Spectral analysis of indefinite Sturm-Liouville operators

Abstract

For a class of indefinite J -nonnegative Sturm-Liouville operators, we present a criterion of similarity to a self-adjoint operator. This criterion is formulated in terms of Weyl-Titchmarsh m-functions. Moreover, using this result, we obtain a criterion, as well as simple sufficient conditions, formulated in terms of the coefficients of a given Sturm-Liouville operator.Key words: Sturm-Liouville operator, J -nonnegative operator, critical point, similarity to a selfadjoint operator. Let∞, and let q and w be real-valued functions such that q, w ∈ L 1 loc (I ) and w(x) > 0 for a.e. x ∈ I . Consider the operatorIn the rest of the paper, we shall assume that the differential expression is singular at the endpoints of the interval I and, moreover, the limit point case prevails at both b and −b, i.e., the operator L := JA is self-adjoint.HereIt is not difficult to observe that A = A * . However, (JA) * = L * = L = JA, that is, A is J -selfadjoint. Our main objective is to answer the question of whether A is similar to a self-adjoint operator.2. Spectral properties of J -self-adjoint Sturm-Liouville operators are studied well enough in the case when the corresponding differential expression is regular. For example, consider the spectral problem −y + q(x)y = z sgn(x)w(x)y, x ∈ (−1, 1), y(−1) = y(1) = 0, (2) assuming that w, q ∈ L 1 (−1, 1) and w > 0 a.e. on (−1, 1) (for the case of general self-adjoint boundary conditions, see [5] and [15]). It is known [6] that the spectrum of (2) is discrete and symmetric with respect to R, and the nonreal part of the spectrum consists of finitely many eigenvalues of finite algebraic and geometric multiplicity. There is a vast literature devoted to the study of the Riesz basis property in the Hilbert space L 2 w (−1, 1) of a system of eigen-and associated functions of (2) (see [2], [4]-[6], [11], [14], [15], and the references therein). The existence of weights w such that the eigen-and associated functions of (2) do not form a Riesz basis was proved by H. Volkmer, and the first explicit examples were constructed later by A. Fleige, N. L. Abasheeva, and S. G. Pyatkov (see [4] and [14]). In [14] (see also [4]) Parfenov proved the following criterion in the case of even weights: If w(x) = w(−x) > 0 for a.e. x ∈ (−1, 1), then the eigen-and associated functions of (2) form a Riesz basis in L 2 w (I ) if and only if there is a t ∈ (0, 1) such that S 0 (t) := lim sup x→0 W (xt) W (x) = 1, W(x) := x 0 w(s) ds.The latter means that the function W is positively increasing at 0. Note that the proof in [14] is based on Pyatkov's interpolation criterion.

Perturbation of eigenvalues of the Klein–Gordon operators

“…• the operator sgn x |x| α − d 2 dx 2 + cδ with α > −1 and δ the Dirac delta, see [15], • the one-dimensional Schrödinger-type operators in a bounded interval with nonself-adjoint Robin-type boundary conditions, see [17], and…”