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arXiv:1004.0453 · Acta Polytechnica

Rectifiable PT -symmetric Quantum Toboggans with Two Branch Points

Miloslav Znojil

Published Jan 5, 2010• 1 citation
arXivHTMLPDF

Abstract

Certain complex-contour (a.k.a. quantum-toboggan) generalizations of Schroedinger’s bound-state problem are reviewed and studied in detail. Our key message is that the practical numerical solution of these atypical eigenvalue problems may perceivably be facilitated via an appropriate complex change of variables which maps their multi-sheeted complex domain of definition to a suitable single-sheeted complex plane.

DOI
10.14311/1277

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•Jan 1, 2020•0 citations
Certain complex-contour (a.k.a. quantum-toboggan) generalizations of Schroedinger’s bound-state problem are reviewed and studied in detail. Our key message is that the practical numerical solution of these atypical eigenvalue problems may perceivably be facilitated via an appropriate complex change of variables which maps their multi-sheeted complex domain of definition to a suitable single-sheeted complex plane.
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.
In this thesis we aim to show that the Schrödinger equation, which is a boundary eigenvalue problem, can have a discrete and real energy spectrum (eigenvalues) even when the Hamiltonian is non-Hermitian.After a brief introduction into non-Hermiticity, we will focus on solving the Schrödinger equation with a special class of non-Hermitian Hamiltonians, namely PTsymmetric Hamiltonians.PT -symmetric Hamiltonians have been discussed by various authors [1, 2, 3, 4, 5] with some of them focusing specifically on obtaining the real and discrete energy spectrum.Various methods for solving this problematic Schrödinger equation will be considered.After starting with perturbation theory, we will move on to numerical methods.Three different categories of methods will be discussed.First there is the shooting method based on a Runge-Kutta solver.Next, we investigate various implementations of the spectral method.Finally, we will look at the Riccati-Padé method, which is a numerical implemented analytical method.PT -symmetric potentials need to be solved along a contour in the complex plane.We will propose modifications to the numerical methods to handle this.After solving the widely documented PT -symmetric Hamiltonian H = p 2 -(ix) N with these methods, we give a discussion and comparison of the obtained results.Finally, we solve another PT -symmetric potential, illustrating the use of paths in the complex plane to obtain a real and discrete spectrum and their influence on the results.
Schrödinger equations for ‘quantum toboggans’ with real energies are given the generalized eigenvalue-problem form H ψ = EW ψ, where H ≠ H † and W ≠ W † ≠ I . The consistent probabilistic interpretation of these models is provided. The new double-series formula for the necessary ad hoc metric Θ = Θ( H , W ) is derived which defines the acceptable inner products in the physical Hilbert space of states. The formula degenerates to the usual single series in the non-tobogganic trivial-weight limit W → I .
We study complex PT-symmetric potentials, with real eigenvalues, corresponding to a complex coordinate shift (x+ic2) of a real even potential. In this case,the rules to achieve a coherent quantum mechanics are known. They allow the calculation of observables, which are found to be independent on c. This result is illustratedby few analytical or semi analytical examples. On the other hand, trying to test thisproperty numerically faces problems linked to the difficulty of finding the proper solutions of the Schrodinger equation. In particular, the large distance behaviour of the ¨wave functions generates instabilities. As an example, we have studied the (x +ic2)4potential.