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