Eigenvalues of the Schrödinger equation for a periodic potential with nonperiodic boundary conditions: A uniform semiclassical analysis

Abstract

A uniform semiclassical expression for the eigenvalues of a one dimensional periodic Schrödinger equation with nonperiodic boundary conditions has been derived. The potential energy function can have any number of symmetric or asymmetric barriers and wells. The treatment is uniform in that the classical turning points can come close together, coalesce, and move into the complex plane as the energy passes through a barrier maximum. A detailed application is made to Mathieu functions of integer order; the equations themselves include the case of fractional order. Approximate semiclassical expressions are derived for the widths of the energy bands and the energy gaps of the periodic Mathieu equation when these quantities are small. The semiclassical results give a physical interpretation to formulas present in the mathematical literature and to the decrease in the splitting of a sequence of avoided crossings with increasing quantum numbers in coupled oscillator systems. Numerical calculations are reported to illustrate the high accuracy of the semiclassical formulas.

Instantons re-examined: Dynamical tunneling and resonant tunneling

“…The last expression corresponds exactly to equation (3.44) of [35] obtained with standard uniform semiclassical analysis. We see on figure 5 that (56) is a very good approximation even when the energies E n get close to γ the energy of the separatrix.…”

Milestones Toward Majorana-Based Quantum Computing

“…Since increasing T "stiffens" the confining potential (in units of E J ), we expect the maximal degeneracy splitting to be parametrically further suppressed as T → 1. For the Josephson-dominated limit E J =E C ≫ 1 considered here, we can compute ΔE max for general T using the same WKB approach [117] as for the cosine potential V SIS valid at T ≪ 1 [52]. The result takes the form…”

Statistical mechanics of Coulomb gases as quantum theory on Riemann surfaces

“…We focus on unequal charges n 1 = n 2 , since the case of n 1 = n 2 reduces to the well-known Hermitian cosine potential 29,30 . For unequal charges the Hamiltonian is non-Hermitian but PT -symmetric, allowing for complex eigenvalues which appear in conjugated pairs 27,28 .…”