Pseudodifferential Operators with Analytic Symbols and Estimates for Eigenfunctions of Schrödinger Operators

Abstract

We study the behavior of eigenfunctions of the Schrödinger operator $–\Delta +\nu$ with potential having power, exponential or super-exponential growth at infinity and discontinuities on manifolds in ${\mathbb{R}}^n$. We use a connection between the domain of analyticity of the main symbol $(|\xi {|}^2+\nu (x){)}^{–1}$ of the parametrix $–\Delta +\nu$ at infinity or near singularities of $\nu$ and the behavior of eigenfunctions at infinity or near singularities of potentials. Our approach is based on a general calculus of pseudodifferential operators with analytic symbols.