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1: 15.18 Physical Applications

… ►The hypergeometric function has allowed the development of “solvable” models for one-dimensional quantum scattering through and over barriers (Eckart (1930), Bhattacharjie and Sudarshan (1962)), and generalized to include position-dependent effective masses (Dekar et al. (1999)). …

2: 36.15 Methods of Computation

… ►(For the umbilics, representations as one-dimensional integrals (§36.2) are used.) …

3: 12.17 Physical Applications

… ►Dean (1966) describes the role of PCFs in quantum mechanical systems closely related to the one-dimensional harmonic oscillator. …

4: 18.39 Applications in the Physical Sciences

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Introduction and One-Dimensional (1D) Systems

►The fundamental quantum Schrödinger operator, also called the Hamiltonian, $\mathscr{H}$, is a second order differential operator of the form … ►These eigenfunctions are the orthonormal eigenfunctions of the time-independent Schrödinger equation …which in one dimensional systems are typically non-degenerate, namely there is only a single eigenfunction corresponding to each ${ϵ}_n$, $n\ge 0$. … ►

1D Quantum Systems with Analytically Known Stationary States

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5: 20.11 Generalizations and Analogs

… ►For specialization to the one-dimensional theta functions treated in the present chapter, see Rauch and Lebowitz (1973) and §21.7(iii). …

6: Bibliography F

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Characteristic values of the integral equation satisfied by the Mathieu functions and its application to a system with chirality-pair interaction on a one-dimensional lattice. Phys. A 190 (3-4), pp. 346–362.

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External Links:

ISSN 0378-4371, Document, MathReview

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6th item.

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7: Bibliography D

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Unification of one-dimensional Fokker-Planck equations beyond hypergeometrics: Factorizer solution method and eigenvalue schemes. Phys. Rev. E (3) 57 (1), pp. 252–275.

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External Links:

ISSN 1063-651X, Document, MathReview (Victor I. Stepanov), ZentralBlatt

Referenced by:

§31.17(ii).

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8: 18.38 Mathematical Applications

… ►Light and Carrington Jr. (2000) review and extend the one-dimensional analysis to solution of multi-dimensional many-particle systems, where the sparse nature of the resulting matrices is highly advantageous. …

9: Bibliography C

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Stationary solutions of the one-dimensional nonlinear Schrödinger equation. I. Case of repulsive nonlinearity. Phys. Rev. A 62 (063610), pp. 1–10.

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Document

Referenced by:

§22.19(iii).

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10: Bibliography S

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Asymptotic Solutions of the One-dimensional Schrödinger Equation. American Mathematical Society, Providence, RI.

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Notes:

Translated from the 1990 Russian original by Vadim Khidekel

External Links:

ISBN 0-8218-0563-3, MathReview (Denise Huet), ZentralBlatt

Referenced by:

§31.13.

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