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Bounds on Schrödinger eigenvalues for polynomial potentials in N dimensions (1997)

Hall, Richard L. ; Saad, Nasser

College Park, Md. : American Institute of Physics (AIP)

Journal of Mathematical Physics 38 (1997), S. 4909-4913

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ISSN: 1089-7658

Source: AIP Digital Archive

Topics: Mathematics , Physics

Notes: If a single particle obeys nonrelativistic QM in RN and has the Hamiltonian H=−Δ+∑q〉0 a(q)rq, a(q)≥0, then the lowest eigenvalue E is given approximately by the semiclassical expression E=minr〉0{(1/r2)+∑q〉0 a(q)(P(q,N)r)q}. It is proved that this formula yields a lower bound when P(q,N)=(Ne/2)1/2(N/qe)1/q[Γ(1+N/2)/Γ(1+N/q)]1/N and an upper bound when P(q,N)=(N/2)1/2[Γ((N+q)/2)/Γ(N/2)]1/q. An extension is made to allow for a Coulomb term when N〉1. The general formula is applied to the examples V(r)=r+r2+r3 and V(r)=r2+r4+r6 in dimensions 1 to 10, and the results are compared to accurate eigenvalues obtained numerically. © 1997 American Institute of Physics.

Type of Medium: Electronic Resource

URL: http://dx.doi.org/10.1063/1.531925

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