Boundary Integral Method for Stationary States of Two-Dimensional Quantum Systems

Abstract

The boundary integral method for calculating the stationary states of a quantum particle in nano-devices and quantum billiards is presented in detail at an elementary level. According to the method, wave functions inside the domain of the device or billiard are expressed in terms of line integrals of the wave function and its normal derivative along the domain's boundary; the respective energy eigenvalues are obtained as the roots of Fredholm determinants. Numerical implementations of the method are described and applied to determine the energy level statistics of billiards with circular and stadium shapes and demonstrate the quantum mechanical characteristics of chaotic motion. The treatment of other examples as well as the advantages and limitations of the boundary integral method are discussed.

RevTeX3.0, 24 pages, 9 EPS figures (included); To be published in Int. J. of Mod. Phys. C

References in corpus (2)

• Transport through Dirty Luttinger Liquids Connected to Reservoirs
• Boundary integral method applied in chaotic quantum billiards

Cited by in corpus (8)

• Boundary element method for resonances in dielectric microcavities
• Spectral Statistics of Rectangular Billiards with Localized Perturbations
• Spectroscopy of drums and quantum billiards: perturbative and non-perturbative results
• An efficient Fredholm method for calculation of highly excited states of billiards
• Fermi gas energetics in low-dimensional metals of spessial geometry
• Eigenvalue Problem in Two Dimensions for an Irregular Boundary II: Neumann Condition
• "Striped" Rectangular Rigid Box with Hermitian and non-Hermitian $\mathcal{P}\mathcal{T}$ Symmetric Potentials
• Classical and Quantum Chaos in the Diamond Shaped Billiard