On the Spectra of Real and Complex Lamé Operators
We study Lamé operators of the form
 L=−d²/dx²+m(m+1)ω²℘(ωx+z₀),
 with m∈N and ω a half-period of ℘(z). For rectangular period lattices, we can choose ω and z0 such that the potential is real, periodic and regular. It is known after Ince that the spectrum of the corresponding Lamé...
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 / W.A. Haese-Hill, M.A. Hallnäs, A.P. Veselov // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 32 назв. — англ. |
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| author_facet | Haese-Hill, W.A. Hallnäs, M.A. Veselov, A.P. |
| citation_txt | On the Spectra of Real and Complex Lamé Operators
 / W.A. Haese-Hill, M.A. Hallnäs, A.P. Veselov // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 32 назв. — англ. |
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| description | We study Lamé operators of the form
L=−d²/dx²+m(m+1)ω²℘(ωx+z₀),
with m∈N and ω a half-period of ℘(z). For rectangular period lattices, we can choose ω and z0 such that the potential is real, periodic and regular. It is known after Ince that the spectrum of the corresponding Lamé operator has a band structure with not more than m gaps. In the first part of the paper, we prove that the opened gaps are precisely the first m ones. In the second part, we study the Lamé spectrum for a generic period lattice when the potential is complex-valued. We concentrate on the m=1 case, when the spectrum consists of two regular analytic arcs, one of which extends to infinity, and briefly discuss the m=2 case, paying particular attention to the rhombic lattices.
|
| first_indexed | 2025-12-07T17:49:29Z |
| format | Article |
| fulltext |
Symmetry, Integrability and Geometry: Methods and Applications SIGMA 13 (2017), 049, 23 pages
On the Spectra of Real and Complex Lamé Operators
William A. HAESE-HILL †, Martin A. HALLNÄS ‡ and Alexander P. VESELOV †
† Department of Mathematical Sciences, Loughborough University,
Loughborough LE11 3TU, UK
E-mail: wahhill@gmail.com, A.P.Veselov@lboro.ac.uk
‡ Department of Mathematical Sciences, Chalmers University of Technology and
the University of Gothenburg, SE-412 96 Gothenburg, Sweden
E-mail: hallnas@chalmers.se
Received April 04, 2017, in final form June 21, 2017; Published online July 01, 2017
https://doi.org/10.3842/SIGMA.2017.049
Abstract. We study Lamé operators of the form
L = − d2
dx2
+m(m+ 1)ω2℘(ωx+ z0),
with m ∈ N and ω a half-period of ℘(z). For rectangular period lattices, we can choose ω
and z0 such that the potential is real, periodic and regular. It is known after Ince that the
spectrum of the corresponding Lamé operator has a band structure with not more than m
gaps. In the first part of the paper, we prove that the opened gaps are precisely the first m
ones. In the second part, we study the Lamé spectrum for a generic period lattice when the
potential is complex-valued. We concentrate on the m = 1 case, when the spectrum consists
of two regular analytic arcs, one of which extends to infinity, and briefly discuss the m = 2
case, paying particular attention to the rhombic lattices.
Key words: Lamé operators; finite-gap operators; spectral theory; non-self-adjoint operators
2010 Mathematics Subject Classification: 34L40; 47A10; 33E10
1 Introduction
The Lamé equation
−d2ψ
dz2
+m(m+ 1)℘(z)ψ = λψ,
where m ∈ N and ℘(x) is Weierstrass’ elliptic function, satisfying
(℘′)2 = 4(℘− e1)(℘− e2)(℘− e3),
is a classical object of 19th century mathematics.
It surprisingly appeared again at the early age of quantum mechanics in the work by Kramers
and Ittmann [15, 16], who discovered that the Lamé equation is closely related to the quantum
top by showing that the corresponding Schrödinger equation is separable in the elliptic coor-
dinate system and that the resulting differential equations are of Lamé form (see, e.g., [12]).
It is interesting that initially the equation was derived by Gabriel Lamé in 1837 as a result of
separation of variables for the Laplace equation in elliptic coordinates, see [17], so one might
argue that its origin was quantum from the very beginning.
Viewed as an equation in the complex domain, its solutions, which have a number of remark-
able properties, were described explicitly by Hermite and Halphen, see, e.g., [31]. Note that
mailto:wahhill@gmail.com
mailto:A.P.Veselov@lboro.ac.uk
mailto:hallnas@chalmers.se
https://doi.org/10.3842/SIGMA.2017.049
2 W.A. Haese-Hill, M.A. Hallnäs and A.P. Veselov
Figure 1. The spectrum of the real Lamé operator (1.1) in them = 1 case, where ∆(λ) is the discriminant
and crosses denote closed gaps.
for x on the real line the potential ℘(x) has singularities. If, however, e1, e2, e3 are all real, one
can make a pure imaginary half-period shift by z0 = ω3 and consider the Lamé operator
L = − d2
dx2
+m(m+ 1)℘(x+ z0), (1.1)
whose potential is real, 2ω1-periodic and regular on the whole of R. Consequently, one can apply
Bloch–Floquet theory, which implies that the spectrum has a band structure, as described, e.g.,
in [25].
Generically, the spectrum of a periodic Schrödinger operator (or Hill operator) on the real
line consists of a countably infinite number of bands. It was Ince [14] who first pointed out
the remarkable fact that the spectrum of L has a band structure with not more than m gaps.
Erdélyi [8] later showed that in fact all m gaps are open.
Nowadays, this is just the simplest example in the large class of finite-gap operators discovered
in the 1970s, see [7, 18, 21]. It turned out that all such operators can be described explicitly in
terms of hyperelliptic Riemann theta functions, with the Lamé operator (1.1) corresponding to
the elliptic case. However, it seems that the question of exactly which gaps in the spectrum are
open has not been explicitly discussed in the literature even in this case.
The first result of this paper, obtained in Section 2, demonstrates that in the Lamé case it is
precisely the first m gaps that are open. Although this fact may not be surprising for experts,
we could not find a rigorous proof in the literature. For m = 1, the result is illustrated in Fig. 1,
showing that all the closed gaps are indeed located in the infinite spectral band.
In the second part of the paper, we consider the Lamé operator with a complex-valued periodic
potential
V (x) = m(m+ 1)ω2℘(ωx+ z0), (1.2)
where ω is any half-period of ℘(x), and the only assumption on z0 ∈ C is that the corresponding
potential is non-singular.
The spectral theory of Schrödinger operators with a complex periodic potential has been
studied in a number of papers, see, e.g., [3, 10, 26, 29, 30] as well as references therein. In
particular, Rofe-Beketov showed that the spectrum of a Schrödinger operator
L = − d2
dx2
+ u(x),
On the Spectra of Real and Complex Lamé Operators 3
with periodic, regular, but complex-valued potential u(x) can be defined as the set of λ ∈ C
such that at least one solution of the equation Lψ = λψ is bounded on the whole real line.
Equivalently, the corresponding Floquet multiplier ρ(λ), given by
ψ(x+ T, λ) = ρ(λ)ψ(x, λ),
where T is a period of u(x), should lie on the unit circle:
|ρ(λ)| = 1.
In the case of the Lamé operator, it follows from work of Gesztesy and Weikard [9, 10, 30],
who studied the general elliptic finite-gap case, that the spectrum consists of finitely many
regular analytic arcs, with precisely one arc extending to infinity. In the general case of an
operator with a periodic complex-valued potential, the question about the relative positions
of the spectral arcs was raised earlier by Pastur and Tkachenko [23], who gave an example in
which the spectral arcs intersect at interior points. Gesztesy and Weikard [9] observed that more
explicit examples are given by the spectrum of the m = 1 Lamé operator for suitable rhombic
period lattices, see also Proposition 3.7 and Figs. 2 and 5 below.
Note that if the shift z0 6= ω3 in the Lamé operator (1.1), we have in general a periodic,
regular, but complex-valued potential. However, it is easy to see that the Floquet multipliers
do not depend on z0, so that the spectrum is the same as in the self-adjoint case. Moreover,
setting ω = ω1, ω3 in (1.2), we have essentially Ince’s result, since the corresponding operator is
equivalent to the previous case.
In Section 3, we consider genuinely complex instances of the Lamé operator, concentrating on
the simplest case m = 1. Although this case was briefly discussed earlier in a wider context by
Gesztesy and Weikard [9] and Batchenko and Gesztesy [1], we believe that the complete picture
is still missing in the literature.
We recall that Hermite’s solutions of the corresponding Lamé equation
−d2ψ
dz2
+ 2℘(z)ψ = λψ, λ = −℘(k),
are given by
ψ(z, k) =
σ(z + k)
σ(z)σ(k)
exp(−ζ(k)z), (1.3)
where k ∈ C and σ(z) and ζ(z) are the Weierstrass σ- and ζ-function, see, e.g., [31]. Since the
solutions have the Floquet property
ψ(z + 2ω, k) = exp(2ηk − 2ζ(k)ω)ψ(z, k),
with η = ζ(ω), they remain bounded on the line z = ωx+ z0, x ∈ R, if and only if
u(k) := Re[ηk − ζ(k)ω] = 0.
By the result of Rofe-Beketov, the corresponding values of λ = −ω2℘(k) constitute the spectrum
of the Lamé operator
L = − d2
dx2
+ 2ω2℘(xω + z0). (1.4)
Thus, the problem is to study the zero level of the real analytic function u(k), k ∈ E× = E \0,
where E = C/L is the corresponding elliptic curve.
4 W.A. Haese-Hill, M.A. Hallnäs and A.P. Veselov
Figure 2. Spectra of the complex Lamé operator (1.4) in rhombic cases with ω = ω1.
We show that u(k) is a Morse function on E× provided the non-degeneracy conditions
η + ωej 6= 0, j = 1, 2, 3,
are satisfied. We note that this condition also appeared in Takemura’s [28] study of analyticity
properties of the eigenvalues of the m = 1 Lamé operator.
Applying Morse theory arguments we show that, under some additional non-singularity as-
sumptions, the spectrum of (1.4) consists of precisely two non-intersecting regular analytic arcs,
with one infinite arc asymptotic to the positive half of the real line shifted by −2ηω, and the
remaining endpoints being −ω2ej , j = 1, 2, 3 (in agreement with [9] and [1]).
The non-degeneracy conditions in general are difficult to control, but in the rhombic case we
prove that there exists exactly one exceptional elliptic curve E∗ with the value of the j-invariant
j∗ ≈ 243.797 (see the top right corner of Fig. 6 for the corresponding period rhombus). The
corresponding spectrum has a tripod structure, see the top right plot in Fig. 2. Note that in this
case the spectrum is a union of three simple regular analytic arcs, which is more than usually
expected in the m = 1 case.
In the rectangular case with ω = ω2 we show that all closed spectral gaps are contained in
the spectral arc extending to infinity. Finally, we briefly discuss the spectrum in the rhombic
Lamé case with m = 2.
On the Spectra of Real and Complex Lamé Operators 5
2 Spectral gaps for the real Lamé operator
The real Lamé operator (1.1) can be conveniently rewritten in the Jacobian form as
Lm = − d2
dx2
+m(m+ 1)k2 sn2(x, k), (2.1)
where 0 < k < 1, m ∈ N and sn(x, k) one of the Jacobi elliptic functions, see, e.g., [31].
It is known after Ince [14] and Erdélyi [8] that its spectrum has a band structure with
exactly m gaps. We are going to explain now which gaps are open.
Theorem 2.1. The spectrum of the Lamé operator (2.1) has precisely the first m gaps open.
To prove this we follow the classical approach of Ince [14], which we explain now following [19].
The Lamé potential m(m+ 1)k2 sn2(x, k) is a real smooth periodic function with period 2K,
where
K =
∫ π/2
0
dφ√
1− k2 sin2 φ
,
so the Lamé operator (2.1) is of Hill type. By the oscillation theorem for Hill operators there
exists a sequence of real numbers
λ0 < λ′1 ≤ λ′2 < λ1 ≤ λ2 < · · · < λ′2n−1 ≤ λ′2n < λ2n−1 ≤ λ2n < · · · , (2.2)
going to infinity, where λj and λ′j correspond to 2K-periodic and 2K-antiperiodic solutions of
the Lamé equation
−d2ψ
dx2
+m(m+ 1)k2 sn2(x, k)ψ = λψ (2.3)
respectively. The intervals (λ2n−1, λ2n) and (λ′2n−1, λ
′
2n) are spectral gaps, and the spectrum of
the Lamé operator (2.1) is the complement to the union of these gaps and the interval (−∞, λ0).
Such a gap is closed if and only if two linearly independent 2K-periodic or 2K-antiperiodic so-
lutions of the Lamé equation (2.3) coexist for λ = λ2n−1 = λ2n or λ = λ′2n−1 = λ′2n, respectively.
So in this language we have to show that all points of coexistence are located in the infinite
spectral band.
To prove this we first transform the Lamé equation (2.3) into an equation of Ince’s type
(1 + a cos 2φ)
d2y
dφ2
+ b(sin 2φ)
dy
dφ
+ (c+ d cos 2φ)y = 0, (2.4)
where a, b, c, d are real parameters with |a| < 1 (see [19]). More precisely, by the substitutions
φ = amx,
where the Jacobi amplitude amx = am(k;x) is determined by
x =
∫ amx
0
dφ√
1− k2 sin2 φ
,
and ψ(x) = y(amx), we arrive at (2.4) with
a = −b =
k2
2− k2
, c =
2λ−m(m+ 1)k2
2− k2
, d =
m(m+ 1)k2
2− k2
. (2.5)
6 W.A. Haese-Hill, M.A. Hallnäs and A.P. Veselov
Note that x = 2K corresponds to φ = π, which is reflected in the fact that the coefficients
in (2.4) are periodic with period π.
If Ince’s equation (2.4) has two linearly independent π-periodic or π-antiperiodic solutions,
then we can find two solutions y1, y2 such that
y1 =
∞∑
n=0
A2n cos 2nφ, y2 =
∞∑
n=1
B2n sin 2nφ, (2.6)
or
y1 =
∞∑
n=0
A2n+1 cos(2n+ 1)φ, y2 =
∞∑
n=0
B2n+1 sin(2n+ 1)φ. (2.7)
Substituting the periodic series into (2.4) with (2.5), we obtain the recurrence relations
Λ0A0 + P (−1)A2 = 0, (2.8a)
2P (0)A0 + Λ1A2 + P (−2)A4 = 0, (2.8b)
P (n− 1)A2n−2 + ΛnA2n + P (−n− 1)A2n+2 = 0, n ≥ 2 (2.8c)
and
Λ1B2 + P (−2)B4 = 0, (2.9a)
P (n− 1)B2n−2 + ΛnB2n + P (−n− 1)B2n+2 = 0, n ≥ 2 (2.9b)
where we have introduced
P (z) =
m(m+ 1)k2
4
− zk2
2
− z2k2, (2.10)
Λn = λ− m(m+ 1)k2
2
−
(
1− k2
2
)
4n2. (2.11)
Similarly, the coefficients in the antiperiodic series should satisfy the recurrence relations
(P ∗(0) + Λ∗0)A1 + P ∗(−1)A3 = 0, (2.12a)
P ∗(n)A2n−1 + Λ∗nA2n+1 + P ∗(−n− 1)A2n+3 = 0, n ≥ 1 (2.12b)
and
(−P ∗(0) + Λ∗0)B1 + P ∗(−1)B3 = 0, (2.13a)
P ∗(n)B2n−1 + Λ∗nB2n+1 + P ∗(−n− 1)B2n+3 = 0, n ≥ 1 (2.13b)
with
P ∗(z) =
m(m+ 1)k2
4
− (2z − 1)k2
4
− (2z − 1)2k2
4
, (2.14)
Λ∗n = λ− m(m+ 1)k2
2
−
(
1− k2
2
)
(2n+ 1)2. (2.15)
At this point, we find it convenient to consider even and odd values of m separately. Letting
m = 2ν, ν ∈ Z≥0,
we have
P (ν) = P ∗(−ν) = 0,
On the Spectra of Real and Complex Lamé Operators 7
so that the infinite tridiagonal matrices corresponding to the recurrence relations (2.8)–(2.9)
and (2.12)–(2.13) become reducible.
In the periodic case, the finite-dimensional parts of these matrices are
Kν,1 =
Λ0 P (−1) 0 · · · · · · 0
2P (0) Λ1 P (−2) 0 · · · 0
0 P (1) Λ2 P (−3) · · · 0
...
. . .
. . .
. . .
. . .
...
0 · · · 0 P (ν − 2) Λν−1 P (−ν)
0 · · · · · · 0 P (ν − 1) Λν
(2.16)
and
Kν,2 =
Λ1 P (−2) 0 · · · 0
P (1) Λ2 P (−3) · · · 0
...
. . .
. . .
. . .
...
0 · · · P (ν − 2) Λν−1 P (−ν)
0 · · · 0 P (ν − 1) Λν
, (2.17)
which correspond to sequences terminating at n = ν in the sense that A2n = B2n = 0 for n > ν.
Introducing the functions
δν,1(λ) = detKν,1(λ), δν,2(λ) = detKν,2(λ),
we let
Sν,1 = {λ : δν,1(λ) = 0}, Sν,2 = {λ : δν,2(λ) = 0},
be the sets of corresponding eigenvalues of the Lamé operator (2.1). Recalling (2.10), it is clear
that all off-diagonal entries of the matrices Kν,1 and Kν,2 are positive. From the general theory
of tridiagonal (Jacobi) matrices (see, e.g., [27]), it follows that Sν,1 and Sν,2 consist of ν + 1
respectively ν real and distinct eigenvalues.
By specializing [19, Theorem 7.3] to the Lamé case, we find that Sν := Sν,1 ∪ Sν,2 contains
the simple part of the periodic spectrum, which we state as the following lemma.
Lemma 2.2. Let m = 2ν, ν ∈ Z≥0. If λ belongs to the periodic spectrum of the Lamé opera-
tor (2.1) and has multiplicity 1, then λ ∈ Sν .
Turning to the the antiperiodic case, we note that the recurrence relations (2.12)–(2.13) have
no solutions given by terminating sequences. Instead, the relevant finite-dimensional matrices
K∗ν,1 =
P ∗(0) + Λ∗0 P (−1) 0 · · · 0
P ∗(1) Λ∗1 P ∗(−2) · · · 0
...
. . .
. . .
. . .
...
0 · · · P ∗(ν − 2) Λ∗ν−1 P ∗(−ν + 1)
0 · · · 0 P ∗(ν − 1) Λ∗ν−1
(2.18)
and
K∗ν,2 =
−P ∗(0) + Λ∗0 P (−1) 0 · · · 0
P ∗(1) Λ∗1 P ∗(−2) · · · 0
...
. . .
. . .
. . .
...
0 · · · P ∗(ν − 2) Λ∗ν−1 P ∗(−ν + 1)
0 · · · 0 P ∗(ν − 1) Λ∗ν−1
(2.19)
8 W.A. Haese-Hill, M.A. Hallnäs and A.P. Veselov
are related to sequences with A2ν−1, B2ν−1 6= 0. Indeed, such solutions of the recurrence relations
(2.12)–(2.13) exist only if the matrices K∗ν,1, K
∗
ν,2 are singular.
Letting
δ∗ν,1(λ) = detK∗ν,1(λ), δ∗ν,2(λ) = detK∗ν,2(λ),
we introduce the sets
S∗ν,1 = {λ : δ∗ν,1(λ) = 0}, S∗ν,2 = {λ : δ∗ν,2(λ) = 0}.
Just as in the periodic case, we see that each of the two sets S∗ν,j consists of ν real and distinct
eigenvalues. Moreover, we infer from [19, Theorem 7.4] that S∗ν := S∗ν,1∪S∗ν,2 contains the simple
part of the antiperiodic spectrum.
Lemma 2.3. Let m = 2ν, ν ∈ Z≥0. If λ belongs to the antiperiodic spectrum of the Lamé
operator (2.1) and has multiplicity 1, then λ ∈ S∗ν .
Since the spectrum of the Lamé operator (2.1) has precisely m = 2ν gaps, the ends of the
gaps must be given by the m largest eigenvalues in Sν and all m eigenvalues in S∗ν . (Recall that
(−∞, λ0) is not counted as a gap.)
We are now ready to complete the proof of Theorem 2.1 for even m. From (2.10)–(2.11) and
(2.16)–(2.17) it easily follows that
δν,j(k) =
ν∏
n=j−1
(
λ− (2n)2
)
+O
(
k2
)
, k → 0,
where j = 1, 2. Similarly, (2.14)–(2.15) and (2.18)–(2.19) readily yield
δ∗ν,j(k) =
ν−1∏
n=0
(
λ− (2n+ 1)2
)
+O
(
k2
)
, k → 0.
This implies that the m gaps close at the points λ = j2, j = 1, . . . ,m, as k → 0. On the
other hand, taking k → 0 in (2.3), we obtain the free Schrödinger equation −ψ′′ = λψ, and
(excluding the constant solution ψ = 1 with λ = 0) the m smallest values of λ for which it
has π-periodic or π-antiperiodic solutions are precisely λ = j2, j = 1, . . . ,m. Using that the
eigenvalues λ = λ(k) ∈ Sν ∪ S∗ν have multiplicity one, a standard argument shows that they
depend analytically on k, see e.g. Chapter XII in [24]. Hence there exists k1 ∈ (0, 1) such that
all points of coexistence are indeed contained in the infinite spectral band for k ∈ (0, k1).
There remains to show that neither of the following two scenarios can occur for k ≥ k1: first,
a point of coexistence crosses one or more open gaps; and, second, an open gap closes while
a point of coexistence turns into an open gap. The first scenario is immediately ruled out by
the interlacing property (2.2), and the second by the analyticity of the eigenvalues.
This concludes the proof in the case of even m. It is straightforward to adapt the arguments
above to handle the case of odd m. In particular, the roles of the periodic and antiperiodic
solutions will then be interchanged, and analogues of Lemmas 2.2–2.3 can once more be obtained
from [19, Theorems 7.3–7.4].
Example 2.4. Consider the simplest nontrivial even case m = 2, which corresponds to the
Lamé operator
L2 = − d2
dx2
+ 6k2 sn2(x, k). (2.20)
On the Spectra of Real and Complex Lamé Operators 9
The edges of its two spectral gaps (λ′1, λ
′
2) and (λ1, λ2), which correspond to the first two
eigenvalues λ′1, λ
′
2 from its antiperiodic spectrum and the second and third eigenvalues λ1, λ2
from its periodic spectrum, are easily computed explicitly.
Indeed, observing that
δ∗1,1 = P ∗(0) + Λ∗0, δ∗1,2 = −P ∗(0) + Λ∗0,
we find that the edges of the first gap are given by
λ′1 = 1 + k2, λ′2 = 1 + 4k2, (2.21)
cf. (2.14)–(2.15). To determine the edges of the second gap, we should find the roots λ of the
polynomials
δ1,1 =
∣∣∣∣ Λ0 P−1
2P0 Λ1
∣∣∣∣ , δ1,2 = Λ1.
Recalling (2.10)–(2.11), we deduce that the two roots of the former polynomial are
λ0 = 2
(
1 + k2 −
√
1− k2 + k4
)
, λ2 = 2
(
1 + k2 +
√
1− k2 + k4
)
, (2.22)
whereas the latter has the single root
λ1 = 4 + k2. (2.23)
The relevant eigenvalues are plotted together with the ground state eigenvalue from the
periodic spectrum in Fig. 3.
In the limit k → 1 the real period goes to infinity and the Lamé potential
6k2 sn2(x, k)→ 6− 6 sech2 x.
The potential u(x) = −6 sech2 x decays exponentially at infinity and is a special example of
a 2-soliton potential, see, e.g., [6]. The corresponding operator
H2 = − d2
dx2
− 6 sech2 x
is known to be reflectionless. Its spectrum has continuous part [0,∞) and discrete part consisting
of the two eigenvalues −1 and −4. Note that for k → 1 the formulae (2.21)–(2.23) show that
the two finite spectral bands degenerate to the eigenvalues λ0 = λ′1 = 2 and λ′2 = λ1 = 5, while
λ2 → 6, in agreement with Fig. 3.
3 The spectrum of the complex Lamé operator
Let E = C/L be a general elliptic curve, where L is a period lattice. To begin with, we do not
impose any reality conditions. Let ℘(z) be the corresponding Weierstrass’ elliptic function,
℘(z + Ω) = ℘(z), Ω ∈ L,
which satisfies the equation
(℘′)2 = 4(℘− e1)(℘− e2)(℘− e3),
see, e.g., [31]. It has poles of second order at all lattice points, which are the only singularities
of ℘(z).
10 W.A. Haese-Hill, M.A. Hallnäs and A.P. Veselov
0.0 0.2 0.4 0.6 0.8 1.0
0
1
2
3
4
5
6
k
λ
(k
)
λ0
λ
1
'
λ
2
'
λ1
λ2
Figure 3. First five eigenvalues from the periodic and antiperiodic spectra of the Lamé operator (2.20).
Let ω be a half-period: 2ω ∈ L, and consider the complex Lamé operator L = L(E , ω,m, z0)
in L2(R) given by
L = − d2
dx2
+m(m+ 1)ω2℘(ωx+ z0), (3.1)
with m ∈ N and z0 ∈ C chosen such that the line z = ωx + z0, x ∈ R, does not contain any
points of L. Note that the potential m(m+1)ω2℘(ωx+z0) is regular and periodic with period 2,
but is in general complex-valued.
From the work of Rofe-Beketov [26] on the spectral theory of non-self-adjoint differential
operators in L2(R) with complex-valued periodic coefficients, it follows that the spectrum of L
can be characterised as the set of λ ∈ C such that the associated Lamé equation
−d2φ
dx2
+m(m+ 1)ω2℘(ωx+ z0)φ = λφ, x ∈ R, (3.2)
has a non-zero bounded solution. Equivalently, the corresponding Floquet multiplier µ(λ),
defined by
φ(x+ 2, λ) = µ(λ)φ(x, λ),
must have absolute value |µ(λ)| = 1.
Proposition 3.1. The spectrum of the complex Lamé operator (3.1) does not depend on the
value of z0 and is determined by the condition (3.6) below.
Proof. We will use the classical results about the Lamé equation
−d2ψ
dz2
+m(m+ 1)℘(z)ψ = λψ (3.3)
in the complex domain z ∈ C. Its solutions were described by Hermite [13] in the following way,
see, e.g., [31, Section 23.7]. First, one notes that the product X of any pair of solutions of (3.3)
satisfies the third order equation
−d3X
dz3
+ 4
(
m(m+ 1)℘(z)− λ
)dX
dz
+ 2m(m+ 1)℘′(z) = 0.
On the Spectra of Real and Complex Lamé Operators 11
Changing variable to
ξ = ℘(z)
yields the algebraic form of this equation:
−4(ξ − e1)(ξ − e2)(ξ − e3)
d3X
dξ3
− 3
(
6ξ2 − g2/2
)d2X
dξ2
+ 4((m2 +m− 3)ξ − λ)
dX
dξ
− 2m(m+ 1)X = 0.
Making a power series ansatz in ξ − e2 (say), it is straightforward to verify that the Lamé
equation (3.3) has two solutions ψ± whose product X is a polynomial in ξ − e2 = ℘(z) − e2 of
the form
X(ξ) = (ξ − e2)m +
m∑
r=1
cr(ξ − e2)m−r,
with some coefficients cr ∈ C. This polynomial can be written in the factorised form
X(ξ) =
m∏
j=1
(ξ − ℘(kj)),
where the kj = kj(λ) ∈ C are determined by this relation up to permutation and up to a sign.
Assume now that ψ± are linearly independent, so that their Wronskian
W
(
ψ+, ψ−
)
≡ C 6= 0. (3.4)
(Otherwise ψ± are one of the 2m + 1 Lamé functions.) The sign of kj can then be fixed by
requiring
dX
dξ
ξ=℘(kj)
=
C
℘′(kj)
.
Finally, by solving (3.4) and d(ψ+ψ−)/dz = dX/dz for d logψ±/dz and integrating, one finds
that (3.3) is satisfied by the two functions
ψ± =
m∏
j=1
(
σ(z ± kj)
σ(z)σ(kj)
)
e
∓
m∑
j=1
ζ(kj)z
. (3.5)
Restricting (3.5) to the line z = ωx + z0 and using the quasi-periodicity of the σ- and ζ-
function we see that (3.2) has the Floquet solutions with Floquet multiplier µ = e2fm ,
fm(λ) = ζ(ω)
m∑
j=1
kj − ω
m∑
j=1
ζ(kj).
Since fm is manifestly independent of z0, it follows that the spectrum of the complex Lamé
operator (3.1), determined by the condition
Re[fm(λ)] = 0, (3.6)
is indeed independent of the value of z0. �
12 W.A. Haese-Hill, M.A. Hallnäs and A.P. Veselov
Unfortunately, the condition (3.6) is very difficult to analyse in general, so we will now focus
on the simplest non-trivial case m = 1. Hermite’s solutions of (3.3) have the form (1.3):
ψ(z, k) =
σ(z + k)
σ(z)σ(k)
exp(−ζ(k)z),
with parameter k related to λ by λ = −℘(k). Restriction to the line z = ωx + z0 gives the
solutions to (3.2) with
λ = −ω2℘(k)
and Floquet multiplier given by
f(k) = ηk − ωζ(k).
In particular, the corresponding solutions are bounded for x ∈ R if and only if
u(k) := Re[f(k)] = 0. (3.7)
To study the solutions of this transcendental equation we will use elementary Morse theory
arguments, see, e.g., [20].
Lemma 3.2. Assuming that
η + ωej 6= 0, j = 1, 2, 3, (3.8)
u(k) is a Morse function on E×, with critical points ±k∗ given by
η + ω℘(k∗) = 0.
Moreover, the index of u(k) at k = ±k∗ is equal to one.
Proof. Given that ω is a half-period, there exist ν1, ν3 ∈ Z such that
ω = ν1ω1 + ν3ω3.
By the quasi-periodicity of ζ(z), we have
u(k + 2ωj)− f(k) = νl Re[ηlωj − ηjωl], l 6= j.
Hence the Legendre relation η1ω3 − η3ω1 = iπ/2 ensures that the right-hand side equals zero,
so that u(k) is a well-defined function on E×.
Writing k = s + it, the critical points of u(k) are the solutions of us = ut = 0, which are
equivalent to
df(k)
dk
=
d(ηk − ωζ(k))
dk
= η + ω℘(k) = 0.
Since ℘(k) is an even function of order two, this equation has precisely two solutions k = ±k∗.
Letting
v(k) = Im[f(k)],
we infer from the Cauchy–Riemann equations that the Hessian matrix
H(u) =
[
uss vtt
vtt −uss
]
,
On the Spectra of Real and Complex Lamé Operators 13
which has eigenvalues
λ± = ±
√
u2ss + v2tt.
We recall that u(k) is a Morse function if H(u) is non-singular at all of its critical points, with
the index being equal to the number of negative eigenvalues. Clearly, H(u) is singular if and
only if uss = vtt = 0 or, equivalently,
d2f(k)
dk2
=
d2(ηk − ωζ(k))
dk2
= ω℘′(k) = 0.
Recalling that the only zeros of ℘′(k) on E× are k = ωj , j = 1, 2, 3, we see that H(u) is singular
only if η + ωej = 0 for some j = 1, 2, 3, which is excluded by the assumption (3.8). �
Using Morse theory, we can thus determine the topological nature of the zero set (3.7). We
assume that (3.8) holds true, so that the critical points of u(k) are non-degenerate, and that the
level set u(k) = 0 is non-singular in the sense that
u(k∗) 6= 0. (3.9)
We note that the lemniscatic case yields an example of a singular level set and the rhombic cases
contain singular examples as well as a degenerate level set, see Figs. 5 and 2.
Proposition 3.3. Under our non-degeneracy and non-singularity assumptions (3.8) and (3.9),
the closure Z̄u ⊂ E of the zero set
Zu =
{
k ∈ E× : u(k) = 0
}
consists of two simple closed curves in E representing the same (non-trivial) homology class.
Proof. By our non-singularity assumption, we can fix the sign of k∗ such that
c∗ := u(k∗) > 0.
For c ∈ R, we let Ec = E×c ⊂ E be the closure of the set
E×c :=
{
k ∈ E× : −∞ < u(k) ≤ c
}
(where, as we shall see below, taking the closure amounts to adding the point k = 0).
First, we prove that each Ec with c < −c∗ is diffeomorphic to a disc. Since the meromorphic
function f(k) = ηk− ωζ(k) is odd and has a simple pole at k = 0 with residue −ω, we can find
neighbourhoods U, V 3 0 and a biholomorphic mapping g : U → V such that
1) g(0) = 0 and g′(0) = 1,
2) g(−k) = −g(k), k ∈ U ,
3) f
(
g−1(z)
)
= −ω/z, z ∈ V .
Letting
ω = α+ iβ, z = x+ iy,
the set Re[−ω/z] ≤ c, z 6= 0, is given by(
x+
α
2c
)2
+
(
y +
β
2c
)2
≤ α2 + β2
4c2
, (3.10)
14 W.A. Haese-Hill, M.A. Hallnäs and A.P. Veselov
where (x, y) 6= (0, 0). By choosing c′ < −c∗ sufficiently small, we can ensure that the disc (3.10)
is contained within V and therefore that it is diffeomorphic to Ec′ . For c′ < c < −c∗, we
infer from [20, Theorem 3.1] that Ec is diffeomorphic to Ec′ , and the assertion follows. (To be
precise, the theorem does not apply as it stands, since u(k) is not defined at k = 0. However,
〈gradu, gradu〉−1 gradu extends to a smooth vector field on a neighbourhood of Ec, c < −c∗,
and the construction of the diffeomorphisms ϕt : E → E is readily adapted to the present case.)
Letting 0 < ε < c∗, the region u−1([−c∗ − ε,−c∗ + ε]) contains no critical point of u other
than −k∗. Since the index of u(k) at k = −k∗ is one, it follows that E−c∗+ε is diffeomorphic
to a disc with a thickened 1-cell attached, cf. [20, Theorem 3.2]. Moreover, increasing c from
−c∗ + ε to 0 does not alter the diffeomorphism type of Ec. Considering the boundary of E0, we
thus arrive at the statement. �
This leads us to the following result, which can also be extracted from [1, Theorem 1.1 and
Example C.1].
Proposition 3.4. Under our assumptions of non-degeneracy (3.8) and non-singularity (3.9),
the spectrum of the complex Lamé operator (3.1) with m = 1 consists of two regular analytic
arcs. Moreover, precisely one arc extends to infinity and the remaining endpoints are −ω2ej,
j = 1, 2, 3.
Proof. By Proposition 3.3, there exist simple closed curves γ1, γ2 such that
Z̄u = γ1 ∪ γ2.
For each j = 1, 2, 3, we deduce from the Legendre relation that ωj ∈ Zu. In addition, Z̄u contains
k = 0 and is invariant under the reflection k → −k, which implies that each curve γj contains
precisely two fixed points. Since the quotient map
E → E/(k → −k) = CP1
is given by k 7→ ℘(k), the image of Z̄u in CP1 under the map k 7→ λ = −ω2℘(k) consists of two
regular analytic arcs connecting two pairs of fixed points. �
The next proposition, which can also be extracted from [1], describes the asymptotic be-
haviour of the infinite spectral arc.
Proposition 3.5. For m = 1, the spectral arc of the complex Lamé operator (3.1) that extends
to infinity is asymptotic to the positive real line shifted by −2ηω.
Proof. Recalling from the proof of Proposition 3.3 the biholomorphic mapping g : U → V , we
observe that any z ∈ V satisfying the spectral condition u
(
g−1(z)
)
= Re[−ω/z] = 0 is of the
form
z = iωs, s 6= 0.
By Properties (1)–(3) of g, we have
g−1(z) = z − η
ω
z3 +O
(
z5
)
.
This implies
λ = −ω2℘
(
g−1(iωs)
)
=
1
s2
− 2ηω +O
(
s2
)
, s→ 0,
from which the assertion clearly follows. �
On the Spectra of Real and Complex Lamé Operators 15
For the remainder of this section, we restrict attention to real elliptic curves E given by
y2 = 4x3 − g2x− g3, g2, g3 ∈ R.
There are two types of such curves, depending on the sign of the discriminant
∆ = g32 − 27g23.
If ∆ > 0, the roots e1, e2, e3 of the polynomial p(x) = 4x3 − g2x − g3 are all real, the
corresponding real curve consists of two ovals (M -curve) and the period lattice L is rectangular.
If ∆ < 0, we have one real root e1 and a pair e2, e3 of complex conjugate roots, e3 = e2. The
corresponding real curve consists of one oval and the period lattice L is rhombic.
We first consider the rectangular case, with basic half-periods
ω1 ∈ (0,∞), ω3 ∈ i(0,∞). (3.11)
The simplest instance of the corresponding Lamé operator (3.1) that is truly complex is obtained
by choosing
ω = ω2 = ω1 + ω3.
For this choice of ω, we have used the software R to numerically compute the spectrum of (3.1)
for g3 = 1 and four different values of g2. The results are displayed in Fig. 4.
We note that the (anti-)periodic solutions of (3.2) are given by k ∈ Zu satisfying
v(k) := Im[f(k)] = p
π
2
, p ∈ Z. (3.12)
By the Legendre relation, v(ω3) = −v(ω1) = π/2 and v(ω2) = 0. Since λ = −ω2℘(ωj), j = 1, 2, 3,
at the endpoints of the spectral arcs, the remaining values of p must correspond to points
contained in one of the two spectral arcs. Following standard terminology for the real Lamé
operator, we call such a point in the spectrum a closed spectral gap. In analogy with Theorem 2.1,
we have the following result.
Theorem 3.6. For m = 1, ω = ω2 and ω1, ω3 of the form (3.11), all closed spectral gaps of the
complex Lamé operator (3.1) are located on the spectral arc extending to infinity.
Our proof of this result relies on a detailed analysis of the lemniscatic case. The corresponding
set Z̄u and spectrum of (3.1) are shown in Fig. 5. The curve k = (1 − i)ω1s, −1 ≤ s ≤ 1, and
spectral arc [0,∞) are deduced analytically in the proof of Proposition 3.7 below, whereas the
remaining curve and spectral arc were obtained numerically using the software R.
Proposition 3.7. Fix ω1 ∈ (0,∞) and let ω3 = iω1 and ω = (1 + i)ω1. Then the spectrum of
the complex Lamé operator (3.1) consists of two analytic arcs, intersecting at the interior point
λ∗ = η2ω2. The spectrum is invariant under complex conjugation and the infinite spectral arc
coincides with [0,∞). Moreover, the tangent lines to the finite spectral arc at its endpoints
λ± = ±iΓ
4(1/4)
16π
(3.13)
are
l±(t) = λ± +
(
Γ4(1/4)
8π2
± i
)2
t, t ∈ R. (3.14)
16 W.A. Haese-Hill, M.A. Hallnäs and A.P. Veselov
Figure 4. Spectra of the complex Lamé operator (3.1) for rectangular period lattices L, m = 1 and
ω = ω2.
Proof. For the lemniscatic ζ- and ℘-function, we recall the identities
ζ(iz) = −iζ(z), ℘(iz) = −℘(z), (3.15)
and special values
η1 = iη3 =
π
4ω1
, e1 = −e3 =
Γ4(1/4)
32πω2
1
, e2 = 0, (3.16)
see, e.g., [22, Section 23.5(iii)].
We note that the spectrum is given by
λ = −ω2
2℘(k) = −2ω2
1i℘(k), k ∈ Zu.
From (3.15)–(3.16), we infer
λ̄ = −2ω2
1i℘(ik̄)
and
u(k) = Re
[
π
4ω1
(1− i)k − (1 + i)ω1ζ(k)
]
= u(ik̄),
On the Spectra of Real and Complex Lamé Operators 17
Figure 5. The set Z̄u, where k = aω1 + bω3, and the spectrum of the complex Lamé operator (3.1) in
the lemniscatic case with m = 1 and ω = ω2.
which clearly implies invariance of the spectrum under complex conjugation. Using the first
identity in (3.15), we find that f((1− i)s) = −f((1− i)s), s ∈ R, which means that
k = (1− i)ω1s ∈ Z̄u, −1 ≤ s ≤ 1.
Since e2 = 0 and λ̄ = λ whenever k = (1− i)ω1s, 0 < |s| ≤ 1, Proposition 3.5 ensures that [0,∞)
coincides with the spectral arc extending to infinity.
The expression (3.13) for the endpoints of the finite spectral arc follows directly from (3.16)
and Proposition 3.4. Following the same line of reasoning as in the proof of Proposition 3.5, and
using standard power series expansions of ζ(z) and ℘(z) around z = ωj , we find parameterisa-
tions such that
λ(t) = −ω2
2ej +
(
3e2j − g2/4
)(
η2ω2 + ej |ω2|2
)2
t+O
(
t2
)
, t→ +0.
(In fact, this requires no reality assumptions on L and any half-period ω and corresponding η
can be substituted for ω2 and η2, respectively.) Specialising to the lemniscatic case, we readily
deduce (3.14).
The fact that the intersection point λ∗ = −ω2℘(k∗) = η2ω2 is not an endpoint follows from
explicit expressions for η2, ω2 and ej , see, e.g., [22]. �
We turn now to the proof of Theorem 3.6. Since Z̄u is invariant under the reflection k → −k,
we may and shall restrict attention to that part of the fundamental period-paralellogram given
by Re k ≥ Im k. From (3.12) and the Legendre relation, we infer that
v(ω2) = 0, v(ω1) = v(−ω3) =
π
2
. (3.17)
Given that k = ±k∗ are the only critical points of f(k), the function v(k) must be monotone on
each component of Zu. Moreover, since f ′(k∗) = 0 and f ′′(k∗) 6= 0, we can find neighbourhoods
U 3 k∗, V 3 0 and a biholomorphic mapping g : U → V such that
f
(
g−1(z)
)
= f(k∗) + z2, z ∈ V,
which yields
u
(
g−1(z)
)
= x2 − y2, v
(
g−1(z)
)
= Im[f(k∗)] + 2xy, z = x+ iy ∈ V.
18 W.A. Haese-Hill, M.A. Hallnäs and A.P. Veselov
Hence, in the neighbourhood U of k = k∗, the zero set Z̄u is given by x = ±y. From these
observations and (3.17), we infer that v(k) is strictly decreasing and increasing as k → k∗ along
the curve connecting 0, ω̄2 and ω1, −ω3, respectively, and that 0 < v(k∗) < π/2. In other words,
all remaining solutions of (3.12) are located on the component extending from k = k∗ towards
k = 0.
This result is readily generalised to all τ := ω3/ω1 ∈ i(0,∞). Indeed, when we alter the value
of τ from τ = i two things can happen: either the two curves remain intersecting or they break
up into two disconnected curves. In the former case we can follow the same line of reasoning
as in the lemniscatic case, and in the latter case we need only note that the closed spectral
gaps cannot move from one curve to the other, since their locations depend continuously on τ .
Applying the mapping k → λ = −ω2℘(k), we thus conclude the proof of Theorem 3.6.
Remark 3.8. We believe that the intersection of the spectral arcs in the rectangular case
with ω = ω2 takes place only in the lemniscatic case, making it the only case in which the
non-degeneracy conditions are violated, but we do not have a rigorous proof of this.
Consider now the case ∆ < 0, corresponding to a rhombic period lattice L. We choose basic
half-periods ω1, ω3 of the form
ω1 ∈ (0,∞), Reω3 =
1
2
ω1, Imω3 > 0, (3.18)
so that ω3 = ω1 − ω3.
Using the software R, we have in Fig. 2 plotted the spectrum of the complex Lamé opera-
tor (3.1) in four different rhombic cases, see Fig. 6 for the corresponding sets Z̄u and period
lattices L. These four cases exemplify the different types of spectra that occur for rhombic
period lattices. In the lower left plot, we have the pseudo-lemniscatic case; and the upper left
and lower right plots correspond to the two real forms of the complex equianharmonic curve.
Proposition 3.9. In the rhombic case (3.18) the spectrum of the complex Lamé operator (3.1)
with ω = ω1 is invariant under λ → λ̄, and the spectral arc extending to infinity coincides
with [−ω2
1e1,∞). Moreover, all closed spectral gaps are located on the spectral arc extending to
infinity.
Proof. Recalling that the spectrum is given by
λ = −ω2
1℘(k), k ∈ Zu,
invariance under complex conjugation follows immediately from
ζ(k̄) = ζ(k), ℘(k̄) = ℘(k).
Combining these identities with 2ω3 − ω1 ∈ i(0,∞), we see that
k = (2ω3 − ω1)s ∈ Z̄u, −1 ≤ s ≤ 1.
Since λ̄ = λ for k = (2ω3 − ω1)s, 0 < s ≤ 1, Proposition 3.5 and 2ω3 − ω1 = ω1 (mod L) imply
that [−ω2
1e1,∞) coincides with the spectral arc extending to infinity.
Observing that the spectrum in the pseudo-lemniscatic case ω1 ∈ (0,∞) and 2ω3 = (1 + i)ω1
(see the lower left plot in Fig. 2) is identical to that in the lemniscatic case, the proof of
Theorem 3.6 is readily adapted to the rhombic case. �
Now let us study the corresponding non-degeneracy conditions
η1 + ω1ej 6= 0, j = 1, 2, 3.
On the Spectra of Real and Complex Lamé Operators 19
Figure 6. The set Z̄u for rhombic period lattices L, m = 1 and ω = ω1.
Theorem 3.10. In the rhombic case with m = 1 the non-degeneracy conditions are violated for
exactly one exceptional curve E∗ uniquely determined by the condition
η1 + ω1e1 = 0. (3.19)
The corresponding spectrum has a tripod structure with three simple analytic arcs joined at 2π/3
angles. The same curve is the bifurcation point for the non-singularity condition (3.9), separating
the cases with intersecting and non-intersecting spectral arcs.
Proof. First of all, since both η1 and ω1 are real the degeneracy condition η1+ω1ej = 0 implies
that the corresponding ej must be real too, so ej = e1.
Recall now that 2ω and 2η are the complete first and second kind elliptic integrals, which
can be also written as integrals
2ω =
∮
γ
dE√
4(E − e1)(E − e2)(E − e3)
, 2η = −
∮
γ
EdE√
4(E − e1)(E − e2)(E − e3)
over the corresponding closed contour γ in the complex domain. In our case the contour γ is
going around e1 and ∞, or, equivalently, around the two complex conjugated roots e2 and e3.
The degeneracy condition (3.19) can be rewritten then as∮
γ
(E − e1)dE√
4(E − e1)(E − e2)(E − e3)
= 0, (3.20)
which also appeared in [1].
20 W.A. Haese-Hill, M.A. Hallnäs and A.P. Veselov
We have to show that there exists exactly one such curve. It would be convenient to use the
parametrisation of the rhombic curves by fixing e2 = i, e3 = −i and e1 = t ∈ R (Weierstrass’s
restriction e1 + e2 + e3 = 0 is violated, but this is not essential). We have to show that Φ(t) = 0
has the only real solution t = t∗, where
Φ(t) :=
∮
γ
(E − t)dE√
4(E − t)(E − i)(E + i)
=
∮
γ
√
E − t
4(E2 + 1)
dE.
Its derivative has the form
d
dt
Φ(t) = −1
2
∮
γ
dE√
4(E2 + 1)(E − t)
,
which is clearly negative since the integrand is positive on the half-line E ∈ (t,∞). Thus Φ(t)
is strictly monotonically decreasing on the whole real line, which proves the uniqueness of the
solution t∗. To show the existence it is enough to check that Φ(−2) > 0 and Φ(2) < 0 (e.g.,
numerically).
Numerical calculations give the value of the j-invariant
j = 1728
g32
g32 − 27g23
of the corresponding elliptic curve to be j∗ ≈ 243.797. One can see the corresponding period
lattice L∗ in the top right corner of Fig. 6.
The tripod structure of the corresponding spectrum follows from a very general result by
Batchenko and Gesztesy [1, Theorem 1.1(v)]. Its image is shown in the top right corner of
Fig. 2.
The equal angles at the intersection point, which is also a general result due to Batchenko
and Gesztesy [1], is readily verified. Indeed, λ is contained in the spectrum if and only if
τ(λ) = trM(λ) ∈ [−2, 2] ⊂ C, where M(λ) is the monodromy matrix. Since M(λ) depends
analytically on λ, we have locally
τ(λ) = a0 + (λ− λ0)k + · · ·
for some k ∈ N. If a0 = ±2, which corresponds to λ0 being an end point of the spectral arc, we
have an k-pod with angles 2π/k, see e.g. tripod in Fig. 2. If a0 ∈ (−2, 2), then we have k arcs
intersecting at λ0 with angles between neighbouring arcs being π/k, see, e.g., the k = 2 cases in
the bottom two plots in Fig. 2.
To study the non-singularity condition recall that the critical points k∗ of the function u(k)
are given by η+ω℘(k∗) = 0. This means that the corresponding value of λ(k∗) = −ω2℘(k∗) = ηω
is real in this case. The non-singularity condition corresponds to the case when this value does
not belong to the spectrum. Since the real part of the spectrum is (−ω2e1,∞) it holds iff
ηω < −ω2e1, which is equivalent to η + ωe1 < 0. �
4 A qualitative analysis of the m = 2 case
Consider now briefly what happens in the case m = 2. The spectral curve of the corresponding
Lamé operator
L = − d2
dx2
+ 6ω2℘(ωx+ z0), (4.1)
On the Spectra of Real and Complex Lamé Operators 21
has the form
w2 =
(
z2 − 3g2
)(
4z3 − 9g2z − 27g3
)
. (4.2)
We restrict the attention to the rhombic case when ∆ = g32 − 27g23 < 0 and ω = ω1. Setting
g3 = 1, this corresponds to g2 ∈ (−∞, 3).
For g2 ∈ (0, 3), the roots of the first factor in the right-hand side of (4.2)
z2 − 3g2 = 0
yield two real endpoints λ = ω2z = ±ω2
√
3g2, whereas the second factor
4z3 − 9g2z − 27g3 = 0
gives one real positive endpoint as well as a complex conjugate pair of endpoints.
Hence we expect that for g2 ∈ (0, 3) the spectrum of (4.1) consists of two real intervals, one
of which extends to infinity, and one arc connecting the two complex endpoints. Moreover, as
g2 → 0, the finite interval collapses to a double point. We can see this in the first two plots
in Fig. 7, produced numerically using the software R. Note that in the special case g2 = 0, the
spectrum consists of only two arcs.
For g2 ∈ (−∞, 0), the first factor yields two purely imaginary endpoints λ = ±iω2
√
−3g2,
which suggests a spectrum consisting of two arcs each connecting two endpoints in the upper-
and lower- half-plane, respectively, and a real interval extending to infinity (see the last plot in
Fig. 7).
5 Concluding remarks
The spectral theory of non-self-adjoint operators, in particular Schrödinger operators with com-
plex potentials, has attracted significant attention in the last two decades, see, e.g., [4, 5, 30].
Nevertheless, it is fair to say that it is still a far less complete theory than in the self-adjoint
case. In particular, we feel there is a need for more examples, which can be studied explicitly
(like the complex harmonic oscillator in [4]).
In our paper we used the example of the classical Lamé equation, whose general solution was
found by Hermite in the 19th century. This example was already briefly discussed in a much
wider framework in [1, 9], but we believe that the complete picture was missing even in the
simplest m = 1 case. We used Hermite’s explicit formulae and Morse theory ideas to study
the structure of the spectrum of complex Lamé operators for two different types of real elliptic
curves with rectangular and rhombic lattices. The rigorous analysis of higher m cases seems
to be difficult since the explicit formulae quickly become quite complicated as m grows, see
[2, 11, 12].
A closely related interesting problem is to study the spectrum of the difference version of the
Lamé operator
L =
θ1(x−mη)
θ1(x)
T η +
θ1(x+mη)
θ1(x)
T−η,
where T η is the shift operator defined by T ηψ(x) = ψ(x+η), and θ1(x, τ) is the odd Jacobi theta
function (see [32] and references therein). When η = p
q is rational we have an operator with
periodic coefficients, in general complex. We expect that the geometry of the corresponding
spectrum (in particular, the band structure in the real case) depends on the arithmetic of p
and q.
22 W.A. Haese-Hill, M.A. Hallnäs and A.P. Veselov
Figure 7. Different types of spectra of the complex Lamé operator (4.1) with g3 = 1, depending on the
sign of g2 ∈ (−∞, 3).
Acknowledgements
We are grateful to Jenya Ferapontov, John Gibbons and Anton Zabrodin for very useful and
encouraging discussions, and especially to Boris Dubrovin, who many years ago asked one of
us (APV) about the position of open gaps in the spectra of Lamé operators. We would like
to thank Professor Gesztesy for his interest in our work and for pointing out further relevant
references, including [1] and [9]. The work of WAH was partially supported by the Department
of Mathematical Sciences at Loughborough University as part of his PhD studies.
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1 Introduction
2 Spectral gaps for the real Lamé operator
3 The spectrum of the complex Lamé operator
4 A qualitative analysis of the m=2 case
5 Concluding remarks
References
|
| id | nasplib_isofts_kiev_ua-123456789-148577 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T17:49:29Z |
| publishDate | 2017 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Haese-Hill, W.A. Hallnäs, M.A. Veselov, A.P. 2019-02-18T16:10:46Z 2019-02-18T16:10:46Z 2017 On the Spectra of Real and Complex Lamé Operators
 / W.A. Haese-Hill, M.A. Hallnäs, A.P. Veselov // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 32 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 34L40; 47A10; 33E10 DOI:10.3842/SIGMA.2017.049 https://nasplib.isofts.kiev.ua/handle/123456789/148577 We study Lamé operators of the form
 L=−d²/dx²+m(m+1)ω²℘(ωx+z₀),
 with m∈N and ω a half-period of ℘(z). For rectangular period lattices, we can choose ω and z0 such that the potential is real, periodic and regular. It is known after Ince that the spectrum of the corresponding Lamé operator has a band structure with not more than m gaps. In the first part of the paper, we prove that the opened gaps are precisely the first m ones. In the second part, we study the Lamé spectrum for a generic period lattice when the potential is complex-valued. We concentrate on the m=1 case, when the spectrum consists of two regular analytic arcs, one of which extends to infinity, and briefly discuss the m=2 case, paying particular attention to the rhombic lattices. We are grateful to Jenya Ferapontov, John Gibbons and Anton Zabrodin for very useful and
 encouraging discussions, and especially to Boris Dubrovin, who many years ago asked one of
 us (APV) about the position of open gaps in the spectra of Lam´e operators. We would like
 to thank Professor Gesztesy for his interest in our work and for pointing out further relevant
 references, including [1] and [9]. The work of WAH was partially supported by the Department
 of Mathematical Sciences at Loughborough University as part of his PhD studies. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications On the Spectra of Real and Complex Lamé Operators Article published earlier |
| spellingShingle | On the Spectra of Real and Complex Lamé Operators Haese-Hill, W.A. Hallnäs, M.A. Veselov, A.P. |
| title | On the Spectra of Real and Complex Lamé Operators |
| title_full | On the Spectra of Real and Complex Lamé Operators |
| title_fullStr | On the Spectra of Real and Complex Lamé Operators |
| title_full_unstemmed | On the Spectra of Real and Complex Lamé Operators |
| title_short | On the Spectra of Real and Complex Lamé Operators |
| title_sort | on the spectra of real and complex lamé operators |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/148577 |
| work_keys_str_mv | AT haesehillwa onthespectraofrealandcomplexlameoperators AT hallnasma onthespectraofrealandcomplexlameoperators AT veselovap onthespectraofrealandcomplexlameoperators |