Borel-Leroy summability of a nonpolynomial potential. (English) Zbl 1176.34109
On \(L^2(-\infty,\infty)\) consider the one-dimensional Schrödinger operator
\[
H(g)=-\frac12\;\frac{d^2}{dx^2}+\frac12\;x^2+\frac{g^{m-1}x^{2m}}{1+\alpha gx^2},
\]
where \(m\) is a positive integer, \(\alpha >0\) and \(g\geq0\). The case \(m=1,2,3\) occurs in physical models of laser theory and quantum field theory. In this paper, the dependence of the eigenvalues on \(g\) for sufficiently small \(g\) is investigated. Using the idea and methods of [G. Auberson, Commun.Math.Phys., 84, 531–546 (1982; Zbl 0508.34042)] established for the cases \(m=2\), it is shown that for all \(m\geq3\) the \(j\)-th eigenvalue \(E_j(g)\) depend analytically on the complexified parameter \(g\) in sectors \(S_j\{g|0<|g|\leq r_j,\, |\arg g|<\theta<\pi\}\), and an asymptotic expansion of \(E_ j(g)\) is given.
Reviewer: Manfred Möller (Johannesburg)
MSC:
34L40 | Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.) |
34E10 | Perturbations, asymptotics of solutions to ordinary differential equations |
34E05 | Asymptotic expansions of solutions to ordinary differential equations |
40G10 | Abel, Borel and power series methods |
34L20 | Asymptotic distribution of eigenvalues, asymptotic theory of eigenfunctions for ordinary differential operators |
Citations:
Zbl 0508.34042References:
[1] | Simon, B., Ann. Phys., 58, 76 (1970) |
[2] | Graffi, S.; Grecchi, V.; Simon, B., Phys. Lett., 32B, 631 (1970) |
[3] | Auberson, G., Commun. Math. Phys., 84, 531 (1982) · Zbl 0508.34042 |
[4] | Auberson, G.; Boissiere, T., Il Nuovo Cimento, 75B, 105 (1983) |
[5] | Varshni, Y. P., Phys. Rev., A36, 3009 (1987) |
[6] | Flessa, J. G.P., Phys. Lett., 83A, 121 (1981) |
[7] | Bleche, M. H.; Leach, P. G.L., J. Phys., A20, 5923 (1987) · Zbl 0656.35023 |
[8] | Mitra, A. K., J. Math. Phys., 19, 2018 (1978) · Zbl 0426.65046 |
[9] | Salam, A.; Strathdee, J., Phys. Rev., D1, 3286 (1970) |
[10] | Fried, M., Phys. Rev., 174, 1725 (1968) |
[11] | da Costa, G. A.T. F.; Gomes, M., J. Math. Phys., 30, 1007 (1989) |
[12] | da Costa, G. A.T. F., J. Math. Phys., 32, 1293 (1991) · Zbl 0728.05057 |
[13] | Kato, T., (Perturbation Theory for Linear Operators (1966), Springer: Springer Berlin-Heidelberg-NY) · Zbl 0148.12601 |
[14] | Reed, M.; Simon, B., (Methods of Modern Mathematical Physics, vol I (Functional Analysis) (1972), Academic Press: Academic Press NY) · Zbl 0242.46001 |
[15] | Reed, M.; Simon, B., (Methods of Modern Mathematical Physics, vol IV (Analysis of Operators) (1978), Academic Press: Academic Press NY) · Zbl 0401.47001 |
[16] | Auberson, G.; Mennessier, G., J. Math. Phys., 22, 2472 (1981) · Zbl 0471.40005 |
[17] | Sokal, A., J. Math. Phys., 21, 261 (1980) · Zbl 0441.40012 |
[18] | da Costa, G. A.T. F., Atas do 55o, Seminário Brasileiro de Análise, 601 (2002) |
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