On the domain of a magnetic Schr\"odinger operator with complex electric potential

TL;DR

This paper reviews spectral properties of magnetic Schrödinger operators with complex and real polynomial potentials, focusing on criteria for selfadjointness, resolvent compactness, and maximal inequalities in L2 spaces.

Contribution

It provides a comparative analysis of spectral criteria for Schrödinger operators with complex and real potentials, including magnetic fields, highlighting conditions for selfadjointness and resolvent properties.

Findings

Criteria for essential selfadjointness established.

Conditions for compactness of the resolvent identified.

Maximal inequalities for operators with polynomial potentials derived.

Abstract

The aim of this paper is to review and compare the spectral properties of (the closed extension of) --$\Delta$ + U (V $\ge$ 0) and --$\Delta$ + iV in L 2 (R^d) for C $\infty$ real potentials U or V with polynomial behavior. The case with magnetic field will be also considered. More precisely, we would like to present the existing criteria for: $\bullet$ essential selfadjointness or maximal accretivity $\bullet$ Compactness of the resolvent. $\bullet$ Maximal inequalities, i.e. the existence of C > 0 such that, $\forall$u $\in$ C^$\infty$\_0 (R ^d), ||u||^2 \_{H^2 (R^d)} + ||U u||^2 \_{L^2 (R^d)}$\le$ C ||(--$\Delta$ + U)u||^2\_{L^2 (R^d)} + ||u||^2\_{L^2 (R^d)}or similar estimates for $-\Delta + i V$.