ABSTRACT:The separability of the Schrö dinger equation for harmonic oscillators in D dimensions and in different coordinate systems (Cartesian, circular, spherical) makes possible the construction of common generating functions for the complete harmonic oscillator wave functions in the corresponding dimensions and coordinates. Explicit forms of such generating functions and their series expansions are presented, and one of their applications is illustrated through the evaluation of transformation brackets.

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An approach to quantum corrals that takes into account that electrons may decay out of the corral by tunneling ͑coherent process͒ and by absorption at the corral boundaries to take into account inelastic ͑inco-herent͒ processes by coupling into the bulk is developed. Our single particle formalism is based on solutions to the two dimensional Schrödinger equation with outgoing boundary conditions involving a complex potential whose real and imaginary parts relate, respectively, to coherent and incoherent processes.

ABSTRACT:The Schrödinger equation for the two-dimensional hydrogen atom is known to be separable and integrable in circular, parabolic, and elliptical coordinates. This makes it possible to construct a common generating function for the complete wave functions of the atom in the respective coordinates. The connections with the corresponding generating function and wave functions for the harmonic oscillator are recognized and applied in this work.