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arXiv:1411.3828 · Canadian Journal of Physics

Quantum star-graph analogues of PT -symmetric square wells: Part II, spectra

Miloslav Znojil

Published Dec 11, 2014• 6 citations
arXivHTMLPDF

Abstract

For the non-Hermitian equilateral q-pointed star-shaped quantum graphs of our previous paper (Can. J. Phys. 90, 1287 (2012) doi:10.1139/p2012-107 ) we show that because of certain dynamical aspects of the model as controlled by the external, rotation-symmetric complex Robin boundary conditions, the spectrum is obtainable in a closed asymptotic-expansion form, in principle at least. Explicit formulae up to the second order are derived for illustration, and a few comments on their consequences are added.

DOI
10.1139/cjp-2014-0475

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For the non-Hermitian equilateral q-pointed star-shaped quantum graphs of our previous paper (Can. J. Phys. 90, 1287 (2012) doi:10.1139/p2012-107 ) we show that because of certain dynamical aspects of the model as controlled by the external, rotation-symmetric complex Robin boundary conditions, the spectrum is obtainable in a closed asymptotic-expansion form, in principle at least. Explicit formulae up to the second order are derived for illustration, and a few comments on their consequences are added.
We recall the solvable [Formula: see text]-symmetric quantum square well on an interval of x ∈ (–L, L) := [Formula: see text] (with an α-dependent non-Hermiticity given by Robin boundary conditions) and generalize it. In essence, we just replace the support interval [Formula: see text] (reinterpreted as an equilateral two-pointed star graph with Kirchhoff matching at the vertex x = 0) with a q-pointed equilateral star graph [Formula: see text] 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.
Abstract We consider PT-symmetric quantum graphs, in which the branching points provide PT-symmetric boundary conditions for the Schrödinger equation on a graph. For such branched quantum wires, we derive general boundary conditions, which keep the Hamiltonian as PT-symmetric with real eigenvalues and positively defined norm of the eigenfunctions. Secular equations for finding the eigenvalues of the quantum graph are derived. Breaking the Kirchhoff rule at the branching points in such systems is shown. Experimental realization of PT-symmetric quantum graphs on branched optical waveguides is discussed.
It is know that PT-symmetric models have real spectra provided the symmetry is not spontaneously broken. Even pseudo-hermitian models have real spectra, which enlarge the the class of non-hermitian models possessing real spectra. We however consider non-hermitian models in higher dimensions which are not necessarily explicit PT-symmetric nor pseudo-hermitian. We show that the models may generate real spectra depending upon the coupling constants of the Hamiltonian. Our models thus further generalize the class of non-hermitian systems, which possess real spectra.