Solving the inverse Sturm–Liouville problem with singular potential and with polynomials in the boundary conditions

Local solvability and stability for the inverse Sturm‐Liouville problem with polynomials in the boundary conditions

“…In recent years, spectral analysis of differential operators with singular coefficients, which are the so‐called distributional coefficients, has attracted much attention of mathematicians (see [17, 22–30]). Some properties of spectrum and solutions of differential equations with singular coefficients were studied in such papers as [22, 24–26].…”

“…The method of spectral mappings has been transferred to the Sturm‐Liouville operators with potentials of the class $$$ {W}_2^{-1} $$$ in [27, 29, 30, 38]. In particular, in [30], we have obtained a constructive solution of an inverse problem for the Sturm‐Liouville equation () with singular potential and the polynomial boundary conditions ().…”

“…The method of spectral mappings has been transferred to the Sturm‐Liouville operators with potentials of the class $$$ {W}_2^{-1} $$$ in [27, 29, 30, 38]. In particular, in [30], we have obtained a constructive solution of an inverse problem for the Sturm‐Liouville equation () with singular potential and the polynomial boundary conditions (). Namely, the inverse problem has been reduced to a linear equation in the Banach space of bounded infinite sequences, and reconstruction formulas for the coefficients $$$ \sigma (x),{r}_1\left(\lambda \right) $$$, and $$$ {r}_2\left(\lambda \right) $$$ have been derived.…”

“…It has been proved in [30] that the spectrum of the boundary value problem $$$ L $$$ is a countable set of eigenvalues. They can be numbered as $$$ {\left\{{\lambda}_n\right\}}_{n\ge 1} $$$ according to their asymptotic behavior: $$$$ {\rho}_n=\sqrt{\lambda_n}=n-p-1+{\mathrm{\varkappa}}_n,\kern0.30em \left\{{\mathrm{\varkappa}}_n\right\}\in {l}_2.$$…”

“…Then, the Weyl function $$$ M\left(\lambda \right) $$$, which will be defined in Section 2, has simple poles at $$$ \lambda ={\lambda}_n,n\ge 1 $$$, and the weight numbers are defined as $$$ {\alpha}_n={\mathrm{Res}}_{\lambda ={\lambda}_n}M\left(\lambda \right),n\ge 1 $$$. They fulfill the asymptotics (see [30]): $$$$ {\alpha}_n=\frac{2}{\pi }+{\mathrm{\varkappa}}_n^0,\kern0.30em \left\{{\mathrm{\varkappa}}_n^0\right\}\in {l}_2. $$$$ …”