Inverse spectral theory for Sturm-Liouville operators with distributional potentials

Abstract

We discuss inverse spectral theory for singular differential operators on arbitrary intervals (a, b) ⊆ ℝ associated with rather general differential expressions of the type true0τf=1rfalse(−false(pfalse[f′+sffalse]false)′+spfalse[f′+sffalse]+qffalse), where the coefficients p, q, r, s are Lebesgue measurable on (a, b) with p−1, q, r, s ∈ L1loc((a,b); dx) and real‐valued with p≠0 and r>0 almost everywhere on (a, b). In particular, we explicitly permit certain distributional potential coefficients. The inverse spectral theory results derived in this paper include those implied by the spectral measure, by two‐spectra, and three‐spectra, as well as local Borg–Marchenko‐type inverse spectral results. The special cases of Schrödinger operators with distributional potentials and Sturm–Liouville operators in impedance form are isolated, in particular.

Mathematical analysis and solution methodology for an inverse spectral problem arising in the design of optical waveguides

“…The unknown refractive index parameters are n + and n ∞ (see equation (38) and Figure 2). The values of the target and initial refractive index proles, n and n (0) , are reported in Table 2.…”

Schrödinger operators with distributional potentials and boundary conditions dependent on the eigenvalue parameter

“…Direct and inverse spectral problems for boundary value problems generated by the equation (1.1) were studied by Savchuk and Shkalikov [25], [27], [28] and Hryniv and Mykytyuk [17], [18]. More general second-order differential expressions were later considered in [7], [8], [12], [24].…”

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