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arXiv · 1205.5211

Quantum star-graph analogues of PT-symmetric square wells

Miloslav Znojil
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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 $α-$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 $α-$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.

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Miloslav Znojil. 2012-09-07. Quantum star-graph analogues of PT-symmetric square wells. https://doi.org/10.1139/p2012-107

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