Spectrum completion and inverse Sturm–Liouville problems

Abstract

Given a finite set of eigenvalues of a regular Sturm–Liouville problem for the equation −y″+qfalse(xfalse)y=λy$$ -{y}^{{\prime\prime} }+q(x)y=\lambda y $$, the potential qfalse(xfalse)$$ q(x) $$ of which is unknown. We show the possibility to compute more eigenvalues without any additional information on the potential qfalse(xfalse)$$ q(x) $$. Moreover, considering the Sturm–Liouville problem with the boundary conditions y′false(0false)−hyfalse(0false)=0$$ {y}^{\prime }(0)- hy(0)=0 $$ and y′false(πfalse)+Hyfalse(πfalse)=0$$ {y}^{\prime}\left(\pi \right)+ Hy\left(\pi \right)=0 $$, where h,0.1emH$$ h,H $$ are some constants, we complete its spectrum without additional information neither on the potential qfalse(xfalse)$$ q(x) $$ nor on the constants h$$ h $$ and H$$ H $$. The eigenvalues are computed with a uniform absolute accuracy. Based on this result, we propose a new method for numerical solution of the inverse Sturm–Liouville problem of recovering the potential from two spectra. The method includes the completion of the spectra in the first step and reduction to a system of linear algebraic equations in the second. The potential qfalse(xfalse)$$ q(x) $$ is recovered from the first component of the solution vector. The approach is based on special Neumann series of Bessel functions representations for solutions of Sturm–Liouville equations possessing remarkable properties and leads to an efficient numerical algorithm for solving inverse Sturm–Liouville problems.

Recovery of Inhomogeneity from Output Boundary Data

“…For example, in [17] (see also [18]), this problem was approached via an iterative algorithm based on the use of properties of the corresponding transmutation operator. In the present work, we apply another approach, which extends the one proposed in [14,15], and consists in converting the knowledge of {g n (L)} N n=0 and {s n (L)} N n=0 into an auxiliary inverse problem for (9) consisting in the recovery of q(x) from one spectrum and a sequence of corresponding multiplier constants. We formulate this problem in the next subsection and show how the knowledge of {g n (L)} N n=0 and {s n (L)} N n=0 allows us to reduce the inverse problem to that auxiliary inverse problem.…”

“…The input data of the inverse problem can be of different nature. For example, in [14] they were the spectral data of the spectral problem on a quantum graph, while in [15] the input data were several first eigenvalues from two spectra for (1). In the present work, we show how the endpoint values of a solution of (1), which satisfies some given initial conditions, serve for recovering the potential q(x), following the scheme described above.…”

“…Second, we develop further the idea proposed in [14,15], which can be formulated as follows. An inverse problem can be efficiently solved by converting the input data into the computed values of the coefficients from the NSBF representations at the endpoint of the interval.…”

Reconstruction techniques for quantum trees

“…Results on the uniqueness and solvability of such problems are well known and can be found, for example, in [22,[26][27][28]. To this problem, we apply the method from [29] which again involves the NSBF representations. This method involves computing certain multiplier constants [30] which relate pairs of Neumann-Dirichlet eigenfunctions associated to the same eigenvalues but normalized at opposite endpoints of the interval.…”

“…In those papers, the system of linear algebraic equations was obtained with the aid of the Gelfand-Levitan integral equation. In [29], another approach, based on the consideration of the eigenfunctions normalized at the opposite endpoints, was developed, and this idea was used in [13,14,34] and is used in the present work when solving the two-spectrum inverse problems on the edges.…”