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Asymptotics of eigenvalues and eigenfunctions of energy-dependent Sturm--Liouville equations

We study asymptotics of eigenvalues, eigenfunctions and norming constants of singular energy-dependent Sturm--Liouville equations with complex-valued potentials. The analysis essentially exploits the integral representation of solutions, which we derive using the connection of the problem under study and a Dirac system of a special form.


Introduction
The main objective of the present paper is to study asymptotics of eigenvalues and eigenfunctions of Sturm-Liouville equations on (0,1) with energy-dependent potentials, viz.
We restrict our attention to such boundary conditions just to concentrate on the ideas and to avoid unnecessary technicalities. Other separated boundary conditions can be treated analogously; in particular, in the last section we formulate some results for the case of the mixed boundary conditions. Energy-dependent Sturm-Liouville equations are of importance in classical and quantum mechanics. For instance, they are used for modelling mechanical systems vibrations in viscous media. The Klein-Gordon equations, which describe the motion of massless particles such as photons, can also be reduced to the form (1.1). The corresponding evolution equations are used to model the interactions between colliding relativistic spinless particles. In such mechanical models the spectral parameter λ is related to the energy of the system, which explains the terminology "energy-dependent" used for the spectral equation (1.1).
Asymptotic behaviour of eigenvalues, eigenfunctions, and other spectral characteristics for usual Sturm-Liouville operators are studied sufficiently well (see, e.g. [10,12,16]). The spectral problem for energydependent Sturm-Liouville equation (1.1) with p ∈ W 1 2 [0, π] and q ∈ L 2 [0, π] was considered by M. Gasymov and G. Guseinov in their short paper [3] of 1981. Some results on asymptotics are formulated there without proofs. Analogous problems under more general boundary conditions were also considered by I. Nabiev in [15]. The asymptotics of eigenvalues and eigenfunctions for Sturm-Liouville equations with singular potentials q ∈ W −1 2 (0, 1) were studied by A. Savchuk and A. Shkalikov in [19].
Our aim in this paper is to investigate the corresponding asymptotics for energy-dependent Sturm-Liouville equations (1.1) under minimal smoothness assumptions on the potentials p and q, including, e.g. the case when q contains Dirac delta-functions and/or Coulumb-like singularities. Our approach consists in establishing a strong connection between the spectral problem (1.1) and the spectral problem for a Dirac system of a special form. We then study the latter and, in particular, derive the integral representation for the solution of (1.1) which, in turn, is a basis for the subsequent asymptotic analysis The paper is organised as follows. In the next section, we rigorously set the spectral problem under study and give main definitions. Connection between the spectral problem (1.1) and that for a special Dirac system is discussed in Section 3. In Section 4, we construct the transformation operator relating the solution of the obtained system with that of the Dirac system with zero potential. Next in Section 5, we derive the asymptotics of eigenvalues, eigenfunctions and the corresponding norming constants for the problem (1.1), (1.2) and justify the factorization formula for its characteristic function. In the last section we formulate analogous results for the spectral problem (1.1) under more general boundary conditions. Appendix A contains main definitions from the spectral theory for operator pencils. We also prove there that the algebraic multiplicity of λ as an eigenvalue of the problem (1.1), (1.2) coincides with that of λ as an eigenvalue of the corresponding operator pencil.
Notations. Throughout the paper, we denote by M 2 = M 2 (C) the linear space of 2 × 2 matrices with complex entries endowed with the Euclidean operator norm. Next, p 0 will stand for 1 0 p(s)ds. The superscript t will signify the transposition of vectors and matrices, e.g. (c 1 , c 2 ) t is the column vector c 1 c 2 .

Preliminaries
In this section, we recall main definitions and formulate the spectral problem under study more rigorously. To start with, introduce the differential expression ℓ(y) := −y ′′ + qy. Since the potential q is a complex-valued distribution, we need to define the action of ℓ in detail. To do this, we use the regularization by quasi-derivative method due to Savchuk and Shkalikov [19,20]. Take a function r from L 2 (0, 1) such that q = r ′ and for every absolutely continuous y introduce its quasi-derivative y [1] := y ′ −ry. Then define ℓ(y) as A straightforward verification shows that so defined ℓ(y) coincides with −y ′′ + qy in the distributional sense. Therefore we can recast the equation (1.1) as (2.1) ℓ(y) + 2λpy = λ 2 y.
A number λ ∈ C is called an eigenvalue of the problem (1.1), (1.2) if equation (2.1) possesses a nontrivial solution satisfying the boundary conditions (1.2). This solution is then called an eigenfunction of the problem (1.1), (1.2) corresponding to λ.
Let y(x, z) be the solution of (2.1) with z instead of λ subject to the initial conditions y(0) = 0, y [1] (0) = 1. This solution exists and is unique [19], so that λ is an eigenvalue of the problem (1.1), (1.2) if and only if it is a zero of the characteristic function ϕ(z) := y(1, z). The corresponding eigenfunction then coincides up to a constant factor with y(·, λ). The multiplicity of λ as a zero of ϕ(z) is called an algebraic multiplicity of the eigenvalue λ of (1.1), (1.2).
As we shall see further ϕ is an analytic nonconstant function, and so the set of its zeros is a discrete subset of C. This shows that the set of eigenvalues of (1.1),(1.2) is discrete. Without loss of generality, we shall make the following standing assumption: (A) 0 is not a zero of the characteristic function ϕ(z), i.e. it is not an eigenvalue of the problem (1.1), (1.2).
In fact, (A) is achieved by shifting the spectral parameter λ if necessary; then for λ 0 such that ϕ(λ 0 ) = 0 the problem (1.1), (1.2) with new spectral parameter µ := λ − λ 0 and with p and q replaced by p + λ 0 and q − 2λ 0 p − λ 2 0 respectively the assumption (A) holds. The spectral problem (1.1), (1.2) can be regarded as the spectral problem for the quadratic operator pencil T (see [14,17]) defined by (2.2) T (λ)y := λ 2 y − 2λpy + y ′′ − qy on the λ-independent domain For the pencil T one can introduce the notions of the spectrum, the eigenvalues and corresponding eigenvectors, their geometric and algebraic multiplicities (see, e.g. [14] and Appendix A). The spectral properties of T were discussed in [17]. In particular, it was proved therein that the spectrum of T consists only of eigenvalues, which can easily be shown to coincide with the eigenvalues of the problem (1.1), (1.2) defined above. In Appendix A, we show that the algebraic multiplicity of λ as an eigenvalue of (1.1), (1.2) coincides with that of λ as an eigenvalue of T .

Reduction to the Dirac system
In this section we reduce equation (1.1) to a λ-linear Dirac-type system of the first order. We shall further use the connection between (1.1) and this system to derive the asymptotics of interest.
The following observation plays an important role in the reduction procedure. Proof. Note firstly that for every complex a, b and every x 0 from [0, 1] the equation ℓ(y) = 0 possesses a unique solution satisfying the conditions y(x 0 ) = a and y [1] (x 0 ) = b (see, e.g. [19]).
Assume that y is a solution of ℓ(y) = 0. We introduce the polar coordinates ρ and θ via y(x) = ρ(x) sin θ(x) and y [1] (x) = ρ(x) cos θ(x). Clearly, the solution y vanishes at some point x 0 from [0, 1] if and only if θ(x 0 ) = πk for some k ∈ Z. The function θ is called the Prüfer angle (see [4,20]) and can be defined to be continuous; it then satisfies the equation Equation (3.1) has the form θ ′ = f (x, θ) with the right-hand side f that is not continuous in x. However, (3.1) is a so-called Caratheodory equation and under the initial condition θ(ξ) = 0 with ξ ∈ [0, 1] it possesses a unique solution θ(x, ξ) (see e.g. [2, Theorem 1.1.2]); this solution depends continuously on ξ (see [2,Theorem 2.8.2]). Therefore the mapping ξ → θ(0, ξ) is continuous and its image I 0 is a compact subset of C containing 0 as a continuious image of a compactum [0, 1]. Note further that for any k ∈ Z the solutions θ k (x, ξ) of the problem (3.1) with θ(ξ) = πk are equal to θ k (x, ξ) = θ(x, ξ) + πk. The images I k of the mappings ξ → θ k (x, ξ) are compact and are the shifts of I 0 by πk along the real axis.
Let us now take any complex number, say θ 0 , outside the union of all the compacta I k , k ∈ Z, and consider the problem (3.1) with θ(0) = θ 0 . In view of the above arguments and uniqueness of the solution of the corresponding initial-value problem, the solution of (3.1) with θ(0) = θ 0 can equal to πk, k ∈ Z, at no point of the interval [0, 1]. Consider the solution y 0 of ℓ(y) = 0 subject to the initial conditions y(0) = 1, y [1] (0) = cot θ 0 . Then y 0 does not vanish on [0, 1], which is the assertion of the lemma.
For λ = 0 consider the functions u 2 := y and u 1 := (y ′ − vy)/λ and recast the spectral equation (1.1) as the following first order system for u 1 and u 2 : we see that the above system is the spectral problem ℓ(P )u = λu for a Dirac differential expression ℓ(P ) acting in L 2 (0, 1) × L 2 (0, 1) via It was shown in [18] that the spectral problem (1.1), (1.2) is closely related to the spectral problem for the Dirac operator D(P ) defined by the differential expression ℓ(P ) on the domain In particular, the nonzero spectra for both problems coincide counting with multiplicity. Moreover, u = (u 1 , u 2 ) t is an eigenfunction of the operator D(P ) corresponding to the eigenvalue λ = 0 if and only if y = u 2 is an eigenfunction of (1.1), (1.2) corresponding to λ and u 1 = (u ′ 2 − vu 2 )/λ. By assumption (A), λ = 0 is not an eigenvalue of (1.1), (1.2); however, it is in the spectrum of D(P ): Proof. A straightforward verification shows that λ = 0 is an eigenvalue of D(P ), and that every corresponding eigenfunction must be collinear to u 0 = (u 1 , u 2 ) t , where u 1 = exp{− v}, u 2 ≡ 0. Therefore the eigenvalue λ = 0 is geometrically simple.
Suppose that the algebraic multiplicity of λ = 0 is greater than one. Then there exists a vector w = (w 1 , w 2 ) t from the domain of D(P ) associated with u 0 , i.e. satisfying the equality D(P )w = u 0 .
requires that w 2 (0) = w 2 (1) = 0 and thus either w 2 is an eigenfunction of (1.1), (1.2) for the eigenvalue λ = 0 or w ≡ 0. The first possibility is ruled out by assumption (A), while the second one is impossible in view of the relation w ′ 2 − vw 2 = u 1 . The contradiction derived shows that no such w exists and finishes the proof.

Transformation operator
In this section we construct the so-called transformation operator relating the solution of the system ℓ(P )u = λu and that of ℓ(P 0 )u = λu with zero matrix potential P 0 (i.e. with matrix potential having all components zero).
Denote by U(x, λ) a 2×2 matrix-valued function satisfying the equation and the initial condition U(0) = I.
This theorem is very similar to the corresponding theorem of [1] and its proof requires only minor modifications.
Proof. Observe that the system (4.1) can be rewritten as The variation of constants method shows that U satisfies the following integral equation: This equation can be solved by the method of successive approximation. Setting we see that the solution of the above equation can formally be given by the sum ∞ n=0 U n . We shall prove below that Observe firstly that the matrix Q anti-commutes with J and therefore e −tJQ (s) =Q(s)e tJ .
Let us estimate the L 2 -norm of K n (x, ·): which yields (4.5). The estimate of the norm K n (x, ·) 2 implies also the convergence of the series K(x, ·) := ∞ n=1 K n (x, ·) in L 2 ((0, 1), M 2 ) with Therefore the first statement of the theorem is proved.
Observe that the 2 × 2 matrix U 0 = e −λxJ is a solution of the system ℓ(P 0 )U = λU with zero potential P 0 . Therefore, in view of the last theorem, the solution U of the problem (4.1) can be obtained from U 0 by means of the transformation operator T = R + K , where R is an operator of multiplication by exp{J x 0 p} and K is an integral operator acting as follows This transformation operator also performs similarity of the corresponding differential expressions, namely, ℓ(P )T = T ℓ(P 0 ).

Asymptotics
In this section, we derive the asymptotics of eigenvalues and eigenfunctions and the corresponding norming constants of the problem (1.1) under the Dirichlet boundary conditions (1.2). We also obtain the factorization of the characteristic function for the problem under study.
In view of this asymptotics, all but finitely many zeros of u 2 (1, λ) are simple.
Next note that u 2 (1, λ) is a characteristic function of the operator D(P ) defined in Section 3, whence λ = 0 is a zero of u 2 (1, λ) of order 1 (see Lemma 3.3). However, under assumption (A) it is not a zero of the characteristic function ϕ(λ). Therefore the set of all the eigenvalues of the problem (1.1), (1.2) coincides with the set of zeros of the function u 2 (1, λ) different from 0. Thus the following theorem holds true.
Next we construct the factorization of the characteristic function ϕ(λ), which allows to determine ϕ(λ) via the eigenvalues of (1.1), (1.2). We use this factorization to derive the formula determining the norming constants of (1.1), (1.2) via two spectra of the equation (1.1) under two types of boundary conditions (see [17]).
Theorem 5.2. Suppose that λ n , n ∈ Z * , are the eigenvalues of the spectral problem (1.1), (1.2). Then the characteristic function ϕ(λ) can be factorized in the following way: Proof. Suppose firstly that p 0 = πl, l ∈ Z. Observe that the function u 2 (1, λ) is given by (5.1) and so it is of exponential type 1. Recall also that, by Lemma 3.3, λ = 0 is a zero of u 2 (1, λ) of order 1. Therefore by Hadamard factorization theorem (see e.g. [21]) we have 1 where A and B are some constants and λ n are the zeros of the function u 2 (1, λ). In view of the asymptotic distribution (5.2), the series λ λn converges. Indeed, π 2 n 2 λ n λ −n · λ n + λ −n π 2 n 2 , and we note that the series ∞ n=1 λn+λ −n π 2 n 2 is absolutely convergent and the sequence π 2 n 2 λnλ −n is uniformly bounded. Therefore we can write To find the values A ′ and B, consider the ratio u 2 (1,λ) sin(λ−p 0 ) and find its limits along the ray λ = re iθ , θ = 0, π. In view of (5.1) and a refined version of the Riemann-Lebesgue lemma [13, Lemma 1.3.1], we have Recall (see e.g. [21]) that the function sin(λ − p 0 ) can be factorized as follows: where ν n are the zeros of sin(λ − p 0 ), i.e. ν n = πn + p 0 . Therefore we have Let us show that A ′ = 0. If A were not 0, then one could choose the direction θ such that ReA ′ re iθ tends to infinity as r → ∞. Next note that by Lemma 5.3 given below the product If p 0 = πl for some l ∈ Z, then πn − λ πn , and so Using this, (5.3) and the arguments analogous to above, we obtain The proof is complete.
Proof. We firstly prove the result analogous to Lemma 3.1. of [5] for sequences of complex numbers. For every ε > 0, set δ = δ(ε) := sup is analytic in the domain D := {z ∈ C | |1 + z| ≥ ε} and tends to zero as |z| → ∞. Using arguments analogous to those in the proof of the mentioned lemma, we obtain that if (a n ) is a sequence of complex numbers from ℓ 2 such that a n converges and |1 + a n | ≥ ε for all n ∈ Z, then Using arguments analogous to those in the proof of one can show that for each ε > 0, there exists δ = δ(ε) such that whenever the sequence (a n ) of numbers from ℓ 2 (Z) with | a n | < ∞ satisfies the inequality |a n + 1| > ε for all n ∈ Z, the following estimate is valid Therefore, for ε such that λn πn > ε Observe that V.p.
p 0 πn = 0. The sequences (λ n ) and (1/n) are from ℓ 2 , so that the series λ n πn is convergent; as a result, the first summand in the righthand side of the last estimate is finite. Since the sequence (1/πn) is from ℓ 2 and p 0 +λn πn 2 ≤ C π 2 n 2 with C := sup n (p 0 +λ n ) 2 , the second summand is finite as well. All these arguments imply that λn πn is convergent and so complete the proof.
Since the sequences (λ n ) and (1/n) belong to ℓ 2 , the series ∞ n=−∞ |λn| |n| is convergent and so the series (5.5) is convergent uniformly in r > R θ for a fixed θ, θ = 0, π. Therefore, which means that the product Note that the vectors v n := (cos(πn + p 0 )x, sin(πn + p 0 )x) t form an orthonormal basis in L 2 (0, 1) × L 2 (0, 1). Next observe that where C := max n∈Z e |λn| . Since the sequence (λ n ) belongs to ℓ 2 , this means that the sequence u n,0 is quadratically close to the orthonormal basis and therefore it is a Bari basis [21]. Then Since u n,0 is quadratically close to v n , the sequence ( L (u n,0 − v n ) ) belongs to ℓ 2 . Next observe that the operator L is of Hilbert-Schmidt class. Therefore ( L v n ) also belongs to ℓ 2 (see e.g. [8, V.2.4.]). All these arguments imply that the sequence ( u n − Ru n,0 ) is from ℓ 2 . Since | u n − 1| ≤ u n − Ru n,0 , we obtain that where (α n ) belongs to ℓ 2 .
Observe also that if p and q are real-valued, the norming constants of (1.1), (1.2) corresponding to real and simple eigenvalues coincide with the norms of eigenvectors of the operator D(P ) (see [6]). Therefore the following holds true. where (α n ) ∈ ℓ 2 .
Using the asymptotics form of u 1 and u 2 , we have Taking into account Theorem 4 from [11], we obtain Theorem 6.1. The eigenvalues of (1.1), (6.1) can be labelled according to their multiplicities as µ n , n ∈ Z, so that they satisfy the following asymptotics (6.2) µ n = π n + 1 2 + p 0 +μ n , with ℓ 2 -sequence (μ n ). In particular, all eigenvalues µ n with large enough |n| are simple.
Analogously to the case of the Dirichlet boundary conditions, we prove the following Theorem 6.2. Let µ n , n ∈ Z, be the eigenvalues of (1.1), (6.1). Then the characteristic function ψ(µ) can be factorized in the following way Theorem 6.3. The eigenfunctions y n of (1.1), (6.1) corresponding to the eigenvalues µ n satisfy the asymptotics where the sequence ( ỹ n (x) ) is from ℓ 2 .
Set τ (λ)y := λ 2 y − 2λpy − ℓ(y). Straightforward verification shows that y j satisfy equalities (A.1) with τ instead of T . Moreover, since λ is a zero of y(1, z) of order m, we have y 1 (1) = · · · = y m−1 (1) = 0, and all the functions y j , j = 0, . . . , m − 1, belong to the domain of T and so form a chain of eigen-and associated vectors of T corresponding to λ. Therefore m does not exceed the algebraic multiplicity of λ as an eigenvalue of T .
Assume that v 0 , . . . , v l is a chain of eigen-and associated vectors corresponding to an eigenvalue λ of T . Then v 0 solves the equation τ (λ)y = 0 and satisfies the boundary conditions (1.2), and thus coincides with y 0 up to a scalar factor. Without loss of generality, we assume that v 0 = y 0 and then show by induction that there exists a sequence (c k ) l k=1 such that To start with, observe that τ ′ (λ)(v 0 − y 0 ) + τ (λ)(v 1 − y 1 ) = 0.
By assumption we obtain Therefore