On the problem of recovery of Sturm--Liouville operator with two frozen arguments

TL;DR

The paper studies the inverse spectral problem for a nonlocal Sturm-Liouville operator with two frozen arguments $a$ and $b$, showing that in general the two spectra do not determine the coefficient pair $(p,q)$. It introduces characteristic function representations $\Delta_j(\lambda)$ and analyzes their linear and bilinear dependence on $p$ and $q$, establishing asymptotics for the eigenvalues and revealing nonuniqueness through explicit constructions. Under the additional condition that $p$ and $q$ vanish on a fixed segment, it proves a uniqueness theorem and derives regularized trace formulas linking spectral data to endpoint Fourier coefficients, with convergence criteria that relax smoothness assumptions. The results extend the inverse spectral theory for nonlocal operators beyond the one frozen argument case and provide concrete criteria for when spectral data suffice to identify the coefficients.

Abstract

Inverse spectral problems consist in recovering operators by their spectral characteristics. The problem of recovering the Sturm-Liouville operator with one frozen argument was studied earlier in works of various authors. In this paper, we study a uniqueness of recovering operator with two frozen arguments and different coefficients p, q by the spectra of two boundary value problems. The case considered here is significantly more difficult than the case of one frozen argument, because the operator is no more a one-dimensional perturbation. We prove that the operator with two frozen arguments, in general case, can not be recovered by the two spectra. For the uniqueness of recovering, one should impose some conditions on the coefficients. We assume that the coefficients p and q equal zero on certain segment and prove a uniqueness theorem. As well, we obtain regularized trace formulae for the two spectra. The result is formulated in terms of convergence of certain series, which allows us to avoid restrictions on the smoothness of the coefficients.

Theorems & Definitions (14)

• Theorem 2.1
• Lemma 3.1
• proof
• Theorem 3.1
• proof
• Example 3.1
• Theorem 4.1
• Lemma 4.1
• proof : Proof of Theorem \ref{['uniqueness theorem']}.
• Theorem 5.1