Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

A subclass of anharmonic oscillators whose eigenfunctions have no recurrence relations


Author: Gary G. Gundersen
Journal: Proc. Amer. Math. Soc. 58 (1976), 109-113
MSC: Primary 34B25
DOI: https://doi.org/10.1090/S0002-9939-1976-0460781-4
MathSciNet review: 0460781
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The equation $ w''(z) + (\lambda - p(z))w(z) = 0(z \in {\mathbf{C}},\lambda \in {\mathbf{R}})$ with a fixed $ p(z) = {a_{2m}}{z^{2m}} + {a_{2m - 2}}{z^{2m - 2}} + \cdots + {a_2}{z^2}({a_i} \geq 0\forall i,{a_{2m}} > 0)$, possesses a set of solutions $ \{ {\psi _n}(z)\} _{n = 0}^\infty $ (with associated $ \{ {\lambda _n}\} _{n = 0}^\infty $) which form a complete orthonormal set for $ {L^2}( - \infty ,\infty )$ (as a real space). Here it is shown that any $ g(z)\psi _N^{(q)}(z)$ (fixed $ N,q \in {\mathbf{N}},g \not\equiv 0$ a polynomial with $ \deg (g) \geq 1$ when $ q = 0$) cannot be expressed as a finite linear combination of $ \{ {\psi _n}(z)\} _{n = 0}^\infty $ when $ \deg (p)$ is a multiple of 4. It is well known that $ g(z)\psi _N^{(q)}(z)$ can always be expressed as such when $ p(z) = {z^2}$ (the Hermite case).


References [Enhancements On Off] (What's this?)

  • [1] W. W. Bell, Special functions for scientists and engineers, Van Nostrand, Princeton, N.J., 1968. MR 46 # 2087a. MR 0302944 (46:2087a)
  • [2] C. M. Bender and T. T. Wu, Phys. Rev. Letters 21 (1968), 406.
  • [3] -, Anharmonic oscillator, Phys. Rev. (2) 184 (1969), 1231-1260. MR 41 #4951. MR 0260323 (41:4951)
  • [4] I. M. Gel'fand and G. E. Shilov, Generalized functions. Vol. 2, Spaces of fundamental and generalized functions, Fizmatgiz, Moscow, 1958; English transl., Academic Press; Gordon and Breach, New York, 1968. MR 21 #5142a; 37 #5693. MR 0230128 (37:5693)
  • [5] G. G. Gundersen, Eigenfunctions and Cauchy problems for anharmonic and harmonic oscillators, Thesis, Rutgers University, 1975.
  • [6] P. F. Hsieh and Y. Sibuya, On the asymptotic integration of second order linear ordinary differential equations with polynomial coefficients, J. Math. Anal. Appl. 16 (1966), 84-103. MR 34 #403. MR 0200512 (34:403)
  • [7] J. S. Rosen, Logarithmic Sobolev inequalities and supercontractivity for anharmonic oscillators, Thesis, Princeton University, 1974.
  • [8] B. Simon, Pointwise bounds on eigenfunctions and wave packets in $ N$-body quantum systems. III, Trans. Amer. Math. Soc. 208 (1975), 317-329. MR 0417597 (54:5647)
  • [9] E. C. Titchmarsh, Eigenfunction expansions associated with second-order differential equations. Part I, 2nd ed. Clarenden Press, Oxford, 1962. MR 8, 458; 31 #426. MR 0176151 (31:426)
  • [10] G. Warner, Harmonic analysis on semi-simple Lie groups I, Springer-Verlag, New York, 1972. MR 0498999 (58:16979)
  • [11] J. Zeitlin, Correspondence between Lie algebra invariant subspaces and Lie group invariant subspaces of representations of Lie groups, Trans. Amer. Math. Soc. 167 (1972), 227-242. MR 45 #6980. MR 0297928 (45:6980)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 34B25

Retrieve articles in all journals with MSC: 34B25


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1976-0460781-4
Keywords: Eigenfunction, eigenvalue, Hermite functions, Hermite polynomials, recurrence relation, complete orthonormal set, anharmonic oscillator, harmonic oscillator, nilpotent Lie algebra, WKB transformation
Article copyright: © Copyright 1976 American Mathematical Society

American Mathematical Society