Abstract
Singular Schrödinger operators on \(L^{2}([0,+\infty))\) with the potential of the form \(\sum_{k=1}^{+\infty}a_{k}\delta_{x_{k}}\), where \(x_{k}\,{>}\,0\) and \(a_{k}\,{\in}\,\mathbb{R}\), are considered. It is constructively proved that every closed semibounded set \(S\subset\mathbb{R}\) can be the essential spectrum of such operator.
This is a preview of subscription content, log in via an institution to check access.
REFERENCES
K. A. Mirzoev, ‘‘Shturm–Liouville operators,’’ Trans. Moscow Math. Soc. 2014, 281–299 (2014). https://doi.org/10.1090/S0077-1554-2014-00234-X
A. S. Kostenko and M. M. Malamud, ‘‘1-D Schrödinger operators with local point interactions on a discrete set,’’ J. Differ. Equations 249, 253–304 (2010). https://doi.org/10.1016/j.jde.2010.02.011
T. Kato, Perturbation Theory for Linear Operators, Grundlehren der Mathematischen Wissenschaften, Vol. 132 (Springer, Berlin, 1966).
A. Savchuk and A. Shkalikov, ‘‘Sturm–Liouville operators with distributional potentials,’’ Trans. Moscow Math. Soc. 2003, 143–192 (2003).
M. Reed and B. Simon, Methods of Modern Mathematical Physics IV: Analysis of Operators (Academic, New York, 1978).
Funding
The work is supported by the Russian Science Foundation, project no. 20-11-20261.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
The author declares that he has no conflicts of interest.
Additional information
Translated by E. Oborin
About this article
Cite this article
Agafonkin, G.A. Reconstruction of the Schrödinger Operator with a Singular Potential on Half-Line by Its Prescribed Essential Spectrum. Moscow Univ. Math. Bull. 78, 203–206 (2023). https://doi.org/10.3103/S0027132223040022
Received:
Published:
Issue Date:
DOI: https://doi.org/10.3103/S0027132223040022