Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Eigenvalue bounds for polynomial central potentials in d dimensions   If a single particle obeys non-relativistic QM in R^d and has the Hamiltonian
H = - Delta + f(r), where f(r)=sum_{i = 1}^{k}a_ir^{q_i}, 2\leq q_i < q_{i+1},
a_i \geq 0$, then the eigenvalues E = E_{n\ell}^{(d)}(\lambda) are given
approximately by the semi-classical expression E = \min_{r > 0}[\frac{1}{r^2} +
\sum_{i = 1}^{k}a_i(P_ir)^{q_i}]. It is proved that this formula yields a lower
bound if P_i = P_{n\ell}^{(d)}(q_1), an upper bound if $P_i =
P_{n\ell}^{(d)}(q_k) and a general approximation formula if P_i =
P_{n\ell}^{(d)}(q_i). For the quantum anharmonic oscillator f(r)=r^2+\lambda
r^{2m},m=2,3,... in d dimension, for example, E = E_{n\ell}^{(d)}(\lambda) is
determined by the algebraic expression
  \lambda={1\over \beta}({2\alpha(m-1)\over mE-\delta})^m({4\alpha \over
(mE-\delta)}-{E\over (m-1)}) where \delta={\sqrt{E^2m^2-4\alpha(m^2-1)}} and
\alpha, \beta are constants. An improved lower bound to the lowest eigenvalue
in each angular-momentum subspace is also provided. A comparison with the
recent results of Bhattacharya et al (Phys. Lett. A, 244 (1998) 9) and Dasgupta
et al (J. Phys. A: Math. Theor., 40 (2007) 773) is discussed.

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The categories of this paper are: gr-qc
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The categories of this paper are: dg-ga
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The categories of this paper are: astro-ph
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The categories of this paper are: chao-dyn
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The categories of this paper are: econ
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The categories of this paper are: q-fin
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The categories of this paper are: alg-geom
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The categories of this paper are: cond-mat
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The categories of this paper are: solv-int
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The categories of this paper are: funct-an
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The categories of this paper are: hep-ph
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The categories of this paper are: bayes-an
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The categories of this paper are: comp-gas
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The categories of this paper are: ao-sci
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The categories of this paper are: chem-ph
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The categories of this paper are: nucl-th