On the existence of the maximum number of isolated eigenvalues for a lattice Schrödinger operator
https://doi.org/10.17586/2220-8054-2025-16-6-737-748
Abstract
This paper presents a detailed spectral analysis of the discrete Schrodinger operator Hγλµ(K), which describes a system of two identical bosons on a two-dimensional lattice, Z2 . The operator’s family is parameterized by the quasi-momentum K ∈ T2 and real interaction strengths: γ for on-site, λ for nearestneighbor, and µ for next-nearest-neighbor interactions. A key finding of our study is that, under specific conditions on the interaction parameters, the operator Hγλµ(K) consistently possesses a total of seven eigenvalues that lie either below the bottom or above the top of its essential spectrum, over all K ∈ T2
Keywords
two-particle system,
discrete Schrodinger operator,
essential spectrum,
bound states,
Fredholm ¨ determinant
Supporting agencies: The authors gratefully acknowledge the support received from the Foundation for Basic Research of the Republic of Uzbekistan (Grant no. FL–9524115052).
About the Authors
S. N. Lakaev
Samarkand State Pedagogical Institute
Uzbekistan
Uzbekistan
Saidakhmat N. Lakaev
140104, Samarkand
D. A. Latipova
Samarkand State University
Uzbekistan
Uzbekistan
Dildora A. Latipova
140104, Samarkand
M. O. Akhmadova
Samarkand State University
Uzbekistan
Uzbekistan
Mukhayyo O. Akhmadova
140104, Samarkand
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Review
For citations:
Lakaev S.N., Latipova D.A., Akhmadova M.O. On the existence of the maximum number of isolated eigenvalues for a lattice Schrödinger operator. Nanosystems: Physics, Chemistry, Mathematics. 2025;16(6):737-748. https://doi.org/10.17586/2220-8054-2025-16-6-737-748
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