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

We consider a nonlocal Schrödinger operator on the interval \((0,2\pi)\) with the periodic boundary conditions and a translation in the free term. The value of the translation is denoted by \(a\) and is treated as a parameter. We show that the resolvent of such an operator is Hölder continuous in this parameter with the exponent \(\frac{1}{2},\) the spectrum of this operator consists of infinitely many discrete eigenvalues accumulating at infinity, and all eigenvalues are continuous in \(a\in[0,2\pi]\) and coincide for \(a=0\) and \(a=2\pi.\) 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 \(n^2,\) where \(n\) in the index counting the eigenvalues, and we find a four-term asymptotics for these eigenvalues for large \(n\) with the error term of order \(O(n^{-3})\), and this term is uniform with respect to \(a.\) We also discuss nontrivial high-frequency phenomena demonstrated by the uniform spectral asymptotics we have found.

DOI 10.1134/S1061920825600552