Quantum star-graph analogues of PT-symmetric square wells

TL;DR

This paper extends a solvable PT-symmetric quantum square well model from a line segment to complex star graphs with multiple edges, revealing families of real energy eigenvalues and proposing a physical interpretation.

Contribution

It generalizes a known PT-symmetric quantum model from an interval to q-pointed star graphs with Robin boundary conditions, deriving a compact secular determinant and analyzing its real eigenvalues.

Findings

Existence of q-independent infinite real energy subfamilies for all q.

Additional q-dependent real energy subfamilies at specific q values.

Compact secular determinant form facilitating spectral analysis.

Abstract

We pick up a solvable ${\cal PT}-$symmetric quantum square well on an interval of $x \in := (-L,L)\mathbb{G}^{(2)}$ (with an $\alpha-$dependent non-Hermiticity given by Robin boundary conditions) and generalize it. In essence, we just replace the support interval $\mathbb{G}^{(2)}$ (reinterpreted as an equilateral two-pointed star graph with the Kirchhoff matching at the vertex $x=0$) by a $q-$pointed equilateral star graph $\mathbb{G}^{(q)}$ endowed with the simplest complex-rotation-symmetric external $\alpha-$dependent Robin boundary conditions. The remarkably compact form of the secular determinant is then deduced. Its analysis reveals that (1) at any integer $q=2,3,...$, there exists the same, $q-$independent and infinite subfamily of the real energies, and (2) at any special $q=2,6,10,...$, there exists another, additional and $q-$dependent infinite subfamily of the real energies. In the spirit of the recently proposed dynamical construction of the Hilbert space of a quantum system, the physical bound-state interpretation of these eigenvalues is finally proposed.