Asymptotic Formulas for Weight Numbers of the Boundary Problem differential operator on a Star-shaped Graph
Main Article Content
Abstract
In this article the boundary value problem differential operator on the graph of a special structure is considered. The graph has edges, joined at one common vertex, and vertices of degree 1. The boundary value problem is set by the Sturm – Liоuvillе differential expression with real-valued potentials, the Dirichlet boundary conditions, and the standard matching conditions. This problem has a countable set of еigеnvаluеs. We consider the so-called weight numbers, being the residues of the diagonal elements of the Weyl matrix in the еigеnvаluеs. These elements are monomorphic functions with simple poles which can be only the еigеnvаluеs. We note that the considered weight numbers generalize the weight numbers of differential operators on a finite interval, equal to the reciprocals to the squared norms of eigenfunсtiоns. These numbers together with the еigеnvаluеs play a role of spectral data for unique reconstruction of operators. We obtain asymрtоtic formulas for the weight numbers using the contour integration, and in the case of the asymptotically close еigеnvаluеs the formulas are got for the sums. The formulas can be used for the analysis of inverse spectral problems on the graphs..
Downloads
Metrics
Article Details
You are free to:
- Share — copy and redistribute the material in any medium or format for any purpose, even commercially.
- Adapt — remix, transform, and build upon the material for any purpose, even commercially.
- The licensor cannot revoke these freedoms as long as you follow the license terms.
Under the following terms:
- Attribution — You must give appropriate credit , provide a link to the license, and indicate if changes were made . You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
- No additional restrictions — You may not apply legal terms or technological measures that legally restrict others from doing anything the license permits.
Notices:
You do not have to comply with the license for elements of the material in the public domain or where your use is permitted by an applicable exception or limitation .
No warranties are given. The license may not give you all of the permissions necessary for your intended use. For example, other rights such as publicity, privacy, or moral rights may limit how you use the material.
