Eigenvalue Problems for Lamé's Differential Equation
The Floquet eigenvalue problem and a generalized form of the Wangerin eigenvalue problem for Lamé's differential equation are discussed. Results include comparison theorems for eigenvalues and analytic continuation, zeros, and limiting cases of (generalized) Lamé-Wangerin eigenfunctions. Algebr...
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| description | The Floquet eigenvalue problem and a generalized form of the Wangerin eigenvalue problem for Lamé's differential equation are discussed. Results include comparison theorems for eigenvalues and analytic continuation, zeros, and limiting cases of (generalized) Lamé-Wangerin eigenfunctions. Algebraic Lamé functions and Lamé polynomials appear as special cases of Lamé-Wangerin functions.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 14 (2018), 131, 21 pages
Eigenvalue Problems for Lamé’s Differential Equation
Hans VOLKMER
Department of Mathematical Sciences, University of Wisconsin-Milwaukee,
P.O. Box 413, Milwaukee, WI, 53201, USA
E-mail: volkmer@uwm.edu
Received August 14, 2018, in final form December 06, 2018; Published online December 12, 2018
https://doi.org/10.3842/SIGMA.2018.131
Abstract. The Floquet eigenvalue problem and a generalized form of the Wangerin eigen-
value problem for Lamé’s differential equation are discussed. Results include comparison
theorems for eigenvalues and analytic continuation, zeros and limiting cases of (generalized)
Lamé–Wangerin eigenfunctions. Algebraic Lamé functions and Lamé polynomials appear
as special cases of Lamé–Wangerin functions.
Key words: Lamé functions; singular Sturm–Liouville problems; tridiagonal matrices
2010 Mathematics Subject Classification: 33E10; 34B30
1 Introduction
The Lamé equation (Arscott [1, Chapter IX]) is
d2w
dz2
+
(
h− ν(ν + 1)k2 sn2(z, k)
)
w = 0, (1.1)
where sn(z, k) is the Jacobian elliptic function with modulus k ∈ (0, 1) (Whittaker and Watson
[22, Chapter XXII]), ν ∈ R and h is the eigenvalue parameter. This equation has regular
singularities at the points z = 2mK + i(2n + 1)K ′, where m, n are integers and K = K(k)
and K ′ = K ′(k) denote complete elliptic integrals. Various eigenvalue problems for the Lamé
equation have been treated in the literature.
The Lamé equation is an even Hill’s equation with fundamental period 2K. The theory of
Hill’s equation is well-known; see, e.g., Arscott [1], Eastham [3] and Magnus and Winkler [14].
Results on the periodic eigenvalue problem specific for the Lamé equation with eigenfunctions
satisfying w(z + 2K) = ±w(z) can be found in [6, Section 15,5], [9, 10] and [18, Chapter 29].
These functions have many applications; see, e.g., [2]. In Section 2 of this paper we will consider
the more general Floquet eigenvalue problem w(z+2K) = eiµπw(z). For the general Hill’s equa-
tion this eigenvalue problem is treated in Eastham [3]. Some results on the Floquet eigenvalue
problem specific for the Lamé equation can be found in Ince [8, Sections 7 and 8].
Wangerin [21] showed that Lamé’s equation appears when Laplace’s equation is separated
in confocal cyclidic coordinates of revolution. Such coordinate systems can be found in Moon
and Spencer [17] and in Miller [16]. They include flat-ring, flat-disk, bi-cyclide and cap-cyclide
coordinates. An outline of Wangerin’s results is given in [6, Section 15.1.3]. In order to obtain
harmonic functions relevant for applications special solutions of the Lamé equation called Lamé–
Wangerin functions were introduced; see Erdélyi [5], [6, p. 88]. The Lamé–Wangerin eigenvalue
problem is obtained when we require that (sn z)1/2w(z) stays bounded at the singularities iK ′
and 2K + iK ′; see Erdélyi [5] and Erdélyi, Magnus and Oberhettinger [6, Section 15.6]. These
eigenfunctions will be defined on the segment (iK ′, 2K + iK ′) but can then be continued ana-
lytically.
mailto:volkmer@uwm.edu
https://doi.org/10.3842/SIGMA.2018.131
2 H. Volkmer
In Section 3 of this paper we consider a more general eigenvalue problem whose eigenfunc-
tions w(z) have the form
w(z) = (z − iK ′)ν+1
∞∑
n=0
qn(z − iK ′)2n
at z = iK ′ and a similar condition at z = 2K + iK ′. We call these eigenfunctions generalized
Lamé–Wangerin functions. Every classical Lamé–Wangerin function is also a generalized Lamé–
Wangerin function but not vice versa unless ν ≥ −1
2 .
The motivation for introducing these functions is as follows. In Section 2 we show that the
eigenvalues of the Floquet eigenvalue problem agree with the eigenvalues of an infinite tridiagonal
matrix F (considered in the Hilbert sequence space `2(Z), Z the set of integers). One is especially
interested in the case that a matrix entry in the diagonal above or below the main diagonal of F
vanishes because then the eigenvalue problem splits in two problems whose eigenvalues are given
by infinite tridiagonal submatrices of F that are only infinite in one direction (the underlying
Hilbert space can be taken as `2(N0), N0 the set of non-negative integers). It turns out that this
special case occurs if and only if ν+µ or ν−µ is an integer (ν is the parameter in Lamé’s equation
and µ is the parameter in Floquet’s condition.) Interestingly, if ν + µ or ν − µ is an integer
then the eigenvalues of one of the submatrices are identical with the eigenvalues of a classical
Lamé–Wangerin problem. This is a simple observation but as far as I know has not been stated
in the literature. Of course, the obvious question is: If the eigenvalues of one submatrix of F
are those for a classical Lamé–Wangerin problem what is the meaning of the eigenvalues of the
complementary submatrix? As we show in this paper, these are the eigenvalues for a generalized
Lamé–Wangerin problem (non-classical except when ν = −1
2).
A second motivation for introducing the generalized Lamé–Wangerin functions is as follows.
Lamé polynomials and algebraic Lamé functions are not special cases of classical Lamé–Wangerin
functions. However, they are special case of generalized Lamé–Wangerin function. We show in
Sections 6 and 7 how Lamé polynomials and algebraic Lamé functions appear in the notation of
generalized Lamé–Wangerin functions. We should mention that we adopt the name “algebraic
Lamé functions” from [6, p. 68]. These functions are called “Lamé–Wangerin functions” in
Lambe [13] and non-meromorphic Lamé functions in Finkel et al. [7].
In Section 5 we compare the eigenvalues of the Floquet and the Lamé–Wangerin problems.
In Sections 6 and 7 we show that algebraic Lamé functions and Lamé polynomials are special
cases of (generalized) Lamé–Wangerin functions. In Section 8 we investigate the number of zeros
of Lamé–Wangerin eigenfunctions. In Section 9 we find the limit of Lamé–Wangerin functions
as k → 0.
I consider some of the results in this paper as new but not all results are new. The treatment
of the generalized Lamé–Wangerin problem is new. The recursions (3.9) and (3.17) are known
from [6] but the “symmetric” recursions (3.12), (3.19) appear to be new. The latter recursions
are used in some proofs and also in Section 6. The results in Sections 4, 5, 8 and 9 are new. Lamé
polynomials and algebraic Lamé functions are well-known, so I make no claim that Sections 6
and 7 contain new results.
2 Floquet solutions
On the real axis z ∈ R, (1.1) is a Hill equation with fundamental period 2K. Let µ ∈ R. We
call h a Floquet eigenvalue if there exists a nontrivial solution w of (1.1) satisfying
w(z + 2K) = eiπµw(z), z ∈ R. (2.1)
Eigenvalue Problems for Lamé’s Differential Equation 3
w(z) is a corresponding Floquet eigenfunction. It is known [3, p. 31] that the eigenvalues are
real and form a sequence converging to ∞. We denote the eigenvalues by
h0(µ, ν, k) ≤ h1(µ, ν, k) ≤ h2(µ, ν, k) ≤ · · · .
The eigenvalues are counted according to multiplicity. If µ is not an integer then
h0(µ, ν, k) < h1(µ, ν, k) < h2(µ, ν, k) < · · · .
Obviously, we have
hm(µ, ν, k) = hm(µ+ 2, ν, k) = hm(−µ, ν, k) = hm(µ,−ν − 1, k). (2.2)
Let w1(z, h, ν, k) and w2(z, h, ν, k) be the solutions of (1.1) satisfying the initial conditions
w1(0) = 1, dw1
dz (0) = 0, w2(0) = 0, dw2
dz (0) = 1. Then Hill’s discriminant D is given by
D(h, ν, k) = w1(2K,h, ν, k) +
dw2
dz
(2K,h, ν, k).
The eigenvalues hm(µ, ν, k) are the solutions of the equation
D(h, ν, k) = 2 cos(µπ); (2.3)
see [3, equation (2.4.4)]. From (2.3) we easily obtain the following result that will be needed
later.
Theorem 2.1. For every m ∈ N0 = {0, 1, 2, . . . }, the function (µ, ν, k) 7→ hm(µ, ν, k) is conti-
nuous on R× R× [0, 1).
Theorem 2.1 can also be inferred from results on Sturm–Liouville theory [12].
Following [6, p. 65] we transform (1.1) by setting
t = 1
2π − am(z, k), (2.4)
where am is Jacobi’s amplitude function. We note that (2.4) establishes a conformal mapping
between the strip |=z| < K ′ and the t-plane cut along the rays mπ ± isL, s ≥ 1, m ∈ Z, where
L := arccosh
1
k
=
1
2
ln
1 + k′
1− k′
, k′ =
√
1− k2.
Then
sn z = cos t, cn z = sin t.
We obtain(
1− k2 cos2 t
)d2w
dt2
+ k2 cos t sin t
dw
dt
+
(
h− ν(ν + 1)k2 cos2 t
)
w = 0. (2.5)
Since am(z + 2K) = am z + π, condition (2.1) becomes
w(t+ π) = e−iπµw(t), t ∈ R.
This condition is equivalent to eiµtw(t) being periodic with period π. Therefore, using Fourier
series, eigenfunctions have the form
w(t) =
∞∑
n=−∞
cne−i(µ+2n)t. (2.6)
4 H. Volkmer
By substituting (2.6) in (2.5), we obtain the three-term recursion
ρncn−1 + (σn − h)cn + τn+1cn+1 = 0, n ∈ Z, (2.7)
where
ρn = −1
4k
2(2n− 1 + µ+ ν)(2n− 2 + µ− ν),
σn = 1
2k
2ν(ν + 1) +
(
1− 1
2k
2
)
(2n+ µ)2,
τn = −1
4k
2(2n+ µ+ ν)(2n− 1 + µ− ν).
This recursion is similar to the one given in [8, equation (7.1)] which is based on Fourier cosine
series instead of the complex form of Fourier series we used. The behavior of solutions {cn}n∈Z
of (2.7) as n→∞ is given by Perron’s rule [19]. If k ∈ (0, 1) we choose n0 so large that ρn 6= 0
and τn+1 6= 0 for n ≥ n0. Then the solutions {cn}n>n0 of equations (2.7) for n ≥ n0 form
a two-dimensional vector space. There exists a recessive solution which is uniquely determined
up to a constant factor with the property
lim
n→∞
cn+1
cn
=
1− k′
1 + k′
< 1. (2.8)
Every solution which is linearly independent of this solution satisfies
lim
n→∞
cn+1
cn
=
1 + k′
1− k′
> 1.
Similar results hold for n→ −∞. We obtain the following theorem.
Theorem 2.2. Let µ, ν ∈ R and k ∈ (0, 1). Then h is one of the eigenvalues hm(µ, ν, k) if and
only if the recursion (2.7) has a nontrivial solution {cn}n∈Z such that
a) either there is n0 such that cn = 0 for n ≥ n0 or {cn} is recessive as n→∞; and
b) either there is n0 such that cn = 0 for n ≤ n0 or {cn} is recessive as n→ −∞.
The expansion (2.6) of a corresponding eigenfunction converges in the strip |=t| < L.
Of course, a nontrivial solution {cn} of (2.7) can be zero for n ≥ n0 or n ≤ n0 only when one
of the numbers ρn or τn vanishes. This happens if and only if at least one of the numbers µ± ν
is an integer. These interesting cases will be discussed in Sections 5, 6 and 7.
Alternatively, we may expand
w(t) =
(
1− k2 cos2 t
)1/2 ∞∑
n=−∞
dne−i(µ+2n)t.
Then we obtain the “adjoint” recursion
τndn−1 + (σn − h)dn + ρn+1dn+1 = 0, n ∈ Z. (2.9)
Theorem 2.2 also holds with (2.9) in place of (2.7).
Eigenvalue Problems for Lamé’s Differential Equation 5
3 Generalized Lamé–Wangerin functions
A (classical) Lamé–Wangerin function w(z) is a nontrivial solution of Lamé’s equation (1.1)
with the property that (sn z)1/2w(z) stays bounded on the segment between the regular singu-
larities K ′ and 2K + iK ′; see [6, Section 15.6]. Such solutions exist only for specific values of h.
If we substitute z = u+ iK ′ then we obtain the singular Sturm–Liouville problem [23]
d2w
du2
+
(
h− ν(ν + 1) sn−2(u, k)
)
w = 0, 0 < u < 2K, (3.1)
with the boundary condition that (snu)−1/2w stays bounded on (0, 2K).
The eigenvalue problem splits into two problems, one for functions that are even with respect
to K + iK ′, that is,
w(K + iK ′ + s) = w(K + iK ′ − s) for −K < s < K, (3.2)
and one for functions which are odd with respect to K + iK ′, that is,
w(K + iK ′ + s) = −w(K + iK ′ − s) for −K < s < K. (3.3)
Without loss of generality, one may assume that ν ≥ −1
2 , and since the exponents at iK ′ and
2K + iK ′ are {ν + 1,−ν}, a Lamé–Wangerin function has the form
w(z) = (z − iK ′)ν+1
∞∑
n=0
qn(z − iK ′)2n (3.4)
for z close to iK ′ with q0 6= 0.
We generalize these eigenvalue problems as follows. Let ν ∈ R, 0 < k < 1. We call h ∈ C an
eigenvalue of the first Lamé–Wangerin problem if (1.1) admits a nontrivial solution w on the
interval (iK ′, 2K+ iK ′) which close to z = iK ′ has the form (3.4) and satisfies w′(K+ iK ′) = 0.
The latter property is equivalent to (3.2). The eigenfunction w will be called a Lamé–Wangerin
function of the first kind. Note that we consider this eigenvalue problem for all real ν not just
for ν ≥ −1
2 . Also note that the condition q0 6= 0 is not required in (3.4) although q0 6= 0 will
hold if ν + 1
2 is not a negative integer.
Similarly, we call h an eigenvalue of the second Lamé–Wangerin problem if (1.1) admits
a nontrivial solution w on the interval (iK ′, 2K + iK ′) which close to z = iK ′ has the form (3.4)
and satisfies w(K + iK ′) = 0. The latter property is equivalent to (3.3). The eigenfunction w
will be called a Lamé–Wangerin function of the second kind.
If ν > −3
2 our eigenvalue problems are included in singular Sturm–Liouville theory (see
also [15]) but this theory does not give us results for ν ≤ −3
2 . We will treat these eigenvalue
problems by a different method developed below.
We substitute
η = e−2it (3.5)
in (2.5). We obtain the Fuchsian equation
k2η(η − η1)(η − η2)
[
d2w
dη2
+
1
2
(
1
η
+
1
η − η1
+
1
η − η2
)
dw
dη
]
+
(
h− k2ν(ν + 1)
(1 + η)2
4η
)
w = 0, (3.6)
6 H. Volkmer
where
η1 :=
1− k′
1 + k′
∈ (0, 1), η2 :=
1 + k′
1− k′
∈ (1,∞).
The differential equation (3.6) has regular singularities at η = 0, η1, η2,∞ with exponents{
−1
2ν,
1
2(ν + 1)
}
,
{
0, 12
}
,
{
0, 12
}
,
{
−1
2ν,
1
2(ν + 1)
}
, respectively. If we combine (2.4) with (3.5)
we obtain
η = e−2i(
1
2
π−am z) = (sn z + i cn z)−2 = (sn z − i cn z)2. (3.7)
Setting z = u+ iK ′ for 0 < u < K this gives
η =
1− dnu
1 + dnu
.
This establishes a bijective increasing map between u ∈ (0,K) and η ∈ (0, η1). Taking into
consideration the behavior of η close to u = 0 and u = K we see that a Lamé–Wangerin
function of the first kind expressed in the variable η is a solution of (3.6) on (0, η1) which close
to η = 0 is of the form
w(η) = η(ν+1)/2
∞∑
n=0
cnη
n, (3.8)
and which is analytic at η = η1. This implies that the radius of convergence of the power series
in (3.8) is ≥ η2. For the coefficients cn we find the recursion(
β
(1)
0 − h
)
c0 + γ1c1 = 0,
αncn−1 +
(
β(1)n − h
)
cn + γn+1cn+1 = 0, n ≥ 1, (3.9)
where
αn = −1
2k
2(n+ ν)(2n− 1),
β(1)n = 1
2k
2ν(ν + 1) +
(
1− 1
2k
2
)
(2n+ ν + 1)2,
γn = −1
2k
2(2n+ 2ν + 1)n.
Note that the equations (3.9) for n ≥ 1 agree with (2.7) when we set µ = ν + 1. The recur-
sion (3.9) is given in [6, Section 15.6(15)].
Using Perron’s rule, we see that h is an eigenvalue of the first Lamé–Wangerin problem if
and only if (3.9) has a nontrivial solution {cn}∞n=0 which is either identically zero for large n
or satisfies (2.8). Of course, a nontrivial solution {cn}∞n=0 of (3.9) can be identically zero for
large n only if one of the numbers αn is zero, that is, if ν is a negative integer.
Alternatively, we may expand a Lamé–Wangerin function of the first kind in the form
w(η) = η(ν+1)/2(η2 − η)1/2
∞∑
n=0
anη
n (3.10)
with the power series having radius ≥ η2. In order to find the recursion for the coefficients an
we transform (3.6) by setting
w(η) = (η2 − η)1/2v(η)
Eigenvalue Problems for Lamé’s Differential Equation 7
to
k2η(η − η1)(η − η2)
[
d2v
dη2
+
1
2
(
1
η
+
1
η − η1
+
3
η − η2
)
dv
dη
]
+
(
h− k2ν(ν + 1)
(1 + η)2
4η
+ 1
4k
2(2η − η1)
)
v = 0. (3.11)
We obtain the recursion(
ε
(1)
0 − h
)
a0 + δ1a1 = 0,
δnan−1 +
(
ε(1)n − h
)
an + δn+1an+1 = 0, n ≥ 1, (3.12)
where
δn = −1
2k
2n(2n+ 2ν + 1),
ε(1)n = 1
2k
2ν(ν + 1) +
(
1− 1
2k
2
)
(2n+ ν + 1)2 + 1
4k
2η1(4n+ 2ν + 3)
= 1
2k
2ν(ν + 1)− k′
(
2n+ 3
2 + ν
)
+
(
1− 1
2k
2
)(
1
4 +
(
2n+ 3
2 + ν
)2)
.
It is a pleasant surprise that, in contrast to (3.9), recursion (3.12) is of self-adjoint form. We
take advantage of this observation and introduce a symmetric operator S = S(1)(ν, k) in the
Hilbert space `2(N0) with the standard inner product. The domain of definition of S is
D(S) =
{
{xn}∞n=0 :
∞∑
n=0
n4|xn|2 <∞
}
and S is defined on D(S) by
S({xj})0 = ε
(1)
0 x0 + δ1x1,
S({xj})n = δnxn−1 + ε(1)n xn + δn+1xn+1, n ≥ 1.
So S is represented by an infinite symmetric tridiagonal matrix.
Theorem 3.1. Let ν ∈ R and k ∈ [0, 1).
(a) S(1)(ν, k) is a self-adjoint operator in `2(N0) with compact resolvent and bounded below.
(b) If k ∈ (0, 1) the eigenvalues of S(1)(ν, k) agree with the eigenvalues of the first Lamé–
Wangerin problem.
(c) If k ∈ (0, 1) the eigenvalues of S(1)(ν, k) are simple.
Proof. (a) We abbreviate S = S(1)(ν, k), and write S = A + B with A = S(1)(ν, 0). So A is
represented by an infinite diagonal matrix with diagonal entries (2n + ν + 1)2, n ∈ N0. It is
clear that A is a positive semi-definite self-adjoint operator with compact resolvent. There are
two constants λ > 0 and c ∈ (0, 1) such that
‖Bx‖ ≤ c‖(A+ λ)x‖ for all x ∈ D(S). (3.13)
To prove this it is convenient to write B = B1+B2+B3 where each Bi has a matrix representation
consisting of only one nonzero “diagonal”, and estimate ‖Bx‖ ≤ ‖B1x‖+ ‖B2x‖+ ‖B3x‖. We
can reach c < 1 because the factor of n2 on the main diagonal of A is 4 while the factors of n2
on the three diagonals of B are −k2, −2k2, −k2, respectively. From (3.13) we obtain that
8 H. Volkmer
T := B(A + λ)−1 is a bounded linear operator with operator norm ‖T‖ ≤ c < 1. Therefore,
1 + T is invertible and
(S + λ)−1 = (A+ λ+B)−1 = (A+ λ)−1(1 + T )−1.
This shows that (S + λ)−1 is a compact operator. Since S is symmetric, we find that S is
self-adjoint; compare [11, Chapter V, Theorem 4.3]. From (3.13) we also obtain that S + λ is
positive definite [11, Chapter V, Theorem 4.11]. Therefore, (a) follows.
(b) h is an eigenvalue of S if and only if the recursion (3.12) has a nontrivial solution {an}∞n=0
with the property that
∞∑
n=0
n4|an|2 < ∞. By Perron’s rule the latter property is equivalent to
an = 0 for large n or {an} is recessive as n→∞.
(c) If k ∈ (0, 1) the eigenvalues of S are simple because the corresponding eigenfunctions of
the first Lamé–Wangerin problem are even with respect to K + iK ′. �
Based on Theorem 3.1 we write the eigenvalues of the first Lamé–Wangerin problem with
k ∈ (0, 1) in the form
H
(1)
0 (ν, k) < H
(1)
1 (ν, k) < H
(1)
2 (ν, k) < · · · .
The Lamé–Wangerin function belonging to H
(1)
m (ν, k) will be denoted by w
(1)
m (z, ν, k). If a nor-
malization is required it will be stated separately. We note that the corresponding eigenvectors
{an}∞n=0 of S when properly normalized form an orthonormal basis in the Hilbert space `2(N0).
The eigenvalues of S(1)(ν, 0) are (2n + ν + 1)2 for n ∈ N0. If we arrange this sequence in
increasing order repeated according to multiplicity we denote these eigenvalues by H
(1)
m (ν, 0).
Explicitly, they are given by the following lemma.
Lemma 3.2. Let p− 1 < ν ≤ p with p ∈ Z. Then, for all m ∈ N0,
H(1)
m (ν, 0) = (2`+ ν + 1)2,
where
` =
m if m+ p ≥ 0,
1
2(m− p) if m+ p < 0, m+ p even,
1
2(−m− p− 1) if m+ p < 0, m+ p odd.
We will need continuity of the eigenvalues H
(1)
m (ν, k).
Theorem 3.3. The function (ν, k) 7→ H
(1)
m (ν, k) is continuous on R× [0, 1) for every m ∈ N0.
Proof. Set S(ν, k) = S(1)(ν, k), A(ν) = S(ν, 0) and S(ν, k) = A(ν) + B(ν, k). Let ν0 > 0
and k0 ∈ (0, 1) be given, and set Ω := [−ν0, ν0] × [0, k0]. Then we can find λ > 0 large
enough and c ∈ (0, 1) such that (3.13) holds uniformly for (ν, k) ∈ Ω. It follows that T (ν, k) :=
B(ν, k)(A(ν) + λ)−1 is a bounded linear operator with operator norm ‖T (ν, k)‖ ≤ c for all
(ν, k) ∈ Ω. As before, we have
(S(ν, k) + λ)−1 = (A(ν) + λ)−1(1 + T (ν, k))−1, (ν, k) ∈ Ω. (3.14)
Suppose we have a sequence (νn, kn) ∈ Ω which converges to
(
ν̂, k̂
)
as n → ∞. Then we can
easily show using the definitions of A and T that∥∥(A(νn) + λ)−1 − (A(ν̂) + λ)−1
∥∥→ 0 as n→∞,
Eigenvalue Problems for Lamé’s Differential Equation 9
and ∥∥T (νn, kn)− T
(
ν̂, k̂
)∥∥→ 0 as n→∞.
Using (3.14) we then obtain that∥∥(S(νn, kn) + λ)−1 −
(
S
(
ν̂, k̂
)
+ λ
)−1∥∥→ 0 as n→∞. (3.15)
If Kn is a sequence of positive definite compact Hermitian operators converging to a positive
definite compact Hermitian operator K with respect to the operator norm, then the mth largest
eigenvalue of K (counted according to multiplicity) converges to the m largest eigenvalue of K as
n→∞ for every m ∈ N0. This follows directly from the minimum-maximum-principle If we set
Kn = (S(νn, kn)+λ)−1, K =
(
S
(
ν̂, k̂
)
+λ
)−1
and use (3.15) we obtain H
(1)
m (νn, kn)→ H
(1)
m
(
ν̂, k̂
)
as n→∞ for every m ∈ N0 as desired. �
A Lamé–Wangerin function of the second kind can be written in the form
w(η) = η(ν+1)/2(η1 − η)1/2(η2 − η)1/2
∞∑
n=0
dnη
n, (3.16)
where the power series
∑
dnη
n has radius ≥ η2. If we set
w(η) = (η − η1)1/2(η − η2)1/2v(η)
in (3.6), we obtain
k2η(η − η1)(η − η2)
[
d2v
dη2
+
1
2
(
1
η
+
3
η − η1
+
3
η − η2
)
dv
dη
]
+
(
3
2k
2η − 1 + 1
2k
2 + h− k2ν(ν + 1)
(1 + η)2
4η
)
v = 0.
This gives the recursion(
β
(2)
0 − h
)
d0 + γ1d1 = 0,
αn+1dn−1 +
(
β(2)n − h
)
dn + γn+1dn+1 = 0, n ≥ 1, (3.17)
where αn, γn are as in (3.9) and
β(2)n = 1
2k
2ν(ν + 1) +
(
1− 1
2k
2
)
(2n+ ν + 2)2.
Note that the equations (3.17) for n ≥ 1 agree with (2.9) when we set µ = ν + 2. The recur-
sion (3.17) is given in [6, Section 15.6(16)].
Alternatively, a Lamé–Wangerin function of the second kind can be written as
w(η) = η(ν+1)/2(η1 − η)1/2
∞∑
n=0
bnη
n, (3.18)
where the power series
∑
bnη
n has radius ≥ η2. If we set
w(η) = (η1 − η)1/2v(η)
in (3.6), we obtain (3.11) with η1, η2 interchanged. This gives the recursion(
ε
(2)
0 − h
)
b0 + δ1b1 = 0,
δnbn−1 +
(
ε(2)n − h
)
bn + δn+1bn+1 = 0, n ≥ 1, (3.19)
10 H. Volkmer
where
δn = −1
2k
2n(2n+ 2ν + 1),
ε(2)n = 1
2k
2ν(ν + 1) +
(
1− 1
2k
2
)
(2n+ ν + 1)2 + 1
4k
2η2(4n+ 2ν + 3)
= 1
2k
2ν(ν + 1) + k′
(
2n+ 3
2 + ν
)
+
(
1− 1
2k
2
)(
1
4 +
(
2n+ 3
2 + ν
)2)
.
For the second Lamé–Wangerin problem we have results parallel to Theorem 3.1, Lem-
ma 3.2 and Theorem 3.3. We denote the eigenvalues of the second Lamé–Wangerin problem
by H
(2)
m (ν, k), and the corresponding Lamé–Wangerin eigenfunctions by w
(2)
m (z, ν, k).
4 Analytic continuation of Lamé–Wangerin functions
In the previous section Lamé–Wangerin functions were defined on the interval (iK ′, 2K + iK ′).
We analytically continue these functions to the strip 0 ≤ =z < K ′ as follows. Using (3.7)
and (3.8) a Lamé–Wangerin function w(1) of the first kind can be written as
w(1)(z) = e−i(ν+1)( 1
2
π−am z)
∞∑
n=0
cn(sn z − i cn z)2n. (4.1)
Since the power series
∑
cnη
n has radius larger than 1 and |η| ≤ 1 for 0 ≤ =z < K ′, the
expansion (4.1) converges in the strip 0 ≤ =z < K ′.
If 0 < η < η1 we have
1
4
k2
(
η1/2 + η−1/2
)2 − 1 =
1
4
k2η−1(η1 − η)(η2 − η). (4.2)
If z is on the segment (iK ′,K + iK ′) and η is given by (3.7) then (4.2) implies
i dn z =
1
2
kη−1/2(η1 − η)1/2(η2 − η)1/2. (4.3)
Therefore, (3.16) implies
w(2)(z) = 2ik−1e−i(ν+2)( 1
2
π−am z) dn z
∞∑
n=0
dn(sn z − i cn z)2n. (4.4)
Again, this expansion is convergent in the strip 0 ≤ =z < K ′.
In order to deal with expansions (3.10) and (3.18) we introduce the function
I1(z) := (dn z + cn z)1/2 (4.5)
also appearing in [8]. This function is analytic in the strip −K ′ < =z < K ′ when the branch of
the root is chosen as follows. The function dn z + cn z does not assume negative values or zero
in the rectangle −2K < <z < 2K, −K ′ < =z < K ′. We choose the principal branch of the root
in (4.5) in this rectangle. We choose positive imaginary roots on the segments (−2K,−2K+iK ′)
and (2K− iK ′, 2K). For other z, I1(z) is determined by I1(z+4K) = −I1(z). A second function
is defined by
I2(z) := −I1(z + 2K) = −(dn z − cn z)1/2.
For 0 < η < η1 we have the identity
(1− η + q)1/2 + (1− η − q)1/2 = 21/2(1− k′)1/2(η2 − η)1/2, (4.6)
Eigenvalue Problems for Lamé’s Differential Equation 11
where
q = k(η1 − η)1/2(η2 − η)1/2
and all roots denote positive roots of positive numbers. For z between iK ′ and K+ iK ′ we have
i cn z =
1
2
(
η−1/2 − η1/2
)
.
Therefore, it follows from (4.3) and (4.6) that the analytic continuation of J1 = η−1/4(η2− η)1/2
to the strip −K ′ < =z < K ′ is given by
(1− k′)1/2J1(z) = ei
1
4
πI1(z) + e−i
1
4
πI2(z). (4.7)
Therefore, the analytic continuation of the Lamé–Wangerin function w(1) of the first kind given
by (3.10) to the strip 0 ≤ =z < K ′ is
w(1)(z) = e−i(ν+
3
2
)( 1
2
π−am z)J1(z)
∞∑
n=0
an(sn z − i cn z)2n.
Similarly, for 0 < η < η1 we have
(1− η + q)1/2 − (1− η − q)1/2 = 21/2(1 + k′)1/2(η1 − η)1/2.
It follows that the analytic continuation of the function J2 = η−1/4(η1 − η)1/2 to the strip
−K ′ < =z < K ′ is given by
(1 + k′)1/2J2(z) = ei
1
4
πI1(z)− e−i
1
4
πI2(z). (4.8)
If a Lamé–Wangerin function w(2) of the second kind is given by (3.18) then its analytic conti-
nuation is
w(2)(z) = e−i(ν+
3
2
)( 1
2
π−am z)J2(z)
∞∑
n=0
bn(sn z − i cn z)2n.
5 Comparison of eigenvalues
Every Lamé–Wangerin function is also a Floquet eigenfunction.
Lemma 5.1. Let ν ∈ R and 0 < k < 1. A Lamé–Wangerin function w(1)(z) of the first kind
satisfies
w(1)(z + 2K) = ei(ν+1)πw(1)(z).
A Lamé–Wangerin function w(2)(z) of the second kind satisfies
w(2)(z + 2K) = eiνπw(2)(z).
Proof. This follows from (4.1) and (4.4). �
If µ + ν or µ − ν is an integer then the Floquet eigenvalues hm(µ, ν) can be expressed in
terms of Lamé–Wangerin eigenvalues H
(j)
m (ν). The properties (2.2) show that it is sufficient to
consider the case µ = ν + 1. Then we have the following result.
Theorem 5.2. Let 0 < k < 1, ν ∈ R, p ∈ N0 and p− 1 < |ν| ≤ p.
12 H. Volkmer
(a) If ν ≥ 0 then
hm(ν + 1, ν) = H(2)
m (−ν − 1), m = 0, 1, . . . , p− 1,
hp+2i+1(ν + 1, ν) = H
(2)
p+i(−ν − 1), i ≥ 0,
hp+2i(ν + 1, ν) = H
(1)
i (ν), i ≥ 0.
(b) If ν < 0 then
hm(ν + 1, ν) = H(1)
m (ν), m = 0, 1, . . . , p− 1,
hp+2i+1(ν + 1, ν) = H
(1)
p+i(ν), i ≥ 0,
hp+2i(ν + 1, ν) = H
(2)
i (−ν − 1), i ≥ 0.
(c) If ν is an integer then
h0(ν + 1, ν) < h1(ν + 1, ν) < · · · < hp(ν + 1, ν),
and, for i ≥ 0,
hp+2i(ν + 1, ν) = hp+2i+1(ν + 1, ν) < hp+2i+2(ν + 1, ν) = hp+2i+3(ν + 1, ν) < · · · .
Proof. (a), (b) Suppose first that ν is not an integer. Then the eigenvalues fm(k) := hm(ν +
1, ν, k) form a strictly increasing sequence. By Lemma 5.1, each eigenvalue gm(k) := H
(1)
m (ν, k)
and Gm(k) := H
(2)
m (−ν − 1, k) is among the f -eigenvalues. This is also true for k = 0. The
sequence {fm(0)}∞m=0 is given by the sequence
{
(2n+ ν + 1)2
}
n∈Z when arranged in increasing
order, {gm(0)} is given by {(2n + ν + 1)2}∞n=0 in increasing order and {Gm(0)} is given by{
(2n + ν + 1)2
}−1
n=−∞ in increasing order. Because of continuity of the functions fm, gm, Gm
(Theorems 2.1 and 3.3) the order of these eigenvalues is the same for all k ∈ [0, 1). An analysis
of the order at k = 0 proves (a) and (b) for noninteger ν. The result extends to integer ν by
continuity.
(c) Let ν be an integer. Then we apply (a) or (b) to ν+ε in place of ν and take limits ε→ 0±.
This proves (c). �
We now compare the eigenvalues H
(j)
m (ν) with H
(j)
m (−ν − 1).
Theorem 5.3. Let p ∈ N0, 0 < k < 1. Let either H = H(1) or H = H(2).
(a) If −p− 3
2 < ν < −p− 1
2 then
Hp(ν) < H0(−ν − 1) < Hp+1(ν) < H1(−ν − 1) < Hp+2(ν) < · · · .
(b) If ν = −p− 1
2 then
Hp−1(ν) < H0(−ν − 1) = Hp(ν) < H1(−ν − 1) = Hp+1(ν) < · · · .
Proof. We consider the eigenvalues Hm = H
(1)
m . The proof for Hm = H
(2)
m is similar.
(a) Let −p− 3
2 < ν < −p− 1
2 . The eigenvalues Hm(ν), m ≥ 0, are pairwise distinct, and the
eigenvalues H`(−ν − 1), ` ≥ 0, are pairwise distinct. The eigenvalues Hm(ν) are also distinct
from H`(−ν− 1) because the corresponding eigenfunctions are linearly independent. Therefore,
using continuity of the functions k 7→ Hm(ν, k) (Theorem 3.3) the order of the eigenvalues
Hm(ν, k), H`(−ν−1, k) must be the same for all k ∈ [0, 1). The sequence {Hm(ν, 0)}∞m=0 agrees
Eigenvalue Problems for Lamé’s Differential Equation 13
with
{
(2n+ ν + 1)2
}∞
n=0
after the latter sequence is arranged in increasing order. Similarly, the
sequence {Hm(−ν−1, 0)}∞m=0 agrees with
{
(2n−ν)2
}∞
n=0
arranged in increasing order. Analysis
of this order at k = 0 implies (a).
(b) If p = 0 then ν = −1
2 and (b) is trivially true because ν = −ν−1. For p ≥ 1 statement (b)
follows from continuity of the functions ν 7→ H(ν, k) and taking one-sided limits as ν → −p− 1
2
in (a). �
We now compare the eigenvalues H
(1)
m (ν) with H
(2)
m (ν).
Theorem 5.4. Let ν ∈ R, 0 < k < 1 and H
(j)
m := H
(j)
m (ν, k).
(a) If ν > −3
2 then
H
(1)
0 < H
(2)
0 < H
(1)
1 < H
(2)
1 < H
(1)
2 < H
(2)
2 < · · · .
(b) If −p− 3
2 < ν < −p− 1
2 with p ∈ N then{
H
(1)
0
H
(2)
0
}
< · · · <
{
H
(1)
p−1
H
(2)
p−1
}
< H(1)
p < H(2)
p < H
(1)
p+1 < H
(2)
p+1 < · · · ,
where, for m = 0, 1, . . . , p− 1,
H(1)
m < H(2)
m if m+ p is even, H(1)
m > H(2)
m if m+ p is odd.
(c) If ν = −p− 1
2 with p ∈ N then
H
(1)
0 = H
(2)
0 < H
(1)
1 = H
(2)
2 < · · · < H
(1)
p−1 = H
(2)
p−1 < H(1)
p < H(2)
p < · · · .
Proof. (a) Let ν > −3
2 . Then the eigenfunctions of the two Lamé–Wangerin problems are
constant multiples of solution (3.4) with q0 6= 0. Therefore, the eigenvalues of the two problems
are mutually distinct. By continuity of the functions k 7→ H
(j)
m (ν, k) (Theorem 3.3) the order of
the eigenvalues H
(j)
m must be the same for all k ∈ [0, 1). Now
H(1)
m (ν, 0) = (2m+ ν + 1)2, H(2)
m (ν, 0) = (2m+ ν + 2)2,
which implies (a).
(b) Let −p− 3
2 < ν < −p− 1
2 with p ∈ N. Again, the eigenvalues of the two Lamé–Wangerin
problems are mutually distinct, and the order of these eigenvalues must be the same for all
k ∈ [0, 1). The sequence
{
H
(j)
m (ν, 0)
}∞
m=0
is the same as {(2n + ν + j)2}∞n=0 but the latter one
has to be ordered increasingly. An analysis of the order leads to the arrangement stated in (b).
(c) Let ν = −p− 1
2 with p ∈ N. Continuity of the functions ν 7→ H
(j)
m (ν, k) and part (b) show
that H
(1)
m = H
(2)
m for m = 0, 1, . . . , p− 1. We know from Theorem 5.3(b) that
H
(j)
m+p(ν) = H(j)
m (−ν − 1), m ≥ 0.
Since −ν − 1 > −3
2 the rest of statement (c) follows from part (a). �
14 H. Volkmer
6 Algebraic Lamé functions
If ν + 1
2 is a nonzero integer then Lamé’s differential equation (1.1) has solutions in finite
terms which are usually called algebraic Lamé functions. These solutions were investigated
in [4, 7, 8, 13]. We obtain these functions as follows.
Let ν = −p − 1
2 with p ∈ N. For j = 1, 2 we introduce the symmetric tridiagonal p by p
matrices
S(j)
p =
ε
(j)
0 δ1 0
δ1 ε
(j)
1
. . .
. . .
0
. . .
. . .
. . . 0
. . .
. . . ε
(j)
p−2 δp−1
0 δp−1 ε
(j)
p−1
,
where
ε(j)n = 1
2k
2
(
p2 − 1
4
)
+ (−1)jk′(2n+ 1− p) +
(
1− 1
2k
2
)(
1
4 + (2n+ 1− p)2
)
,
δn = k2n(p− n).
The coefficient δp vanishes in (3.12), (3.19). Therefore, if (a0, a1, . . . , ap−1)
t is an eigenvector
of S
(j)
p and an := 0 for n ≥ p then (3.10), (3.18) are Lamé–Wangerin functions of the first and
second kind, respectively.
We note that S
(1)
p is the mirror image of S
(2)
p with respect to the anti-diagonal, that is, we
have
ε(1)n = ε
(2)
p−1−n, δn = δp−n.
It follows that S
(1)
p and S
(2)
p have the same eigenvalues and the corresponding eigenvectors are in-
verse to each other, that is, if (a0, a1, . . ., ap−1)
t is an eigenvector for S
(1)
p then (ap−1, ap−2, . . ., a0)
t
is an eigenvector for S
(2)
p belonging to the same eigenvalue. According to Theorem 5.4(c), the
common eigenvalues of S
(j)
p are
H(1)
m (−p− 1
2) = H(2)
m
(
−p− 1
2
)
, m = 0, 1, . . . , p− 1.
If (a0, a1, . . . , ap−1)
t is a (real) eigenvector of S
(1)
p then
w(1) = η−
1
2
p+ 1
4 (η2 − η)1/2
p−1∑
n=0
anη
n, (6.1)
w(2) = η−
1
2
p+ 1
4 (η1 − η)1/2
p−1∑
n=0
ap−n−1η
n (6.2)
are solutions of (3.11). These are algebraic Lamé functions expressed in the variable η. We
note that the functions w(1) and w(2) are essentially Heun polynomials. For if we set w =
η−
1
2
p+ 1
4 (ηj − η)1/2v(s) and η = η1s, then we obtain the Heun equation for v(s) and
p−1∑
n=0
an(η1s)
n
is a Heun polynomial.
Eigenvalue Problems for Lamé’s Differential Equation 15
If we substitute (3.7) in (6.1), (6.2) and use the functions J1(z), J2(z) defined in (4.7), (4.8)
we obtain
w(1)(z) = J1(z)
p−1∑
n=0
an(sn z − i cn z)2n−p+1,
w(2)(z) = J2(z)
p−1∑
n=0
an(sn z + i cn z)2n−p+1.
We know from Lemma 5.1 that
w(1)(z + 2K) = i(−1)pw(1)(z), w(2)(z + 2K) = i(−1)p+1w(2)(z).
Moreover, we have
(1 + k′)1/2w(1)(z̄) = −i(1− k′)1/2w(2)(z),
and, for x ∈ R,
w(1)(x) = w(1)(2K − x), w(2)(x) = −w(2)(2K − x),
which shows that the real part of w(1)(x) is a function even with respect to x = K while the
imaginary part of w(1)(x) is odd with respect to x = K.
One should notice that if ν = −p − 1
2 and h is an eigenvalue of the matrix S
(j)
p then all
(nontrivial) solutions of Lamé’s equation qualify as “algebraic Lamé functions”. We picked the
basis of two solutions, one even and one odd with respect to z = K + iK ′. Ince [8] considered
the basis of even and odd solutions (with respect to z = 0) while Erdélyi [4] has the basis of
even or odd solutions with respect to z = K.
In the simplest case ν = −3
2 we have
H
(1)
0
(
−3
2
)
= H
(2)
0
(
−3
2
)
= 1
4
(
1 + k2
)
and
w
(1)
0 (z) = J1(z), w
(2)
0 (z) = J2(z).
If ν = −5
2 then
H
(1)
0
(
−5
2
)
= H
(2)
0
(
−5
2
)
= 5
4
(
1 + k2
)
−
(
1− k2 + k4
)1/2
,
H
(1)
1
(
−5
2
)
= H
(2)
1
(
−5
2
)
= 5
4
(
1 + k2
)
+
(
1− k2 + k4
)1/2
.
If we choose a0 = −k2, a1 = 3
4k
2 + 9
4 −
1
2k
2η1 −H(1)
m
(
−5
2
)
, m = 0, 1, then
w(1)
m (z) = J1(z)(a0(sn z + i cn z) + a1(sn z − i cn z)),
w(2)
m (z) = J2(z)(a0(sn z − i cn z) + a1(sn z + i cn z)).
7 Lamé polynomials
Let ν = −p − 1 with p ∈ N0. It is well-known [1, Chapter IX] that there are 2p + 1 distinct
values of h for which (1.1) admits nontrivial solutions which are polynomials in cn z, sn z, dn z.
In our notation these values of h are
H(1)
m (−p− 1), m = 0, 1, . . . , p (7.1)
16 H. Volkmer
and
H(2)
m (−p− 1), m = 0, 1, . . . , p− 1. (7.2)
Since αp+1 = 0 in (3.9), the numbers (7.1) are the eigenvalues of the p+ 1 by p+ 1 tridiagonal
matrix
T
(1)
p+1 =
β0 γ1 0
α1 β1
. . .
. . .
0
. . .
. . .
. . . 0
. . .
. . . βp−1 γp
0 αp βp
,
where
αn = 1
2k
2(p+ 1− n)(2n− 1),
βn = 1
2k
2p(p+ 1) +
(
1− 1
2k
2
)
(2n− p)2,
γn = 1
2k
2(2p+ 1− 2n)n.
If (c0, c1, . . . , cp)
t is an eigenvector of T
(1)
p+1 then
w = η−p/2
p∑
n=0
cnη
n
is a solution of (3.6). After substituting (3.7) we obtain
w =
p∑
n=0
cn(sn z − i cn z)2n−p.
Indeed, since (sn z − i cn z)−1 = sn z + i cn z, these solutions are polynomials in cn z, sn z. The
matrix T
(1)
p+1 has the symmetries
αn = γp−n+1, βn = βp−n.
Therefore, the space of symmetric vectors {cn}pn=0 (cn = cp−n, n = 0, 1, . . . , p), as well as the
space of antisymmetric vectors is invariant under T
(1)
p+1. Thus eigenvectors of T
(1)
p+1 will lie in one
of these invariant subspaces.
If p is even we find 1
2p + 1 Lamé polynomials of the form P
(
sn2 z
)
where P is a poly-
nomial of degree 1
2p if we use symmetric eigenvectors, and 1
2p Lamé polynomials of the form
cn z sn zP
(
sn2 z
)
where P is a polynomial of degree 1
2p−1 if we use antisymmetric eigenvectors.
If p is odd we find 1
2(p + 1) Lamé polynomials of the form sn zP
(
sn2 z
)
where P is a poly-
nomial of degree 1
2(p − 1) if we use symmetric eigenvectors, and 1
2(p + 1) Lamé polynomials
of the form cn zP
(
sn2 z
)
where P is a polynomial of degree 1
2(p − 1) if we use antisymmetric
eigenvectors.
Similarly, Lamé–Wangerin functions of the second kind belonging to the eigenvalues (7.2) are
Lamé polynomials that have the factor dn z.
Eigenvalue Problems for Lamé’s Differential Equation 17
8 Zeros of Lamé–Wangerin functions
We first determine the number of zeros of Lamé–Wangerin functions w
(j)
m (z) in the open interval
(iK ′,K + iK ′).
Theorem 8.1. Let j = 1, 2, m ∈ N0, ν ∈ R, k ∈ (0, 1).
(a) If ν > −3
2 then w
(j)
m has exactly m zeros in (iK ′,K + iK ′).
(b) If −p− 3
2 < ν ≤ −p− 1
2 , p ∈ N, then w
(j)
m has exactly max{0,m−p} zeros in (iK ′,K+iK ′).
Proof. Consider j = 1. The proof for j = 2 similar. Let P be the set of all real numbers
different from −p − 1
2 for all p ∈ N. For h ∈ R, ν ∈ P , let w(z, h, ν) be the solution of (1.1)
given locally at z = iK ′ by (3.4) with q0 = 1. Then one can show that (z− iK ′)−ν−1w(z, h, ν) is
continuous for z in [iK ′,K+iK ′], h ∈ R, ν ∈ P (k fixed). If we set wm(z, ν) = w
(
z,H
(1)
m (ν, k), ν
)
then (z − iK ′)−ν−1wm(z, ν) is continuous for z ∈ [iK ′,K + iK ′] and ν ∈ P . This implies that
the number of zeros of wm(·, ν) in (iK ′,K + iK ′) is finite and it is locally constant as a function
of ν.
(a) follows by considering ν = 0:
w(1)
m (z, 0, k) = (−1)m sin
(
(2m+ 1)
π
2K
(z − iK ′)
)
. (8.1)
(b) Suppose −p− 3
2 < ν ≤ −p− 1
2 with p ∈ N. Let m = 0, 1, . . . , p. By Theorem 5.3(a), we
have Hm(ν) ≤ H0(−ν − 1). By (a), w0(·,−ν − 1) has no zeros in (iK ′,K + iK ′). Therefore, by
Sturm comparison, wm(·, ν) also has no zeros in this interval.
Now consider m > p. If ν = −p − 1
2 then, by Theorem 5.3(b), Hm(ν) = Hm−p(−ν − 1).
Therefore, by (a), wm(·, ν) has m − p zeros in (iK ′,K + iK ′). If −p − 3
2 < ν < p − 1
2 then, by
Theorem 5.3(a), Hm(ν) < Hm−p(−ν − 1). Therefore, wm(·, ν) can have at most m− p zeros in
(iK ′,K + iK ′). If ν = −p− 1
2 we just showed that there are m− p zeros. By continuity, there
are exactly m− p zeros. For the latter step Lamé–Wangerin functions should be normalized by
the initial conditions w(K + iK ′) = 1, w′(K + iK ′) = 0. �
Now we look for zeros of Lamé–Wangerin functions in the strip 0 ≤ =z < K ′.
Lemma 8.2. A Lamé–Wangerin function which is not a Lamé polynomial has no zeros on the
real axis.
Proof. If µ is not an integer then a nontrivial Floquet solution w(z), z ∈ R, of (1.1) with
w(z+2K) = eiµπw(z) does not have zeros on the real axis. This is because the conjugate of w(z)
is a Floquet solution with conjugate multiplier e−iµπ, and eiµπ, e−iµπ are distinct. So w(z) and
its conjugate function are linearly independent. It follows from Lemma 5.1 that Lamé–Wangerin
functions have no zeros on the real axis if ν is not an integer.
Suppose that ν is an integer, and w(z) is a Lamé–Wangerin function belonging to the eigen-
value H
(1)
m (ν). Suppose that w(z0) = 0. with z0 ∈ R. Using (3.8) and the substitution (2.4) we
have
w(t) =
∞∑
n=0
cne−it(2n+ν+1)
and this function has a zero at t0 ∈ R. The coefficients cn are real so the functions
<w(t) =
∞∑
n=0
cn cos(2n+ ν + 1)t, t ∈ R (8.2)
=w(t) = −
∞∑
n=0
cn sin(2n+ ν + 1)t, t ∈ R (8.3)
18 H. Volkmer
both vanish at t = t0. The functions (8.2), (8.3) are both solutions of the differential equa-
tion (2.5) with the same values for h and ν. Since they have a common zero these solutions
must be linearly dependent. Now <w(t) is an even function and =w(t) is odd. So one of the
functions <w(t), =w(t) must vanish identically. This implies that cn = 0 for large enough n and
so w(z) is a Lamé polynomial. The proof is similar for Lamé–Wangerin function of the second
kind. �
According to (3.10) we write a Lamé–Wangerin function of the first kind as
w(1)
m = η(ν+1)/2(η2 − η)1/2v(1)m (η, ν, k),
where
v(1)m (η, ν, k) =
∞∑
n=0
anη
n
is given by a power series with radius ≥ η2 > 1. Similarly, we write a Lamé–Wangerin function
of the second kind as
w(2)
m = η(ν+1)/2(η1 − η)1/2v(2)m (η, ν, k).
Theorem 8.3. Let m ∈ N0, ν ∈ R, k ∈ (0, 1).
(a) Suppose that −m − ν 6∈ N, and choose ` ∈ N0 such that H
(1)
m (ν, 0) = (2` + ν + 1)2;
see Lemma 3.2. Then v
(1)
m (·, ν, k) has exactly ` zeros in the unit disk |η| < 1 counted by
multiplicity.
(b) Suppose that −m − ν − 1 6∈ N, and choose ` ∈ N0 such that H(2)(ν, 0) = (2` + ν + 2)2.
Then v
(2)
m (·, ν, k) has exactly ` zeros in the unit disk |η| < 1 counted by multiplicity.
Proof. We prove only (a). The proof of (b) is similar. We normalize the Lamé–Wangerin
functions wm(z) of the first kind by setting wm(K + iK ′) = 1. Then wm(z, ν, k) is the solution
of (1.1) with h = H
(1)
m (ν, k) determined by the initial conditions w(K+iK ′) = 1, w′(K+iK ′) = 0.
By continuous parameter dependence of solutions of initial value problems of linear differential
equations, and using Theorem 3.3, we obtain that wm(z, ν, k) is a continuous function of (z, ν, k)
for z ∈ R, ν ∈ R, k ∈ (0, 1). Since |η| = 1 is in correspondence with z ∈ [0, 2K), we see that
vm(η, ν, k) is a continuous function of |η| = 1, ν ∈ R, k ∈ (0, 1). We want to apply Rouché’s
theorem to the homotopy s 7→ vm(η, sν, k) for s ∈ [0, 1]. If vm(η, sν, k) 6= 0 on the unit circle
|η| = 1 for all s ∈ [0, 1], then vm(·, sν, k) has the same number of zeros in |η| < 1 for s ∈ [0, 1].
Suppose that ν > −m − 1. By Lemma 8.2, the function vm(η, sν, k) has no zeros on the
unit circle |η| = 1 for 0 ≤ s ≤ 1 and so the number of zeros of vm(·, ν, k) in the open unit disk
agrees with that of vm(·, 0, k). It follows from (8.1) that the number of zeros of vm(·, ν, k) in
the open unit disk is equal to m. Under our assumption on (ν,m) we have ` = m, so we obtain
statement (a) for ν > −m− 1.
Now we assume that −p − 1 < ν < −p with p ∈ N and m < p. We use similar homotopies
to show that the number of zeros of vm(·, ν, k) may depend on p and m but not on ν, k ∈ (0, 1).
So we consider ν = −p − 1
2 . Then w is an algebraic Lamé function and vm(η) =
p−1∑
n=0
anη
n is
a polynomial. Let kn ∈ (0, 1) be a sequence converging to 0. Since the vector (a0, a1, . . . , ap−1)
t is
an eigenvector of the matrix S
(1)
p from Section 6 it is easy to see that when properly normalized
the eigenvectors belonging to kn converge to the vector (a0, . . . , ap−1)
t with all components
equal to 0 except a` = 1. Therefore, under the new normalization vm
(
η,−p − 1
2 , kn
)
converges
uniformly to η` as n→ 0+. By Rouché’s theorem, we obtain the desired statement.
This completes the proof. �
Eigenvalue Problems for Lamé’s Differential Equation 19
Using the map (3.7) the unit disk |η| < 1 can be related to a domain in the z-plane. Consider
the rectangle
Q = {z ∈ C : 0 < <z < 2K, 0 < =z < K ′}.
The function z 7→ η is a conformal map from Q onto the unit disk |η| < 1 with a branch cut
along the interval (−1, η1). If z starts at z = 0 and moves clockwise around the boundary of Q,
then η starts at η = −1 and moves in the mathematically positive direction along the unit circle
returning to η = −1 when z = 2K. Then η moves from η = −1 to η = 0 when z reaches z = iK ′.
Then η moves to η1 when z = K + iK ′ and returns to η = 0, then to η = −1. It follows that
the set
Q̃ = {z : 0 ≤ <z ≤ K, 0 < =z ≤ K ′} ∪ {z : K < <z < 2K, 0 < =z < K ′}
is mapped bijectively onto the open unit disk |η| < 1.
Theorem 8.3(a) can be extended to include Lamé polynomials. If ν = −m − 1,−m − 2, . . .
then let `1 be the smallest nonnegative integer n satisfying H
(1)
m (ν, 0) = (2n + ν + 1)2 and `2
the largest such integer. Then vm has `1 zeros in the open unit disk |η| < 1 and `2 zeros in the
closed unit disk |η| ≤ 1. This follows from the known location of zeros of Lamé polynomials [1,
Section 9.4]. Similarly, Theorem 8.3(b) can be extended.
9 The limit k → 0 of Lamé–Wangerin functions
Substituting u = 2K
π s in (3.1), we obtain the differential equation
d2w
ds2
+
4K2
π2
(
h− ν(ν + 1) sn−2
(
2K
π
s
))
w = 0, 0 < s < π. (9.1)
In (9.1) we set h = H
(1)
m (ν, k) and take w = w
(1)
m (s, ν, k) as the corresponding Lamé–Wangerin
eigenfunction normalized by the initial condition
w
(π
2
)
= 1,
dw
ds
(π
2
)
= 0.
By Theorem 3.3, H
(1)
m (ν, k) → H
(1)
m (ν, 0) = (2` + ν + 1)2 as k → 0+, where ` ∈ N0 is chosen
according to Lemma 3.2. As k → 0+ we see that w
(1)
m (s, ν, k) converges to the solution W
(1)
m (s, ν)
of the differential equation
d2W
ds2
+
(
(2`+ ν + 1)2 − ν(ν + 1)
sin2 s
)
W = 0 (9.2)
satisfying the initial conditions
W
(π
2
)
= 1,
dW
ds
(π
2
)
= 0.
The convergence
w(1)
m (s, ν, k)→W (1)
m (s, ν) as k → 0+
is uniform on compact subintervals of (0, π). Differential equation (9.2) appears in the theory
of Gegenbauer polynomials [20, equation (4.7.11)]. We find that
W (1)
m (s, ν) = (sin s)ν+1F
(
−`, `+ ν + 1; 1
2 ; cos2 s
)
,
20 H. Volkmer
where F denotes the hypergeometric function. Equivalently, using Gegenbauer polynomials
G
(λ)
n (x) we have [20, equation (4.7.30)]
W (1)
m (s, ν) = (sin s)ν+1(−1)`
(
`+ ν
`
)−1
G
(ν+1)
2` (cos s).
The binomial coefficient may vanish but the formula remains valid if we take limits ν → ν0 at
exceptional values ν = ν0.
Similarly, let w
(2)
m (s, ν, k) be the solution of (9.1) with h = H
(2)
m (ν, k) satisfying the initial
conditions
w
(π
2
)
= 0,
dw
ds
(π
2
)
= 1.
We choose ` ∈ N0 such that H
(2)
m (ν, 0) = (2`+ ν + 2)2. Then we obtain
w(2)
m (s, ν, k)→W (2)
m (s, ν) = −(sin s)ν+1 cos sF
(
−`, `+ ν + 2; 3
2 ; cos2 s
)
as k → 0+ uniformly on compact subintervals of (0, π). In terms of Gegenbauer polynomials we
have
W (2)
m (s, ν) = (sin s)ν+1(−1)`+1
(
2(ν + 1)
(
`+ ν + 1
`
))−1
G
(ν+1)
2`+1 (cos s).
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Eigenvalue Problems for Lamé’s Differential Equation 21
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https://doi.org/10.1017/CBO9780511608759
1 Introduction
2 Floquet solutions
3 Generalized Lamé–Wangerin functions
4 Analytic continuation of Lamé–Wangerin functions
5 Comparison of eigenvalues
6 Algebraic Lamé functions
7 Lamé polynomials
8 Zeros of Lamé–Wangerin functions
9 The limit k0 of Lamé–Wangerin functions
References
|
| id | nasplib_isofts_kiev_ua-123456789-209873 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-05T00:14:17Z |
| publishDate | 2018 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Volkmer, H. 2025-11-28T09:37:42Z 2018 Eigenvalue Problems for Lamé's Differential Equation / H. Volkmer // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 23 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 33E10; 34B30 arXiv: 1808.04877 https://nasplib.isofts.kiev.ua/handle/123456789/209873 https://doi.org/10.3842/SIGMA.2018.131 The Floquet eigenvalue problem and a generalized form of the Wangerin eigenvalue problem for Lamé's differential equation are discussed. Results include comparison theorems for eigenvalues and analytic continuation, zeros, and limiting cases of (generalized) Lamé-Wangerin eigenfunctions. Algebraic Lamé functions and Lamé polynomials appear as special cases of Lamé-Wangerin functions. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Eigenvalue Problems for Lamé's Differential Equation Article published earlier |
| spellingShingle | Eigenvalue Problems for Lamé's Differential Equation Volkmer, H. |
| title | Eigenvalue Problems for Lamé's Differential Equation |
| title_full | Eigenvalue Problems for Lamé's Differential Equation |
| title_fullStr | Eigenvalue Problems for Lamé's Differential Equation |
| title_full_unstemmed | Eigenvalue Problems for Lamé's Differential Equation |
| title_short | Eigenvalue Problems for Lamé's Differential Equation |
| title_sort | eigenvalue problems for lamé's differential equation |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/209873 |
| work_keys_str_mv | AT volkmerh eigenvalueproblemsforlamesdifferentialequation |