Eigenvalue bounds for polynomial central potentials in d dimensions

  title={Eigenvalue bounds for polynomial central potentials in d dimensions},
  author={Q. Katatbeh and R. Hall and N. Saad},
  journal={Journal of Physics A},
If a single particle obeys non-relativistic QM in Rd and has the Hamiltonian H = −Δ + f(r), where , then the eigenvalues E = E(d)nl(λ) are given approximately by the semi-classical expression . It is proved that this formula yields a lower bound if Pi = P(d)nl(q1), an upper bound if Pi = P(d)nl(qk) and a general approximation formula if Pi = P(d)nl(qi). For the quantum anharmonic oscillator f(r) = r2 + λr2m, m = 2, 3, ... in d dimension, for example, E = E(d)nl(λ) is determined by the algebraic… Expand
3 Citations

Tables from this paper

On some polynomial potentials in d-dimensions
The d-dimensional Schrodinger's equation is analyzed with regard to the existence of exact solutions for polynomial potentials. Under certain conditions on the interaction parameters, we show thatExpand
Development of the perturbation theory using polynomial solutions
The number of quantum systems for which the stationary Schrodinger equation is exactly solvable is very limited. These systems constitute the basic elements of the quantum theory of perturbation. TheExpand
Exact and approximate solutions to Schrödinger’s equation with decatic potentials
The one-dimensional Schrödinger’s equation is analysed with regard to the existence of exact solutions for decatic polynomial potentials. Under certain conditions on the potential’s parameters, weExpand


Bounds on Schrodinger eigenvalues for polynomial potentials in N dimensions
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 expressionExpand
Envelope theory in spectral geometry
It is shown that the discrete spectrum of Schrodinger Hamiltonians of the form H=−Δ+vf may be represented by the semiclassical expression Enl=minr≳0 {K(f)nl(r) + vf(r)}. The K functions are found toExpand
Kinetic potentials in quantum mechanics
Suppose that the Hamiltonian H=−Δ+vf(r) represents the energy of a particle which moves in an attractive central potential and obeys nonrelativistic quantum mechanics. The discrete eigenvaluesExpand
Generalized comparison theorems in quantum mechanics
This paper is concerned with the discrete spectra of Schrodinger operators H = −Δ + V, where V(r) is an attractive potential in N spatial dimensions. Two principal results are reported for the bottomExpand
Summation of perturbation series of eigenvalues and eigenfunctions of anharmonic oscillators.
  • M. A. Núñez
  • Mathematics, Medicine
  • Physical review. E, Statistical, nonlinear, and soft matter physics
  • 2003
The calculations of this work show that the ordinary Padé approximants from the sole un-normalized E(R) series for the octic oscillator give accurate results with small or large lambda. Expand
A geometrical theory of energy trajectories in quantum mechanics
Suppose f(r) is an attractive central potential of the form f(r)=∑ki=1 g(i)( f(i)(r)), where {f(i)} is a set of basis potentials (powers, log, Hulthen, sech2) and {g(i)} is a set of smooth increasingExpand
A Convergent Renormalized Strong Coupling Perturbation Expansion for the Ground State Energy of the
Abstract The Rayleigh–Schrodinger perturbation series for the energy eigenvalue of an anharmonic oscillator defined by the Hamiltonian Ĥ ( m ) ( β )= p 2 + x 2 + βx 2 m with m =2, 3, 4, … divergesExpand
New exact solutions for polynomial oscillators in large dimensions
A new type of exact solvability is reported. The Schrodinger equation is considered in a very large spatial dimension D 1 and its central polynomial potential is allowed to depend on 'many' ( = 2q)Expand
Extended continued fractions and energies of the anharmonic oscillators
We describe the analytic solution to the Schrodinger eigenvalue problem for the class of the central potentials V(r)=∑δ∈Zaδrδ, where a−2>−1/4, amax δ >0, Z is an arbitrary finite set of the integerExpand
Eigenvalues of ?x2m anharmonic oscillators
The ground state as well as excited energy levels of the generalized anharmonic oscillator defined by the Hamiltonian Hm = − d2/dx2+x2+ λx2m, m = 2,3, …, have been calculated nonperturbatively usingExpand