukas SCHIMMER c
a) Department of Mathematics, Imperial College London, London SW7 2AZ, UK
E-mail: a.laptev@imperial.ac.uk
b) Saint Petersburg State University, Saint Petersburg, Russia
c) Institut Mittag–Leffler, The Royal Swedish Academy of Sciences, 182 60 Djursholm, Sweden
E-mail: lukas.schimmer@kva.se
Received September 12, 2021, in final form November 25, 2021; Published online December 06, 2021
https://doi.org/10.3842/SIGMA.2021.105
Abstract. We prove sharp Lieb–Thirring type inequalities for the eigenvalues of a class of
one-dimensional functional difference operators associated to mirror curves. We furthermore
prove that the bottom of the essential spectrum of these operators is a resonance state.
Key words: Lieb–Thirring inequality; functional difference operator; semigroup property
2020 Mathematics Subject Classification: 47A75; 81Q10
To our friend and coauthor Leon Takhtajan
on the occasion of his 70th birthday
1 Introduction
Let P be the self-adjoint quantum mechanical momentum operator on L2(R), i.e., P = −i d
dx and
for b > 0 denote by U(b) the Weyl operator U(b) = exp(−bP ). By using the Fourier transform
ψ̂(k) = (Fψ)(k) =
∫
R
e−2πikxψ(x) dx
we can write the domain of U(b) as
dom(U(b)) =
{
ψ ∈ L2(R) : e−2πbkψ̂(k) ∈ L2(R)
}
.
Equivalently, dom(U(b)) consists of those functions ψ(x) which admit an analytic continuation
to the strip {z = x+ iy ∈ C : 0 < y < b} such that ψ(x+ iy) ∈ L2(R) for all 0 ≤ y < b and there
is a limit ψ(x+ ib− i0) = limε→0+ ψ(x+ ib− iε) in the sense of convergence in L2(R), which we
will denote simply by ψ(x+ib). The domain of the inverse operator U(b)−1 can be characterised
similarly.
For b > 0 we define the operator W0(b) = U(b) + U(b)−1 = 2 cosh(bP ) on the domain
dom(W0(b)) =
{
ψ ∈ L2(R) : 2 cosh(2πbk)ψ̂(k) ∈ L2(R)
}
.
The operator W0(b) is self-adjoint and unitarily equivalent to the multiplication operator
2 cosh(2πbk) in Fourier space. Its spectrum is thus absolutely continuous covering the inter-
val [2,∞) doubly.
This paper is a contribution to the Special Issue on Mathematics of Integrable Systems: Classical and Quan-
tum in honor of Leon Takhtajan.
The full collection is available at https://www.emis.de/journals/SIGMA/Takhtajan.html
mailto:a.laptev@imperial.ac.uk
mailto:lukas.schimmer@kva.se
https://doi.org/10.3842/SIGMA.2021.105
https://www.emis.de/journals/SIGMA/Takhtajan.html
2 A. Laptev and L. Schimmer
Let V ≥ 0, V ∈ L1(R) now be a real-valued potential function. The scalar inequality
2 cosh(2πbk)− 2 ≥ (2πbk)2 implies the operator inequality
W0(b)− 2 ≥ −b2 d2
dx2
(1.1)
on dom(W0(b)). By Sobolev’s inequality, we can conclude that the operator
WV (b) =W0(b)− V
is symmetric and bounded from below on the common domain of W0(b) and V . We can thus
consider its Friedrichs extension, which we continue to denote by WV (b). This operator acts as
(WV (b)ψ)(x) = ψ(x+ ib) + ψ(x− ib)− V (x)ψ(x).
Furthermore, by an application of Weyl’s theorem (in a version for quadratic forms) and Rellich’s
lemma together with the fact that the form domain ofW0(b) is continuously embedded in H1(R)
(as discussed at the beginning of Section 4) the spectrum ofWV (b) consists of essential spectrum
[2,∞) and discrete finite-multiplicity eigenvalues below. Details of this argument in the similar
case of a Schrödinger operator can be found in the upcoming book [2, Proposition 4.14].
We will show that the discrete spectrum satisfies a version of Lieb–Thirring inequalities for
1/2-Riesz means. When formulating the main result of the paper it is convenient to parametrise
the eigenvalues (repeated with multiplicities) as λj = −2 cos(ωj), where ωj ∈ [0, π] for λj ∈
[−2, 2] and ωj ∈ i[0,∞) for λj ≤ −2. Note that in the latter case λj = −2 cosh(|ωj |).
Theorem 1.1. Let V ≥ 0 and let V ∈ L1(R). If WV (b) ≥ −2, then the discrete eigenvalues
λj = −2 cos(ωj) ∈ [−2, 2) (repeated with multiplicities) satisfy∑
j≥1
sinωj
ωj
≤ 1
2πb
∫
R
V (x) dx. (1.2)
The constant in the inequality (1.2) is sharp in the sense that there is a potential V such that (1.2)
becomes equality.
Remark 1.2. Note that Theorem 1.1 does not allow to estimate eigenvalues below −2. In fact,
from the proof of this theorem, the case of one eigenvalue below −2 could be included in the
inequality (1.2). We expect that the inequality holds true for all eigenvalues below −2. However,
the method we use in the proof prevents us from including all eigenvalues due to oscillating
properties of the resolvent (W0(b)− λ)−1 for λ < −2.
Lieb–Thirring inequalities were first established for Schrödinger operators in [15]. For a one-
dimensional Schrödinger operator − d
dx2−V on L2(R) with negative eigenvalues µ1 ≤ µ2 ≤· · ·< 0,
these bounds state that for any γ ≥ 1/2 there is a constant Lγ > 0 such that∑
j≥1
|µj |γ ≤ Lγ
∫
R
V (x)γ+1/2 dx (1.3)
for all V ≥ 0, V ∈ Lγ+1/2(R). The condition γ ≥ 1/2 is optimal. Inequality (1.1) implies that∑
j≥1
|λj − 2|γ ≤ Lγ
b
∫
R
V (x)γ+1/2 dx (1.4)
for all eigenvalues λj ≤ 2 of WV (b). Under the additional assumption WV (b) ≥ −2, our
bound (1.2) presents an improvement of (1.4) for γ = 1/2. This can be seen from the fact
A Sharp Lieb–Thirring Inequality for Functional Difference Operators 3
that for γ = 1/2 the sharp constant in (1.3) is given by L1/2 = 1/2 [7] and from the strict
inequality
|λj − 2|
1
2 = |2 cosωj + 2|
1
2 <
π sinωj
ωj
for ωj ∈ [0, π). The difference of the terms above vanishes as ωj → π, implying that (1.4) is
asymptotically optimal for small coupling. While the necessity of γ ≥ 1/2 in the Lieb–Thirring
inequality for Schrödinger operators does not allow us to conclude that (1.4) fails for 0 ≤ γ < 1/2,
we will prove the following.
Theorem 1.3. Let b > 0. If V ∈ L1(R) with
∫
R V dx > 0, then WV (b) has at least one
eigenvalue below 2. Furthermore, if 0 ≤ γ < 1/2, then there is no constant Lγ such that (1.4)
holds for all compactly supported V . This conclusion holds even under the assumption that
WV (b) ≥ −2.
The study of different properties of functional difference operators WV (b) was considered
before. In the case when −V = V0 = e2πbx is an exponential function, the operator WV0(b) first
appeared in the study of the quantum Liouville model on the lattice [1] and plays an important
role in the representation theory of the non-compact quantum group SLq(2,R). The spectral
analysis of this operator was first studied in [9], see also [17]. In the case when−V = 2 cosh(2πbx)
the spectrum of WV (b) is discrete and converges to +∞. Its Weyl asymptotics were obtained
in [13]. This result was extended to a class of growing potentials in [14]. More information on
spectral properties of functional difference operators can be found in papers [4, 5, 10, 11, 16].
The proof method of Theorem 1.1 is similar to the proof of the sharp Lieb–Thirring inequa-
lity (1.3) for a one-dimensional Schrödinger operator in the case γ = 1/2 as presented in [6].
It relies on a property of convolutions of the resolvent kernels of the operator under consideration.
Such a semigroup property was also recently established for Jacobi operators where it was again
used to prove sharp Lieb–Thirring type inequalities [12]. With a different proof (not using
the convolution property) the sharp inequalities for the Schrödinger operator and the Jacobi
operator were first obtained in [7] and in [8], respectively. Despite formal similarity to the case
of Jacobi operators, it is still surprising that the proof method works for functional difference
operators WV (b). These operators could be considered as differential operators of infinite order
since the symbol cosh(2πbk) can be written as an infinite Taylor series of symbols of even degree
w.r.t. the variable k.
2 Free resolvent
SinceW0(b) ≥ 2 we conclude thatW0(b)−λ is an invertible operator for λ < 2. Let λ = −2 cos(ω)
with ω ∈ [0, π] if λ ∈ [−2, 2] and ω ∈ i[0,∞) if λ < −2. Then in Fourier space the inverse of
W0(b)− λ is given by the multiplication operator (2 cosh(2πbk) + 2 cos(ω))−1.
Applying the inverse Fourier transform F−1 to (2 cosh(2πbk)+2 cos(ω))−1 we find the kernel
of the free resolvent Gλ = (W0(b)− λ)−1 that is
Gλ(x, y) = Gλ(x− y) =
1
2b sinω
sinh
(
ω
b (x− y)
)
sinh
(
π
b (x− y)
) . (2.1)
Remark 2.1. Note that Gλ(x− y) is an even and positive kernel for ω ∈ [0, π] and it becomes
oscillating if ω ∈ i(0,∞). This fact is one of the reasons why we are able to study Lieb–Thirring
inequalities only for the eigenvalues λj ∈ [−2, 2]. This interval contains all of the discrete
spectrum if the potential V is “small” enough. However, if V generates eigenvalues lying in
(−∞,−2), then the oscillating property of the Green’s function prevents us from obtaining the
desired inequality for all eigenvalues.
4 A. Laptev and L. Schimmer
Note that the value of Gλ on the diagonal x = y takes the form
Gλ(0) =
1
2πb
ω
sinω
(2.2)
and we can see the relation between the right-hand side of (2.2) and the expression in the
left-hand side of (1.2). Due to our parameterisation of the spectral parameter, the convergence
λ→ 2− implies ω → π− and thus
Gλ(0) ∼
1
2b
1√
1− cos2 ω
∼ 1
2b
1√
2− λ
as λ→ 2−.
If λ→ −∞, then ω → i∞ and
Gλ(0) ∼
1
πb
|λ|−1 log |λ|.
In [17] L. Faddeev and L.A. Takhtajan studied the resolvent in a slightly different form
Gλ(x, y) =
σ
sinh
(
πiκ
σ
)( e−2πiκ(x−y)
1− e−4πiσ(x−y)
+
e2πiκ(x−y)
1− e4πiσ(x−y)
)
,
which coincides with (2.1) with σ = i/2b, λ = 2 cosh(2bπκ) and κ = ω−π
2πib . It was pointed out
that the free resolvent can be written using the analogues of the Jost solutions
f−(x,κ) = e−2πiκx and f+(x,κ) = e2πiκx
that appear in the theory of one-dimensional Schrödinger operators. Namely
Gλ(x− y) =
2σ
C(f−, f+)(κ)
(
f−(x,κ)f+(y,κ)
1− e
πi
σ′ (x−y)
+
f−(y,κ)f+(x,κ)
1− e−
πi
σ′ (x−y)
)
,
where σ′σ = −1/4 and where C(f, g) is the so-called Casorati determinant (a difference analogue
of the Wronskian) of the solutions of the functional-difference equation
C(f, g)(x,κ) = f(x+ 2σ′,κ)g(x,κ)− f(x,κ)g(x+ 2σ′,κ).
For the Jost solutions C(f−, f+)(x,κ) = 2 sinh
(
πiκ
σ
)
.
The equality (W0(b)−λ)G(x− y) = δ(x− y) could be interpreted as an equation of distribu-
tions. Since the functions f±(x, k) are Jost solutions, the distribution defined by (W0(b)− λ)×
G(x − y) is supported only at x = y, and its singular part coincides with the singular part of
the distribution
− 2σσ′
πiC(f−, f+)(κ)
(
f−(x+ 2σ′,κ)f+(y,κ)− f−(y,κ)f+(x+ 2σ′,κ)
x− y − i0
+
f−(x− 2σ′,κ)f+(y,κ)− f−(y,κ)f+(x− 2σ′,κ)
x− y + i0
)
in the neighbourhood of x = y. This singular part is equal to
−2σσ′
πi
(
1
x− y − i0
− 1
x− y + i0
)
= δ(x− y),
where the authors used the Sokhotski–Plemelj formula. This formula is similar to the respective
formula for a Schrödinger operator when the Dirac δ-function appears by differentiating a step
function.
A Sharp Lieb–Thirring Inequality for Functional Difference Operators 5
3 Proof of inequality (1.2)
3.1 Some auxiliary results
We first collect some results from [6] verbatim. Let A be a compact operator on a Hilbert
space G and let us denote
∥A∥n =
n∑
j=1
√
λj(A∗A),
where λj(A
∗A) are the eigenvalues of A∗A in decreasing order. Then by Ky Fan’s inequality
(see for example [3, Lemma 4.2]) the functionals ∥ · ∥n, n = 1, 2, . . . , are norms and thus for any
unitary operator Y in G we have
∥Y ∗AY ∥n = ∥A∥n.
Definition 3.1. Let A, B be two compact operators on G. We say that A majorises B or
B ≺ A, iff
∥B∥n ≤ ∥A∥n, for all n ∈ N.
Lemma 3.2. Let A be a nonnegative compact operator acting in G, {Y (k)}k∈R be a family of
unitary operators on G, and let g(k) dk be a probability measure on R. Then the operator
B =
∫
R
Y (k)∗AY (k)g(k) dk
is majorised by A.
Proof. This is a simple consequence of the triangle inequality
∥B∥n ≤
∫
R
∥Y ∗(k)AY (k)∥ng(k) dk = ∥A∥n
∫
R
g(k) dk = ∥A∥n. ■
Let λj = −2 cosωj ≤ 2 be the eigenvalues of W0(b) − V with V ≥ 0. In order to slightly
simplify the notations it is convenient to write
λj = −2 cos
(√
θj
)
with θj ∈
(
−∞, π2
]
and ω2
j = θj .
Let us denote by Kλ the Birman–Schwinger operator
Kλ = V 1/2GλV
1/2. (3.1)
Let µj(Kλ) be the eigenvalues (in decreasing order) of the Birman–Schwinger operator Kλ
defined in (3.1). Then due to the Birman–Schwinger principle we have
1 = µj(Kλj
). (3.2)
Let us define the operator
Lθ :=
1
G−2 cos
√
θ(0)
K−2 cos
√
θ,
6 A. Laptev and L. Schimmer
where G−2 cos
√
θ(0) =
1
2πb
√
θ
sin
√
θ
is given in (2.2). Then from (3.2) we obtain
∑
j≥1
1
Gλj
(0)
=
∑
j≥1
1
Gλj
(0)
µj(Kλj
) =
∑
j≥1
µj(Lθj ).
The integral kernel of the operator Lθ is given by
√
V (x)gπ2,θ(x− y)
√
V (y), where
gπ2,θ(x) :=
π√
θ
sinh
(√
θ
b x
)
sinh
(
π
b x
) .
Consider a more general function
gφ,θ(x) :=
√
φ
√
θ
sinh
(√
θ
b x
)
sinh
(√φ
b x
) .
Since gφ,θ(0) = 1 its Fourier transform ĝφ,θ = F(gφ,θ) satisfies the equation∫
R
ĝφ,θ(k) dk = 1.
Moreover, for any −∞ < θ < φ with 0 < φ < π2 we have
ĝφ,θ(k) = F
(√
φ
√
θ
sinh
(√
θ
b x
)
sinh
(√φ
b x
))(k) = 2π sin
(
π
√
θ√
φ
)
√
θ
b
2 cosh
(
2π2bk√
φ
)
+ 2 cos
(
π
√
θ√
φ
) ,
and the right-hand side is positive. Thus ĝφ,θ dk is a probability measure for such values.
Note also that importantly
gπ2,θ(x)
gπ2,θ′(x)
=
√
θ′√
θ
sinh
(√
θ
b x
)
sinh
(√
θ′
b x
) = gθ′,θ(x)
and therefore(
ĝπ2,θ′ ∗ ĝθ′,θ
)
(k) = ĝπ2,θ(k).
This is the interesting convolution/semigroup property mentioned in the introduction. In the
special case −∞ < θ < 0 = θ′ analogous computations lead to the same result with ĝ0,θ(k) =
χ[−1,1]
(
2πbk/
√
|θ|
)
πb/
√
|θ|.
Lemma 3.3 (monotonicity). For (θ, θ′) such that −∞ < θ ≤ θ′ and 0 ≤ θ′ < π2 we have
Lθ ≺ Lθ′.
Proof. Let Y (k) : L2(R) → L2(R) be the unitary multiplication operator
(Y (k)ψ)(x) = e−2πikxψ(x)
and let T be the projection onto V 1/2, i.e.,
(Tψ)(x) = V 1/2(x)
∫
R
V 1/2(y)ψ(y) dy.
A Sharp Lieb–Thirring Inequality for Functional Difference Operators 7
Using Y (k′ + k′′) = Y (k′)Y (k′′) and Lemma 3.2 we obtain
Lθ =
∫
R
Y (k)∗TY (k)ĝπ2,θ(k) dk
=
∫
R
∫
R
Y (k)∗TY (k)ĝπ2,θ′(k
′)ĝθ′,θ(k − k′) dk′dk
=
∫
R
Y (k′′)∗
(∫
R
Y (k′)∗TY (k′)ĝπ2,θ′(k
′) dk′
)
Y (k′′)ĝθ′,θ(k
′′) dk′′ ≺ Lθ′ ,
where we have used that ĝθ′,θ dk is a probability measure. ■
Remark 3.4. With a slight abuse of notations, Lemma 3.3 says that Lλ ≺ Lλ′ for any λ < 2
as long as λ ≤ λ′ and −2 ≤ λ′ < 2.
3.2 Proof of inequality (1.2)
We now enumerate the eigenvalues of the operator WV (b) belonging to the interval [−2, 2) such
that −2 ≤ λ1 ≤ λ2 ≤ λ3 ≤ · · · repeated with multiplicity. By using the monotonicity established
in Lemma 3.3 we have a sequence of inequalities
1
Gλ1(0)
= 2πb
sinω1
ω1
= µ1(Lθ1) ≤ µ1(Lθ2),
2∑
j=1
1
Gλj
(0)
= 2πb
2∑
j=1
sinωj
ωj
≤
2∑
j=1
µj(Lθ2) ≤
2∑
j=1
µj(Lθ3),
3∑
j=1
1
Gλj
(0)
= 2πb
3∑
j=1
sinωj
ωj
≤
3∑
j=1
µj(Lθ3) ≤
3∑
j=1
µj(Lθ4), etc.
Note that we do not use any assumptions on the multiplicities of the eigenvalues, other than
their finiteness. Furthermore, by Lemma 3.3 the same results also hold true if a single eigenvalue
is below −2. Continuing the above process and noting that the trace of Lθ is
∫
R V dx for all θ,
we finally obtain∑
j≥1
sinωj
ωj
≤ 1
2πb
∫
R
V (x) dx.
The proof is complete.
Remark 3.5. Note that 2 cosh(2πbk)−2
b2
→ (2πk)2 tends to the symbol of the second derivative as
b→ 0 and that Wb2V (b) ≥ −2 for sufficiently small b. We thus expect that it should be possible
to recover the Lieb–Thirring inequality (1.3) for a Schrödinger operator with the sharp constant
L1/2 = 1/2 from Theorem 1.1.
4 Sharpness of inequality (1.2)
Similarly to the case of Schrödinger operators, we aim to prove that the Lieb–Thirring in-
equality becomes an equality for Dirac-delta potentials. To this end let c > 0 and consider
the potential Vc(x) = cδ(x). To properly define WVc(b), we first note that the quadratic form
⟨ψ, (W0(b)− 2)ψ⟩ can be written as
⟨ψ, (W0(b)− 2)ψ⟩ =
∫
R
∣∣2 sinh(πbk)ψ̂(k)∣∣2 dk =
∫
R
|ψ(x+ ib/2)− ψ(x− ib/2)|2 dx. (4.1)
8 A. Laptev and L. Schimmer
This can be seen by introducing the self-adjoint operatorD(b) = U(b/2)−U(b/2)−1 = 2 sinh
(
bP
2
)
and checking that D(b)2 =W0(b)− 2 either directly or by means of the identity cosh(2πbk)− 1
= 2 sinh(πbk)2. The form domain of W0(b) is thus dom(D(b)) = dom(W0(b/2)) ⊂ H1(R) and
on this domain Sobolev’s inequality yields that
|ψ(0)|2 ≤ ε
∫
R
|ψ′(x)|2 dx+
1
ε
∫
R
|ψ(x)|2 dx
≤ ε
b2
∫
R
∣∣2 sinh(πbk)ψ̂(k)∣∣2 dk + 1
ε
∫
R
|ψ(x)|2 dx
for any choice of ε > 0. The KLMN theorem thus allows us to define W0(b) − Vc. As a rank
one perturbation of the operator W0(b) the potential Vc generates no more than one eigenvalue
below the continuous spectrum [2,∞).
In Fourier space the eigenequation (W0(b)− cδ)ψc = λψc becomes
2 cosh(2πbk)ψ̂c(k)− cψc(0) = λψ̂c(k)
by means of the formal identity F(δψc) = ψc(0). Writing again λ = −2 cosω we obtain
ψ̂c(k) =
cψc(0)
2 cosh(2πbk) + 2 cosω
(4.2)
and therefore
ψc(x) = cψc(0)G−2 cosω(x) =
cψc(0)
2b sinω
sinh
(
ω
b x
)
sinh
(
π
b x
) . (4.3)
Of course we could have seen this immediately by using the equation for the Green’s function
(W0(b) + 2 cosω)G−2 cosω(x) = δ(x).
Letting x→ 0 in (4.3) we find
1 =
c
2b sinω
ω
π
or equivalently
sinω
ω
=
c
2πb
. (4.4)
Since sin
√
θ√
θ
is a monotone decreasing function of θ = ω2 ∈
(
−∞, π2
]
that takes all values in
[0,∞), for any c > 0 there is a unique solution ωc to (4.4) and vice versa. If c/(2πb) < 1 then
ωc ∈ (0, π) and otherwise ωc ∈ i[0,∞). Since
∫
Vc dx = c, the identity (4.4) can be rewritten as
sinω
ω
=
1
2πb
∫
R
Vc(x) dx
showing that the Lieb–Thirring inequality is satisfied for potentials −cδ with a single eigenvalue
that can be placed anywhere in (−∞, 2) by choosing c > 0 suitably.
Remark 4.1. If we choose the normalising constant ψ(0) > 0 then the eigenfunction defined
in (4.3)
ψc(x) =
cψ(0)
2b sinωc
sinh
(
ωc
b x
)
sinh
(
π
b x
)
A Sharp Lieb–Thirring Inequality for Functional Difference Operators 9
is positive assuming that the coupling constant c is small enough satisfying the inequality
c/(2πb) ≤ 1 and thus ωc ∈ [0, π). Note that if c/(2πb) = 1 then ωc = 0 and
ψc(x) =
πψ(0)x
b sinh
(
π
b x
) > 0.
However, if the coupling constant c > 2πb then ωc ∈ i(0,∞) and hence
ψc(x) =
cψ(0)
2b sinh |ωc|
sin
( |ωc|
b x
)
sinh
(
π
b x
)
is an oscillating function and in particular has an infinite number of zeros. This contradicts
a possible conjecture that the eigenfunction for the lowest eigenvalue is strictly positive.
Open problem. Assume that the discrete spectrum σd(WV (b)) of the operator WV (b) satisfies
the property σd(WV (b)) ⊂ [−2, 2). Is it true that the eigenfunction corresponding to the lowest
eigenvalue could be chosen strictly positive?
5 Necessity of γ ≥ 1/2
The following argument is similar to that presented in the upcoming book [2, Propositions 4.41
and 4.42] for the case of a Schrödinger operator. For ε > 0 let ψε(x) = 1/ cosh(2εx/b). If ε is
sufficiently small, say ε ≤ ε0, then ψε ∈ dom(W0(b)). Using (4.1) we compute that
⟨ψε, (W0(b)− 2)ψε⟩ =
b sin2 ε
2ε
∫
R
∣∣∣∣ 2 sinhx
cos2 ε cosh2 x+ sin2 ε sinh2 x
∣∣∣∣2 dx ≤ Cbε (5.1)
for a constant C > 0 independent of ε ≤ ε0. For any potential V ∈ L1(R) it holds that
⟨ψε, V ψε⟩ →
∫
R V dx as ε→ 0 by dominated convergence and thus for sufficiently small ε
⟨ψε, (WV (b)− 2)ψε⟩ < 0.
By the min-max principle this proves the first part of Theorem 1.3.
For the second assertion of the theorem we choose more specifically the compactly supported
potential V (x) = cχ[−1/2,1/2](x/b). By Sobolev’s inequality WV (b) ≥ −2 for sufficiently small
c ≤ c0 such that all the discrete eigenvalues of WV (b) are contained in [−2, 2). Furthermore
∥ψε∥2 = b/ε and, since tanhx ≥ x/2 for 0 ≤ x ≤ 1,
⟨ψε, V ψε⟩ = cb
∫ 1/2
−1/2
| cosh(2εx)|−2 dx =
cb tanh ε
ε
≥ 1
2
cb (5.2)
for ε ≤ 1. We now choose ε = cδ. If δ ≤ min(ε0/c0, 1/c0) such that ε ≤ min(ε0, 1), then (5.1)
and (5.2) both hold and
⟨ψε, (WV (b)− 2)ψε⟩
∥ψε∥2
≤ Cε2 − 1
2
cε = c2δ
(
Cδ − 1
2
)
.
Choosing δ < min(ε0/c0, 1/c0, 1/2C) we can conclude by the min-max principle that WV (b)− 2
has a negative eigenvalue λ1 ≤ −c2δ
(
1
2 − Cδ
)
. If a Lieb–Thirring inequality (1.4) were to hold
for γ < 1/2 then for some finite Lγ
c2γδγ
(
1
2
− Cδ
)γ
≤ Lγ
b
∫
R
V (x)γ+
1
2 dx = Lγc
γ+ 1
2 ,
which is clearly a contradiction if c→ 0.
10 A. Laptev and L. Schimmer
Acknowledgements
A. Laptev was partially supported by RSF grant 18-11-0032. L. Schimmer was supported by
VR grant 2017-04736 at the Royal Swedish Academy of Sciences. The authors would like to
thank the anonymous referees for their useful comments to improve the article.
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1 Introduction
2 Free resolvent
3 Proof of inequality (1.2)
3.1 Some auxiliary results
3.2 Proof of inequality (1.2)
4 Sharpness of inequality (1.2)
5 Necessity of gamma >= 1/2
References
|
| id | nasplib_isofts_kiev_ua-123456789-211422 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-14T16:00:25Z |
| publishDate | 2021 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Laptev, Ari Schimmer, Lukas 2026-01-02T08:29:34Z 2021 A Sharp Lieb-Thirring Inequality for Functional Difference Operators. Ari Laptev and Lukas Schimmer. SIGMA 17 (2021), 105, 10 pages 1815-0659 2020 Mathematics Subject Classification: 47A75; 81Q10 arXiv:2109.05465 https://nasplib.isofts.kiev.ua/handle/123456789/211422 https://doi.org/10.3842/SIGMA.2021.105 We prove sharp Lieb-Thirring type inequalities for the eigenvalues of a class of one-dimensional functional difference operators associated with mirror curves. We furthermore prove that the bottom of the essential spectrum of these operators is a resonance state. A. Laptev was partially supported by an RSF grant 18-11-0032. L. Schimmer was supported by a VR grant 2017-04736 at the Royal Swedish Academy of Sciences. The authors would like to thank the anonymous referees for their useful comments to improve the article. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications A Sharp Lieb-Thirring Inequality for Functional Difference Operators Article published earlier |
| spellingShingle | A Sharp Lieb-Thirring Inequality for Functional Difference Operators Laptev, Ari Schimmer, Lukas |
| title | A Sharp Lieb-Thirring Inequality for Functional Difference Operators |
| title_full | A Sharp Lieb-Thirring Inequality for Functional Difference Operators |
| title_fullStr | A Sharp Lieb-Thirring Inequality for Functional Difference Operators |
| title_full_unstemmed | A Sharp Lieb-Thirring Inequality for Functional Difference Operators |
| title_short | A Sharp Lieb-Thirring Inequality for Functional Difference Operators |
| title_sort | sharp lieb-thirring inequality for functional difference operators |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211422 |
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