arXiv subjects
Publications and source records attributed to Vyacheslav Pivovarchik.
We study characteristic functions and describe asymptotics of the eigenvalues for the spectral Sturm-Liouville problem on graphs equipped with Robin-Kirhhoff boundary conditions. Also, we show how to recover the coefficients in the Robin conditions for the quantum graphs provided the shape of the graphs and some Robin eigenvalues are known.
We consider a scattering problem generated by the Sturm-Liouville equation on a tree which consists of an equilateral compact subtree and a half-infinite lead attached to its root. We assume that the potential on the lead is identically zero while the potentials on the finite edges are real. We show how to find the shape of the tree using the S-function of the scattering problem and the eigenvalues of the operators associated with the compact tree.
Let v_1 and v_2 be two distinct vertices of a tree T_0. Let ϕ_N^{(i)} (i=1,2) be the characteristic functions of the Sturm-Liouville problem on T_0 rooted at v_i with Neumann conditions at the root and let ϕ_D^{(i)} (i=1,2) be the characteristic functions of the Sturm-Liouville problem on T_0 with Dirichlet conditions at the root. We prove that if attaching any tree to T_0 at the vertices v_1 and v_2 leads to cospectral trees and d(v_1)=d(v_2) then ϕ_N(λ)^{(1)}\equiv ϕ_N(λ)^{(2)} and ϕ_D(λ)^{(1)}\equiv ϕ_D(λ)^{(1)} (which means that the scattering is the same at v_1 and v_2).
Spectral problems are considered generated by the Sturm-Liouville equation on equilateral trees with the Dirichlet boundary conditions at the pendant vertices and continuity and Kirchhoff's conditions at the interior vertices. It is proved that there are no co-spectral (i.e., having the same spectrum of such problem) among equilateral trees of less or equal 8 vertices. All co-spectral trees of 9 vertices are presented.
We show how to find the shape of an equilateral caterpillar tree using the spectra of the Neumann and the Dirichlet problems generated by the Sturm-Liouville equation on this tree. We prove that in the case of a caterpillar tree the spectra of the Neumann and Dirichlet problems uniquely determine the shape of the tree.
We show how to find the shape of an equilateral tree using the spectra of the Neumann and the Dirichlet problems generated by the Sturm-Liouville equation. In case of snowflake trees the spectra of the Neumann and Dirichlet problems uniquely determine the shape of the tree.
We consider a scattering problem generated by the Sturm-Liouville equation on a tree which consists of a equilateral compact subtree with a lead (a half-infinite edge) attached to this compact subtree. We assume that the potential on the lead is zero identically and the potentials on the finite edges are L_2-functions. We show how to find the shape of the tree using the S-function of the scattering problem.
We propose a simple method for resolution of co-spectrality of Schrödinger operators on metric graphs. Our approach consists of attaching a lead to them and comparing the $S$-functions of the corresponding scattering problems on these (non-compact) graphs. We show that in several cases -- including general graphs on at most 6 vertices, general trees on at most 9 vertices, and general fuzzy balls -- eigenvalues and scattering data are together sufficient to distinguish co-spectral metric graphs.
Spectral problems are considered generated by the Sturm-Liouville equation on connected simple equilateral graphs with the Neumann and Dirichlet boundary conditions at the pendant vertices and continuity and Kirchhoff's conditions at the interior vertices. We highlight the cases where the first and the second terms of the asymptotics of the eigenvalues uniquely determine the shape of the graph or of its interior subgraph.
The spectrum of a selfadjoint quadratic operator pencil of the form $λ^2M-λG-A$ is investigated where $M\geq 0$, $G\geq 0$ are bounded operators and $A$ is selfadjoint bounded below is investigated. It is shown that in the case of rank one operator $G$ the eigenvalues of such a pencil are of two types. The eigenvalues of one of these types are independent of the operator $G$. Location of the eigenvalues of both types is described. Examples for the case of the Sturm-Liouville operators $A$ are given.
For a graph consisting of parallel connected subgraphs we express the characteristic function of the boundary value problem with generalized Neumann conditions at both joining points via characteristic functions of different boundary problems on the subgraphs.
In [V. Barcilon Explicit solution of the inverse problem for a vibrating string. J. Math. Anal. Appl. {\bf 93} (1983) 222-234] two boundary value problems were considered generated by the differential equation of a string $$ y^{\prime\prime}+λp(x)y=0, \ \ 0\leq x \leq L<+\infty \eqno{(*)} $$ with continuous real function $p(x)$ (density of the string) and the boundary conditions $y(0)=y(L)=0$ the first problem and $y^{\prime}(0)=y(L)=0$ the second one. In the above paper the following formula was stated $$ p(0)={1}{L^2μ_1}\mathop{\prod}\limits_{n=1}^{\infty}{λ_n^2}{μ_n μ_{n+1}} \eqno{(**)} $$ where $\{λ_k\}_{k=1}^{\infty}$ is the spectrum of the first boundary value problem and $\{μ_k\}_{k=1}^{\infty}$ of the second one. Rigorous proof of (**) was given in [C.-L. Shen On the Barcilon formula for the string equation with a piecewise continuous density function. Inverse Problems {\bf 21}, (2005) 635--655] under more restrictive conditions of piecewise continuity of $p^{\prime}(x)$. In this paper (**) was deduced using $$ p(0)=\lim\limits_{λ\to +\infty}({ϕ(L,-λ)}{λ^{{1}{2}}ψ(L,-λ)})^2 \eqno{(***}) $$ where $ϕ(x,λ)$ is the solution of (*) which satisfies the boundary conditions $ϕ(0)-1=ϕ^{\prime}(0)=0$ and $ψ(x,λ)$ is the solution of (*) which satisfies $ψ(0)=ψ^{\prime}(0)-1=0$. In our paper we prove that (***) is true for the so-called M.G. Krein's string which may have any nondecreasing mass distribution function $M(x)$ with finite nonzero $M^{\prime}(0)$. Also we show that (**) is true for a wide class of strings including those for which $M(x)$ is a singular function, i.e. $M^{\prime}(x)=p(x)\mathop{=}\limits^{a.e.}0$.
We solve two inverse spectral problems for star graphs of Stieltjes strings with Dirichlet and Neumann boundary conditions, respectively, at a selected vertex called root. The root is either the central vertex or, in the more challenging problem, a pendant vertex of the star graph. At all other pendant vertices Dirichlet conditions are imposed; at the central vertex, at which a mass may be placed, continuity and Kirchhoff conditions are assumed. We derive conditions on two sets of real numbers to be the spectra of the above Dirichlet and Neumann problems. Our solution for the inverse problems is constructive: we establish algorithms to recover the mass distribution on the star graph (i.e. the point masses and lengths of subintervals between them) from these two spectra and from the lengths of the separate strings. If the root is a pendant vertex, the two spectra uniquely determine the parameters on the main string (i.e. the string incident to the root) if the length of the main string is known. The mass distribution on the other edges need not be unique; the reason for this is the non-uniqueness caused by the non-strict interlacing of the given data in the case when the root is the central vertex. Finally, we relate of our results to tree-patterned matrix inverse problems.