The Sturm–Liouville Operator with Rapidly Growing Potential and the Asymptotics of its Spectrum

Abstract

In this paper, we study the asymptotic behavior of the discrete spectrum of the Sturm–Liouville operator given on the positive real semiline by the expression $-y′′ +q(x)y$ and the zero boundary condition $y(0) \cos{\alpha} + y'(0) \sin{\alpha} = 0$, for rapidly growing potentials $q(x)$. The asymptotics of the eigenvalues of the operator for the classes of potentials are obtained, which characterize the rate of their growth at infinity.