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bound-state problems

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1: 33.22 Particle Scattering and Atomic and Molecular Spectra
For scattering problems, the interior solution is then matched to a linear combination of a pair of Coulomb functions, F ( η , ρ ) and G ( η , ρ ) , or f ( ϵ , ; r ) and h ( ϵ , ; r ) , to determine the scattering S -matrix and also the correct normalization of the interior wave solutions; see Bloch et al. (1951). For bound-state problems only the exponentially decaying solution is required, usually taken to be the Whittaker function W η , + 1 2 ( 2 ρ ) . …
2: 18.39 Applications in the Physical Sciences
Bound state solutions to the relativistic Dirac Equation, for this same problem of a single electron attracted by a nucleus with Z protons, involve Laguerre polynomials of fractional index. …
3: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
Bounded and Unbounded Linear Operators
, of unit multiplicity, unless otherwise stated. … The bound states are in the negative energy discrete spectrum, and the scattering states are in the positive energy continuous spectrum, 𝝈 c = [ 0 , ) , or, said more simply, in the continuum. … … If T is a bounded operator then its spectrum is a closed bounded subset of . …
4: Bibliography G
  • R. D. M. Garashchuk and J. C. Light (2001) Quasirandom distributed bases for bound problems. J. Chem. Phys. 114 (9), pp. 3929–3939.
  • A. G. Gibbs (1973) Problem 72-21, Laplace transforms of Airy functions. SIAM Rev. 15 (4), pp. 796–798.
  • R. G. Gordon (1968) Error bounds in equilibrium statistical mechanics. J. Math. Phys. 9, pp. 655–663.
  • R. G. Gordon (1969) New method for constructing wavefunctions for bound states and scattering. J. Chem. Phys. 51, pp. 14–25.
  • H. W. Gould (1960) Stirling number representation problems. Proc. Amer. Math. Soc. 11 (3), pp. 447–451.
  • 5: Bibliography L
  • A. Laforgia (1991) Bounds for modified Bessel functions. J. Comput. Appl. Math. 34 (3), pp. 263–267.
  • L. J. Landau (2000) Bessel functions: Monotonicity and bounds. J. London Math. Soc. (2) 61 (1), pp. 197–215.
  • N. N. Lebedev, I. P. Skalskaya, and Y. S. Uflyand (1965) Problems of Mathematical Physics. Revised, enlarged and corrected English edition; translated and edited by Richard A. Silverman. With a supplement by Edward L. Reiss, Prentice-Hall Inc., Englewood Cliffs, N.J..
  • D. Lemoine (1997) Optimal cylindrical and spherical Bessel transforms satisfying bound state boundary conditions. Comput. Phys. Comm. 99 (2-3), pp. 297–306.
  • E. M. Lifshitz and L. P. Pitaevskiĭ (1980) Statistical Physics, Part 2: Theory of the Condensed State. Pergamon Press, Oxford.
  • 6: Bibliography B
  • P. Baldwin (1991) Coefficient functions for an inhomogeneous turning-point problem. Mathematika 38 (2), pp. 217–238.
  • A. O. Barut and L. Girardello (1971) New “coherent” states associated with non-compact groups. Comm. Math. Phys. 21 (1), pp. 41–55.
  • W. G. Bickley (1935) Some solutions of the problem of forced convection. Philos. Mag. Series 7 20, pp. 322–343.
  • Å. Björck (1996) Numerical Methods for Least Squares Problems. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA.
  • K. R. Brownstein (2000) Criterion for Existence of a Bound State In One Dimension. American Journal of Physics 68 (2), pp. 160–161.
  • 7: Bibliography S
  • R. Shail (1978) Lamé polynomial solutions to some elliptic crack and punch problems. Internat. J. Engrg. Sci. 16 (8), pp. 551–563.
  • J. A. Shohat and J. D. Tamarkin (1970) The Problem of Moments. 4th edition, American Mathematical Society Mathematical Surveys, Vol. 1, American Mathematical Society, Providence, RI.
  • B. Simon (1976) The Bound State of Weakly Coupled Schrödinger Operators in One and Two Dimensions. Annals of Physics 97 (2), pp. 279–288.
  • B. D. Sleeman (1966a) Some Boundary Value Problems Associated with the Heun Equation. Ph.D. Thesis, London University.
  • I. N. Sneddon (1966) Mixed Boundary Value Problems in Potential Theory. North-Holland Publishing Co., Amsterdam.