Bounds on Schr{umlt o}dinger eigenvalues for polynomial potentials in N dimensions

Journal Article · Wed Oct 01 04:00:00 EDT 1997 · Journal of Mathematical Physics

DOI:https://doi.org/10.1063/1.531925· OSTI ID:554215

Hall, R L; Saad, N ^[1]

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1. Department of Mathematics and Statistics, Concordia University, 1455 de Maisonneuve Boulevard West, Montreal, Quebec H3G 1M8 (Canada)

If a single particle obeys nonrelativistic QM in R{sup N} and has the Hamiltonian H={minus}{Delta}+{summation}{sub q{gt}0}a(q)r{sup q}, a(q){ge}0, then the lowest eigenvalue E is given approximately by the semiclassical expression E=min{sub r{gt}0}{l_brace}(1/r{sup 2})+{summation}{sub q{gt}0}a(q)(P(q,N)r){sup q}{r_brace}. It is proved that this formula yields a lower bound when P(q,N)=(Ne/2){sup 1/2}(N/qe){sup 1/q}[{Gamma}(1+N/2)/{Gamma}(1+N/q)]{sup 1/N} and an upper bound when P(q,N)=(N/2){sup 1/2}[{Gamma}((N+q)/2)/{Gamma}(N/2)]{sup 1/q}. An extension is made to allow for a Coulomb term when N{gt}1. The general formula is applied to the examples V(r)=r+r{sup 2}+r{sup 3} and V(r)=r{sup 2}+r{sup 4}+r{sup 6} in dimensions 1 to 10, and the results are compared to accurate eigenvalues obtained numerically. {copyright} {ital 1997 American Institute of Physics.}

• OSTI ID: 554215
• Journal Information: Journal of Mathematical Physics, Journal Name: Journal of Mathematical Physics Journal Issue: 10 Vol. 38; ISSN JMAPAQ; ISSN 0022-2488
• Country of Publication: United States
• Language: English

Related Subjects

66 PHYSICS
EIGENVALUES
HAMILTONIANS
POLYNOMIALS
POTENTIALS
SCHROEDINGER EQUATION
SEMICLASSICAL APPROXIMATION