Quantum star-graph analogues of PT-symmetric square wells

Abstract

We pick up a solvable $\mathcal{P}\mathcal{T}-$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.

20 pp, 1 figure

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