Eigenvalue Problems for Lam\'e's Differential Equation

The Floquet eigenvalue problem and a generalized form of the Wangerin eigenvalue problem for Lam\'e's differential equation are discussed. Results include comparison theorems for eigenvalues and analytic continuation, zeros and limiting cases of (generalized) Lam\'e-Wangerin eigenfunctions. Algebraic Lam\'e functions and Lam\'e polynomials appear as special cases of Lam\'e-Wangerin functions.


Introduction
The Lamé equation [ where sn(z, k) is the Jacobian elliptic function with modulus k ∈ (0, 1) [14, Ch. 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 Lame-Wangerin eigenvalue problem is obtained when we require that (sn z) 1/2 w(z) stays bounded at the singularities iK ′ and 2K + iK ′ ; see [4, §15.6]. In Section 3 of this paper we consider the more general eigenvalue problem whose eigenfunctions w(z) have the form at z = iK ′ and a similar condition at z = 2K + iK ′ .
In Section 5 we compare the eigenvalues of these problems. We show that every (generalized) Lamé-Wangerin function is also a Floquet eigenfunction. In Sections 6 and 7 we show that algebraic Lamé functions and Lamé polynomials are special cases of 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.
Of course, some results of this paper are well-known but there are new results. The treatment of the generalized Lamé-Wangerin problem is new. In particular, we show in Theorem 3 that Lamé-Wangerin eigenvalues and eigenfunctions are determined by an infinite formally symmetric tridiagonal matrix operator. In the theory of the Lamé equation several finite and infinite tridiagonal matrices appear, but they are usually not symmetric. Also, the treatment of algebraic Lamé functions in Section 6 is quite different from treatments in Erdélyi [3], Ince [6] and Lambe [9].

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 (2.1) w(z + 2K) = e iπµ w(z), z ∈ R.
Let w 1 (z, h, ν, k) and w 2 (z, h, ν, k) be the solutions of (1.1) satisfying the initial conditions w 1 (0) = 1, dw 1 dz (0) = 0, w 2 (0) = 0, dw 2 dz (0) = 1. Then Hill's discriminant D is given by where am is Jacobi's amplitude function. We note that (2.5) establishes a conformal mapping between the strip |ℑz| < K ′ and the t-plane cut along the rays mπ ± isL, s ≥ 1, m ∈ Z, where We obtain Since am(z + 2K) = am z + π, condition (2.1) becomes This condition is equivalent to e iµt w(t) being periodic with period π. Therefore, using Fourier series, eigenfunctions have the form By substituting (2.8) in (2.6), we obtain the three-term recursion . The behavior of solutions {c n } n∈Z of (2.9) as n → ∞ is given by Perron's rule [12]. If k ∈ (0, 1) we choose n 0 so large that ρ n = 0 and τ n+1 = 0 for n ≥ n 0 . Then the solutions {c n } n>n 0 of equations (2.9) for n ≥ n 0 form a two-dimensional vector space. There exists a recessive solution which is uniquely determined up to a constant factor with the property Every solution which is linearly independent of this solution satisfies Similar results hold for n → −∞. We obtain the following theorem.
Theorem 2. Let µ, ν ∈ R and k ∈ (0, 1). Then h is one of the eigenvalues h m (µ, ν, k) if and only if the recursion (2.9) has a nontrivial solution {c n } n∈Z such that a) either there is n 0 such that c n = 0 for n ≥ n 0 or {c n } is recessive as n → ∞; and b) either there is n 0 such that c n = 0 for n ≤ n 0 or {c n } is recessive as n → −∞.
The expansion (2.8) of a corresponding eigenfunction converges in the strip |ℑt| < L.
Of course, a nontrivial solution {c n } of (2.9) can be zero for n ≥ n 0 or n ≤ n 0 only when one of the numbers ρ n or τ n 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 5,6,7. Alternatively, we may expand Then we obtain the "adjoint" recursion Theorem 2 also holds with (2.13) in place of (2.9).
The eigenvalue problem splits into two problems, one for functions that are even with respect to K + iK ′ , that is, and one for functions which are odd with respect to K + iK ′ , that is, Without loss of generality, one may assume that ν ≥ − 1 2 , and since the exponents at K ′ and 2K + iK ′ are {ν + 1, −ν}, a Lamé-Wangerin function has the form for z close to iK ′ with q 0 = 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 q 0 = 0 is not required in (3.4) although q 0 = 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 [10]) 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 in (2.6). We obtain the Fuchsian equation . The differential equation (3.6) has regular singularities at η = 0, Setting z = u + iK ′ for 0 < u < K this gives This establishes a bijective increasing map between u ∈ (0, K) and η ∈ (0, η 1 ). 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, η 1 ) which close to η = 0 is of the and which is analytic at η = η 1 . This implies that the radius of convergence of the power series in (3.8) is ≥ η 2 . For the coefficients c n we find the recursion Note that the equations (3.9) for n ≥ 1 agree with (2.9) when we set µ = ν + 1. The recursion (3.9) is given in [4, §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 {c n } ∞ n=0 which is either identically zero for large n or satisfies (2.10). Of course, a nontrivial solution {c n } ∞ n=0 of (3.9) can be identically zero for large n only if one of the numbers α n 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 a n η n with the power series having radius ≥ η 2 . In order to find the recursion for the coefficients a n we transform (3.6) by setting We obtain the recursion δ n a n−1 + (ǫ (1) n − h)a n + δ n+1 a n+1 = 0, n ≥ 1, 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 ℓ 2 (N 0 ) with the standard inner product. The domain of definition of S is 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 ∈ N 0 . 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 To prove this it is convenient to write B = B 1 + B 2 + B 3 where each B i has a matrix representation consisting of only one nonzero "diagonal", and We can reach c < 1 because the factor of n 2 on the main diagonal of A is 4 while the factors of n 2 on the three diagonals of B are −k 2 , −2k 2 , −k 2 , respectively. From (3.13) we obtain that T := B(A + λ) −1 is a bounded linear operator with operator norm T ≤ c < 1. Therefore, 1 + T is invertible and This shows that (S + λ) −1 is a compact operator. Since S is symmetric, we find that S is self-adjoint; compare [8, Ch. V, Thm 4.3]. From (3.13) we also obtain that S + λ is positive definite [8, Ch. V, Thm 4.11]. Therefore, (a) follows. (b) h is an eigenvalue of S if and only if the recursion (3.12) has a nontrivial solution {a n } ∞ n=0 with the property that ∞ n=0 n 4 |a n | 2 < ∞. By Perron's rule the latter property is equivalent to a n = 0 for large n or {a n } 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 we write the eigenvalues of the first Lamé-Wangerin problem with k ∈ (0, 1) in the form . If a normalization is required it will be stated separately. We note that the corresponding eigenvectors {a n } ∞ n=0 of S when properly normalized form an orthonormal basis in the Hilbert space ℓ 2 (N 0 ).
The eigenvalues of S (1) (ν, 0) are (2n+ν +1) 2 for n ∈ N 0 . 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.
We will need continuity of the eigenvalues H (1) m (ν, k).
Let ν 0 > 0 and k 0 ∈ (0, 1) be given, and set Ω : 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 Suppose we have a sequence (ν n , k n ) ∈ Ω which converges to (ν,k) as n → ∞. Then we can easily show using the definitions of A and T that Using (3.14) we then obtain that If K n 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 ∈ N 0 . This follows directly from the minimum-maximum-principle m (ν,k) as n → ∞ for every m ∈ N 0 as desired.
A Lamé-Wangerin function of the second kind can be written in the form where the power series d n η n has radius ≥ η 2 . If we set in (3.6), we obtain This gives the recursion (β n − h)d n + γ n+1 d n+1 = 0, n ≥ 1, where α n , γ n are as in (3.9) and Alternatively, a Lamé-Wangerin function of the second kind can be written as where the power series b n η n has radius ≥ η 2 . If we set in (3.6), we obtain (3.11) with η 1 , η 2 interchanged. This gives the recursion (ǫ

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 ≤ ℑz < K ′ as follows. Using (3.7) and (3.8) a Lamé-Wangerin function w (1) of the first kind can be written as Since the power series c n η n has radius larger than 1 and |η| ≤ 1 for 0 ≤ ℑz < K ′ , the expansion (4.1) converges in the strip 0 ≤ ℑz < K ′ .
If a Lamé-Wangerin function w (2) of the second kind is given by (3.19) then its analytic continuation is

Comparison of eigenvalues
Every Lamé-Wangerin function is also a Floquet eigenfunction.
If µ + ν or µ − ν is an integer then the Floquet eigenvalues h m (µ, ν) can be expressed in terms of Lamé-Wangerin eigenvalues H (j) m (ν). The properties (2.3) show that it is sufficient to consider the case µ = ν + 1. Then we have the following result.
We now compare the eigenvalues H   We now compare the eigenvalues H Theorem 9. Let ν ∈ R, 0 < k < 1 and H (j) where, for m = 0, 1, . . . , p − 1, 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 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).
m (ν, k) and part (b) show that H 2 the rest of statement (c) follows from part (a).

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 [3], [6], [9]. We obtain these functions as follows.
We note that S (1) p is the mirror image of S (2) p with respect to the antidiagonal, that is, we have It follows that S (1) p and S (2) p have the same eigenvalues and the corresponding eigenvectors are inverse to each other, that is, if (a 0 , a 1 , . . . , a p−1 ) t is an eigenvector for S (1) p then (a p−1 , a p−2 , . . . , a 0 ) t is an eigenvector for S If (a 0 , a 1 , . . . , a p−1 ) t is a (real) eigenvector of S (1) p then a n η n , (6.1) 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 (η j − η) 1/2 v(s) and η = η 1 s, then we obtain the Heun equation for v(s) and p−1 n=0 a n (η 1 s) n is a Heun polynomial. If we substitute (3.7) in (6.1), (6.2) and use the functions J 1 (z), J 2 (z) defined in (4.8), (4.11) we obtain We know from Lemma 6 that Moreover, we have and, for x ∈ R, 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. The algebraic Lamé functions which are even or odd with respect to z = K were considered in [3].
In the simplest case ν = − 3 2 we have H
Therefore, the space of symmetric vectors {c n } p n=0 (c n = c p−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 2 p + 1 Lamé polynomials of the form P (sn 2 z) where P is a polynomial of degree 1 2 p if we use symmetric eigenvectors, and 1 2 p Lamé polynomials of the form cn z sn zP (sn 2 z) where P is a polynomial of degree 1 2 p − 1 if we use antisymmetric eigenvectors. If p is odd we find 1 2 (p + 1) Lamé polynomials of the form sn zP (sn 2 z) where P is a polynomial of degree 1 2 (p − 1) if we use symmetric eigenvectors, and 1 2 (p + 1) Lamé polynomials of the form cn zP (sn 2 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.
Proof. If µ is not an integer then a nontrivial Floquet solution w(z), z ∈ R, of (1.1) with w(z + 2K) = e iµπ w(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 e iµπ , e −iµπ are distinct. So w(z) and its conjugate function are linearly independent. It follows from Lemma 6 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 eigenvalue H (1) m (ν). Suppose that w(z 0 ) = 0. with z 0 ∈ R. Using (3.8) and the substitution (2.5) we have and this function has a zero at t 0 ∈ R. The coefficients c n are real so the functions both vanish at t = t 0 . The functions (8.2), (8.3) are both solutions of the differential equation (2.6) with the same values for h and ν. Since they have a common zero these solutions must be linearly dependent. Now ℜw(t) is an even function and ℑw(t) is odd. So one of the functions ℜw(t), ℑw(t) must vanish identically. This implies that c n = 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.
The binomial coefficient may vanish but the formula remains valid if we take limits ν → ν 0 at exceptional values ν = ν 0 .