On eigenfunction approximations for typical non-self-adjoint Schroedinger operators

TL;DR

This work develops a global JWKB framework to approximate eigenfunctions of 1D non-self-adjoint Schrödinger operators, addressing the challenge of spectral instability and non-orthogonal eigenfunctions. By solving the eikonal equation and constructing leading-order JWKB modes $y(x)$, the authors obtain explicit, globally defined approximants and show how to combine multiple JWKB modes to capture complex eigenfunction structures. They validate the approach with numerical experiments on a harmonic-oscillator-like operator and dilation-analytic potentials, demonstrating that eigenfunctions can be well-approximated by linear combinations of a small number of JWKB modes and that a cut-off procedure yields $L^2$-admissible functions. The results reveal a two-mode structure for higher eigenvalues and provide practical procedures for selecting mode centers, truncating tails, and computing coefficients, suggesting broad applicability to similar spectral problems and to resonances of dilation-analytic operators.

Abstract

We construct efficient approximations for the eigenfunctions of non-self-adjoint Schroedinger operators in one dimension. The same ideas also apply to the study of resonances of self-adjoint Schroedinger operators which have dilation analytic potentials. In spite of the fact that such eigenfunctions can have surprisingly complicated structures with multiple local maxima, we show that a suitable adaptation of the JWKB method is able to provide accurate lobal approximations to them.

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