Spectral properties of the iterated Laplacian with a potential in a punctured domain

Abstract

In the work we derive regularized trace formulas which were established in papers of Kanguzhin and Tokmagambetov for the Laplace and m-Laplace operators in a punctured domain with the fixed iterating order m ∈ N. By using techniques of Sadovnichii and Lyubishkin, the authors in that papers described regularized trace formulae in the spatial dimension d = 2. In this note one claims that the formulas are also true for more general operators in the higher spatial dimensions, namely, 2 ≤ d ≤ 2m. Also, we give the further discussions on a development of the analysis associated with the operators in punctured domains. This can be done by using so called 'nonharmonic' analysis.

A second regularized trace formula for a higher order differential operator

“…And then, Lidskii and Sadovnicii 15‐17 indicated a technique for computing trace formulas of general problems including ordinary differential equations on a finite interval. The subject has a very broad theory, and we recommend the list of studies in previous works 18‐21 for more and further information.…”