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21: 18.38 Mathematical Applications
The solved Schrödinger equations of §18.39(i) involve shape invariant potentials, and thus are in the family of supersymmetric or SUSY potentials. SUSY leads to algebraic simplifications in generating excited states, and partner potentials with closely related energy spectra, from knowledge of a single ground state wave function. …
22: Bibliography H
  • R. L. Hall, N. Saad, and K. D. Sen (2010) Soft-core Coulomb potentials and Heun’s differential equation. J. Math. Phys. 51 (2), pp. Art. ID 022107, 19 pages.
  • D. R. Herrick and S. O’Connor (1998) Inverse virial symmetry of diatomic potential curves. J. Chem. Phys. 109 (1), pp. 11–19.
  • 23: Bibliography W
  • R. S. Ward (1987) The Nahm equations, finite-gap potentials and Lamé functions. J. Phys. A 20 (10), pp. 2679–2683.
  • M. I. Weinstein and J. B. Keller (1985) Hill’s equation with a large potential. SIAM J. Appl. Math. 45 (2), pp. 200–214.
  • 24: Bibliography B
  • E. Bank and M. E. H. Ismail (1985) The attractive Coulomb potential polynomials. Constr. Approx. 1 (2), pp. 103–119.
  • M. V. Berry (1966) Uniform approximation for potential scattering involving a rainbow. Proc. Phys. Soc. 89 (3), pp. 479–490.
  • A. Bhattacharjie and E. C. G. Sudarshan (1962) A class of solvable potentials. Nuovo Cimento (10) 25, pp. 864–879.
  • J. C. Bronski, L. D. Carr, B. Deconinck, J. N. Kutz, and K. Promislow (2001) Stability of repulsive Bose-Einstein condensates in a periodic potential. Phys. Rev. E (3) 63 (036612), pp. 1–11.
  • 25: Bibliography E
  • C. Eckart (1930) The penetration of a potential barrier by electrons. Phys. Rev. 35 (11), pp. 1303–1309.
  • 26: Bibliography N
  • J. Negro, L. M. Nieto, and O. Rosas-Ortiz (2000) Confluent hypergeometric equations and related solvable potentials in quantum mechanics. J. Math. Phys. 41 (12), pp. 7964–7996.
  • 27: Bibliography S
  • M. J. Seaton (2002a) Coulomb functions for attractive and repulsive potentials and for positive and negative energies. Comput. Phys. Comm. 146 (2), pp. 225–249.
  • B. Simon (1973) Resonances in n -body quantum systems with dilatation analytic potentials and the foundations of time-dependent perturbation theory. Ann. of Math. (2) 97, pp. 247–274.
  • I. N. Sneddon (1966) Mixed Boundary Value Problems in Potential Theory. North-Holland Publishing Co., Amsterdam.
  • C. Snow (1952) Hypergeometric and Legendre Functions with Applications to Integral Equations of Potential Theory. National Bureau of Standards Applied Mathematics Series, No. 19, U. S. Government Printing Office, Washington, D.C..
  • 28: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
    The sum of the kinetic and potential energies give the quantum Hamiltonian, or energy operator; often also referred to as a Schrödinger operator. …
    Example 2: Radial 3D Schrödinger operators, including the Coulomb potential
    which appear in the quantum theory of binding or scattering of a particle in a spherically symmetric potential V ( r ) in three dimensions, and where r [ 0 , ) . …Unlike in the example in the paragraph above, in 3-dimensions a “dip below zero, or a potential well” in V ( r ) does not always correspond to the existence of a discrete part of the spectrum. …
    29: Bibliography D
  • L. Dekar, L. Chetouani, and T. F. Hammann (1999) Wave function for smooth potential and mass step. Phys. Rev. A 59 (1), pp. 107–112.
  • R. Dutt, A. Khare, and U. P. Sukhatme (1988) Supersymmetry, shape invariance, and exactly solvable potentials. Amer. J. Phys. 56, pp. 163–168.
  • 30: Bibliography R
  • S. Ritter (1998) On the computation of Lamé functions, of eigenvalues and eigenfunctions of some potential operators. Z. Angew. Math. Mech. 78 (1), pp. 66–72.