Florian Leben | Searcharxiv

arXiv2023

Limit point and limit circle trichotomy for Sturm-Liouville problems with complex potentials

The limit point and limit circle classification of real Sturm-Liouville problems by H. Weyl more than 100 years ago was extended by A.R. Sims around 60 years ago to the case when the coefficients are complex. Here, the main result is a collection of various criteria which allow us to decide to which class of Sims' scheme a given Sturm-Liouville problem with complex coefficients belongs. This is subsequently applied to a second order differential equation defined on a ray in $\mathbb C$ which is motivated by the recent intensive research connected with $\mathcal P \mathcal T$-symmetric Hamiltonians.

math.CA↗

arXiv2019

Operator based approach to PT-symmetric problems on a wedge-shaped contour

We consider a second-order differential equation $$ -y''(z)-(iz)^{N+2}y(z)=λy(z), \quad z\in Γ$$ with an eigenvalue parameter $λ\in \mathbb{C}$. In $\mathcal{PT}$ quantum mechanics $z$ runs through a complex contour $Γ\subset \mathbb{C}$, which is in general not the real line nor a real half-line. Via a parametrization we map the problem back to the real line and obtain two differential equations on $[0,\infty)$ and on $(-\infty,0].$ They are coupled in zero by boundary conditions and their potentials are not real-valued. The main result is a classification of this problem along the well-known limit-point/ limit-circle scheme for complex potentials introduced by A.R.\ Sims 60 years ago. Moreover, we associate operators to the two half-line problems and to the full axis problem and study their spectra.

math-ph↗