Quantum star-graph analogues of P T -symmetric square wells

Abstract

We recall the solvable -symmetric quantum square well on an interval of x ∈ (-L, L):= (2) (with an α-dependent non-Hermiticity given by Robin boundary conditions) and generalize it. In essence, we just replace the support interval (2) (reinterpreted as an equilateral two-pointed star graph with Kirchhoff matching at the vertex x = 0) with a q-pointed equilateral star graph (q) endowed with the simplest complex-rotation-symmetric external α-dependent Robin boundary conditions. The remarkably compact form of the secular determinant is then deduced. Its analysis reveals that (i) at any integer q = 2, 3,., there exists the same q-independent and infinite subfamily of the real energies, and (ii) at any special q = 2, 6, 10,., there exists another, additional, 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. © 2012 Published by NRC Research Press.