relation to quantum eigenvalues

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1: 25.17 Physical Applications

§25.17 Physical Applications

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2: Bibliography W

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Lectures on Quantum Mechanics. Cambridge University Press, Cambridge, UK.

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ISBN 978-1-107-02872-2

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§18.39(ii).

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Group Theory and its Application to the Quantum Mechanics of Atomic Spectra. Pure and Applied Physics. Vol. 5, Academic Press, New York.

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Expanded and improved ed. Translated from the German by J. J. Griffin.

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ISBN 0-12-750550-4, MathReview (A. S. Wightman), ZentralBlatt

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§34.12.

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Hypergeometric Series, Recurrence Relations and Some New Orthogonal Polynomials. Ph.D. Thesis, University of Wisconsin, Madison, WI.

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§16.4(iii), §16.4(iii), §16.4(iii).

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Computation with Recurrence Relations. Pitman, Boston, MA.

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ISBN 0-273-08508-5, MathReview (S. Conde), ZentralBlatt

Referenced by:

§13.29(iv), §16.25, §2.9(iii), §3.10(iii), §3.6(iii), §3.6(v), §3.6(vii).

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Elliptic genera and quantum field theory. Comm. Math. Phys. 109 (4), pp. 525–536.

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ISSN 0010-3616, Document, MathReview (J. Stasheff), ZentralBlatt

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1st item.

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3: 18.38 Mathematical Applications

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Supersymmetric Quantum Mechanics (SUSY)

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EOP’s, Painlevé Transcendents, and Quantum Mechanics

►EOP’s are the subject of recent work on rational solutions to the fourth Painlevé equation, see Clarkson (2003a) and Marquette and Quesne (2016),where use of Hermite EOP’s makes a connection to quantum mechanics. …

4: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ► $u_{\lambda }\in 𝒟(T)$, corresponding to distinct eigenvalues, are orthogonal: i. … ►The special form of (1.18.28) is especially useful for applications in physics, as the connection to non-relativistic quantum mechanics is immediate: $-\frac{d^2}{{dx}^2}$ being proportional to the kinetic energy operator for a single particle in one dimension, $q(x)$ being proportional to the potential energy, often written as $V(x)$, of that same particle, and which is simply a multiplicative operator. The sum of the kinetic and potential energies give the quantum Hamiltonian, or energy operator; often also referred to as a Schrödinger operator. … ►Should an eigenvalue correspond to more than a single linearly independent eigenfunction, namely a multiplicity greater than one, all such eigenfunctions will always be implied as being part of any sums or integrals over the spectrum. …

5: Bibliography R

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Normal limit theorems for symmetric random matrices. Probab. Theory Related Fields 112 (3), pp. 411–423.

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ISSN 0178-8051, Document, MathReview (Alexei Khorunzhy), ZentralBlatt

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§35.9.

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On the computation of Lamé functions, of eigenvalues and eigenfunctions of some potential operators. Z. Angew. Math. Mech. 78 (1), pp. 66–72.

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ISSN 0044-2267, Document, MathReview Entry, ZentralBlatt

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§29.20(ii).

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Calculation of $3n$-$j$ symbols by Labarthe’s method. International Journal of Quantum Chemistry 63 (1), pp. 57–64.

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Document

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§34.13.

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The similarity solution for the Korteweg-de Vries equation and the related Painlevé transcendent. Proc. Roy. Soc. London Ser. A 361, pp. 265–275.

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FullText, MathReview (A. D. Brjuno), ZentralBlatt

Referenced by:

§32.11(ii), §32.17, §32.3(ii).

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On Simple Waves with Profiles in the form of some Special Functions—Chebyshev-Hermite, Mathieu, Whittaker—in Two-phase Media. In Differential Operators and Related Topics, Vol. I (Odessa, 1997), Operator Theory: Advances and Applications, Vol. 117, pp. 313–322.

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ISBN 3-7643-6287-1, MathReview Entry, ZentralBlatt

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5th item.

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6: Bibliography V

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An integral transform involving Heun functions and a related eigenvalue problem. SIAM J. Math. Anal. 17 (3), pp. 688–703.

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ISSN 0036-1410, Document, MathReview (William L. Perry), ZentralBlatt

Referenced by:

§31.10(i).

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Quantum Theory of Angular Momentum. World Scientific Publishing Co. Inc., Singapore.

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ISBN 9971-50-107-4, MathReview (M. A. Rashid), ZentralBlatt

Referenced by:

§16.24(iii), §34.1, §34.11, §34.13, §34.13, §34.14, §34.2, §34.3(iii), §34.3(vi), §34.4, §34.4, §34.5(iii), §34.5(vi), §34.7(ii), §34.7(iii), §34.8, §34.9, Ch.34.

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Representation of Lie Groups and Special Functions. Volume 3: Classical and Quantum Groups and Special Functions. Mathematics and its Applications (Soviet Series), Vol. 75, Kluwer Academic Publishers Group, Dordrecht.

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Notes:

Translated from the Russian by V. A. Groza and A. A. Groza

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ISBN 0-7923-1493-X, MathReview (P. D. F. Ion), ZentralBlatt

Referenced by:

§18.38(iii), Ch.35.

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Integral relations for Lamé functions. SIAM J. Math. Anal. 13 (6), pp. 978–987.

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ISSN 0036-1410, Document, MathReview (F. M. Arscott), ZentralBlatt

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§29.8, §29.8.

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On the growth of convergence radii for the eigenvalues of the Mathieu equation. Math. Nachr. 192, pp. 239–253.

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ISSN 0025-584X, Document, MathReview (M. E. Muldoon), ZentralBlatt

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§28.6(i).

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7: Bibliography L

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Quantum Mechanics: Non-relativistic Theory. Pergamon Press Ltd., Oxford.

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Notes:

Revised second edition. Course of Theoretical Physics, Vol. 3. Translated from the Russian by J. B. Sykes and J. S. Bell

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MathReview, ZentralBlatt

Referenced by:

§9.16.

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Solutions to a generalized spheroidal wave equation: Teukolsky’s equations in general relativity, and the two-center problem in molecular quantum mechanics. J. Math. Phys. 27 (5), pp. 1238–1265.

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ISSN 0022-2488, Document, MathReview (Basilis C. Xanthopoulos), ZentralBlatt

Referenced by:

§30.12, §31.17(ii).

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Recurrence relations for hypergeometric functions of unit argument. Math. Comp. 45 (172), pp. 521–535.

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ISSN 0025-5718, FullText, MathReview (S. Conde), ZentralBlatt

Referenced by:

§16.4(iii).

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Kinetic Theory: Classical, Quantum, and Relativistic Descriptions. third edition, Springer, New York.

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ISBN 0387955518

Referenced by:

§18.39(iii).

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Monotonic sequences related to zeros of Bessel functions. Numer. Algorithms 49 (1-4), pp. 221–233.

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ISSN 1017-1398, Document, FullText, MathReview (Javier Segura), ZentralBlatt

Referenced by:

§10.21(iv), §10.21(iv).

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8: Bibliography D

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The constrained quantum mechanical harmonic oscillator. Proc. Cambridge Philos. Soc. 62, pp. 277–286.

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Document, MathReview (T. Erber)

Referenced by:

§12.11(i), §12.11(i), §12.17.

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Computational properties of three-term recurrence relations for Kummer functions. J. Comput. Appl. Math. 233 (6), pp. 1505–1510.

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ISSN 0377-0427, Document, FullText, MathReview (Richard B. Paris), ZentralBlatt

Referenced by:

§13.29(iv), §13.29(iv).

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

Referenced by:

§31.17(ii).

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Quantum Fields and Strings: A Course for Mathematicians. Vol. 1, 2. American Mathematical Society, Providence, RI.

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Notes:

Material from the Special Year on Quantum Field Theory held at the Institute for Advanced Study, Princeton, NJ, 1996–1997

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ISBN 0-8218-1198-3, MathReview, ZentralBlatt

Referenced by:

§21.9.

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Asymptotic approximations for the Jacobi and ultraspherical polynomials, and related functions. Methods Appl. Anal. 6 (3), pp. 21–56.

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ISSN 1073-2772, Pdf, MathReview (Walter Van Assche), ZentralBlatt

Referenced by:

§15.12(iii), §18.15(i), §18.15(vi).

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9: 18.39 Applications in the Physical Sciences

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§18.39(i) Quantum Mechanics

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The Quantum Coulomb Problem

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b) The Bohr Quantum Number

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10: Bibliography B

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Moment equations for probability distributions in classical and quantum mechanics. Phys. Rev. A 58 (3), pp. 1799–1809.

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Document

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§24.18.

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Quantum mechanics of one- and two-electron atoms. Springer-Verlag, Berlin.

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External Links:

MathReview (L. Van Hove), ZentralBlatt

Referenced by:

§18.39(ii), §18.39(ii).

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Tables Relating to the Radial Mathieu Functions. Vol. 1: Functions of the First Kind. U.S. Government Printing Office, Washington, D.C..

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MathReview (C. J. Bouwkamp)

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1st item.

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Tables Relating to the Radial Mathieu Functions. Vol. 2: Functions of the Second Kind. U.S. Government Printing Office, Washington, D.C..

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MathReview (C. J. Bouwkamp)

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2nd item, 1st item.

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Concerning the zeros of some functions related to Bessel functions. J. Mathematical Phys. 10 (9), pp. 1729–1744.

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Document, MathReview (C. V. Coffman), ZentralBlatt

Referenced by:

§10.21(xi).

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