Uniform Spectral Asymptotics for the Schrödinger Operator with Translation in Free Term and Periodic Boundary Conditions

Abstract: We consider a nonlocal Schrödinger operator on the interval with the periodic boundary conditions and a translation in the free term. The value of the translation is denoted by and is treated as a parameter. We show that the resolvent of such an operator is Hölder continuous in this parameter with the exponent the spectrum of this operator consists of infinitely many discrete eigenvalues accumulating at infinity, and all eigenvalues are continuous in and coincide for and Our main result is a uniform spectral asymptotics for the operator under consideration. Namely, we show that sufficiently large eigenvalues separate into pairs, each is located in the vicinity of the point where in the index counting the eigenvalues, and we find a four-term asymptotics for these eigenvalues for large with the error term of order, and this term is uniform with respect to We also discuss nontrivial high-frequency phenomena demonstrated by the uniform spectral asymptotics we have found. DOI 10.1134/S1061920825600552 © Pleiades Publishing, Ltd. 2025.