Sturm–Liouville operators

Abstract

Let (a, b) ⊂ R be a finite or infinite interval, let p 0 (x), q 0 (x), and p 1 (x), x ∈ (a, b), be real-valued measurable functions such that p 0 , p −1 0 , p 2 1 p −1 0 , and q 2 0 p −1 0 are locally Lebesgue integrable (i.e., lie in the space L 1 loc (a, b)), and let w(x), x ∈ (a, b), be an almost everywhere positive function. This paper gives an introduction to the spectral theory of operators generated in the space L 2 w (a, b) by formal expressions of the formwhere all derivatives are understood in the sense of distributions. The construction described in the paper permits one to give a sound definition of the minimal operator L 0 generated by the expression l[f ] in L 2 w (a, b) and include L 0 in the class of operators generated by symmetric (formally self-adjoint) second-order quasi-differential expressions with locally integrable coefficients. In what follows, we refer to these operators as Sturm-Liouville operators. Thus, the well-developed spectral theory of second-order quasi-differential operators is used to study Sturm-Liouville operators with distributional coefficients. The main aim of the paper is to construct a Titchmarsh-Weyl theory for these operators. Here the problem on the deficiency indices of L 0 , i.e., on the conditions on p 0 , q 0 , and p 1 under which Weyl's limit point or limit circle case is realized, is a key problem. We verify the efficiency of our results for the example of a Hamiltonian with δ-interactions of intensities h k centered at some points x k , where2010 Mathematics Subject Classification. Primary 34B24; Secondary 34B20, 34B40. Key words and phrases. Second-order quasi-differential operator, minimal operator, maximal operator, Sturm-Liouville theory, deficiency index, limit point-limit circle, linear differential equation with distributional coefficients, finite-difference equation, Jacobi matrix.

Solving An Inverse Problem For The Sturm-Liouville Operator With A Singular Potential By Yurko's Method

“…The basic results of inverse problem theory were obtained for operators induced by the Sturm-Liouville expression ℓy := −y ′′ +q(x)y with regular (i.e., square integrable) potential q (see [1,2,3,4]). In recent years, spectral analysis of differential operators with singular coefficients from spaces of distributions has attracted much attention of mathematicians (see [5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32]). Properties of spectral characteristics and solutions of differential equations with singular coefficients were studied in [5,6,7,8,9,10,11,12,13,14].…”

“…In recent years, spectral analysis of differential operators with singular coefficients from spaces of distributions has attracted much attention of mathematicians (see [5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32]). Properties of spectral characteristics and solutions of differential equations with singular coefficients were studied in [5,6,7,8,9,10,11,12,13,14]. Some aspects of inverse problem theory for differential operators with singular coefficients have been investigated in [15,16,17,18,19,20,21,22,23,24,…”

Spectral Theory of Infinite Quantum Graphs

“…This effect was discovered by C. Shubin Christ and G. Stolz [104, pp. 495-496] in the special case |e k | = 1/k and α k = −(2k + 1), k ∈ N. For further details and results we refer to [66], [81]. ♦ Our main aim is to find a boundary relation Θ α parameterizing the operator H α in terms of the boundary triplet Π G given by (2.13)-(2.15).…”

Hamiltonian Systems and Sturm–Liouville Equations: Darboux Transformation and Applications

“…The matrix function F in (1.3) is the 2 × 2 Shin-Zettl matrix of general form (see, e.g., § 2 in [14] or in [15]). We note that Shin-Zettl differential expressions were introduced in [50,53] and were actively studied in regularization and spectral theories (see the books [1,54], papers [14,15], recent surveys [37,55] and various references therein). The Lagrange-symmetric case ω = ω, p = p, q = q, r 1 = −r 2 (1.…”