Interlace properties for the real and imaginary parts of the wave functions of complex-valued potentials with real spectrum

TL;DR

This paper investigates the properties of wave functions for complex potentials with real spectra, revealing interlacing zeros of their real and imaginary parts and how the imaginary component influences probability density profiles.

Contribution

It establishes new mathematical properties of wave functions in complex potentials, including interlacing zeros and the influence of the imaginary part on probability densities, with proofs and applications to various potentials.

Findings

Zeros of real and imaginary parts interlace, ensuring non-null probability densities.

The imaginary part's profile determines maxima and minima distribution of probability densities.

Continuity and zero integral of the imaginary potential are conjectured to control density extrema.

Abstract

Some general properties of the wave functions of complex-valued potentials with real spectrum are studied. The main results are presented in a series of lemmas, corollaries and theorems that are satisfied by the zeros of the real and imaginary parts of the wave functions on the real line. In particular, it is shown that such zeros interlace so that the corresponding probability densities $\rho(x)$ are never null. We find that the profile of the imaginary part $V_I(x)$ of a given complex-valued potential determines the number and distribution of the maxima and minima of the related probability densities. Our conjecture is that $V_I(x)$ must be continuous in $\mathbb R$, and that its integral over all the real line must be equal to zero in order to get control on the distribution of the maxima and minima of $\rho(x)$. The applicability of these results is shown by solving the eigenvalue equation of different complex potentials, these last being either ${\cal PT}$-symmetric or not invariant under the ${\cal PT}$-transformation.