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Double wells: Perturbation series summable to the eigenvalues and directly computable approximations. (English) Zbl 0645.35071

We give a rigorous proof of the analyticity of the eigenvalues of the double-well Schrödinger operators and of the associated resonances. We specialize the Rayleigh-Schrödinger perturbation theory to such problems, obtaining an expression for the complex perturbation series uniquely related to the eigenvalues through a summation method. By an approximation we obtain new series expansions directly computable, still summable, which, in the case of the Herbst-Simon model, can be given in an explicit form.

MSC:

35P05 General topics in linear spectral theory for PDEs
35J10 Schrödinger operator, Schrödinger equation
35B20 Perturbations in context of PDEs
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[1] Brézin, E., Parisi, G., Zinn-Justin, J.: Perturbation theory at large orders for potentials with degenerate minima. Phys. Rev.D16, 408 (1977)
[2] Caliceti, E., Graffi, S., Maioli, M.: Perturbation theory of odd anharmonic oscillators. Commun. Math. Phys.75, 51 (1980) · Zbl 0446.47044
[3] Caliceti, E., Grecchi, V., Maioli, M.: The distributional Borel summability and the large coupling ?4 lattice fields. Commun. Math. Phys.104, 163 (1986) · Zbl 0648.40007
[4] Caliceti, E., Grecchi, V., Maioli, M.: Erratum to ?The distributional Borel summability and the large coupling ?4 lattice fields?. Commun. Math. Phys. (in press) · Zbl 0656.40010
[5] Crutchfield II, W. Y.: No horn of signularities for double-well anharmonic oscillators. Phys. Lett.77B, 109 (1978)
[6] Graffi, S., Grecchi, V.: Resonance in stark effect and perturbation theory. Commun. Math. Phys.62, 83 (1978) · Zbl 0432.40007
[7] Graffi, S., Grecchi, V.: The Borel sum of the double well perturbation series and the Zinn-Justin conjecture. Phys. Lett.121B, 410 (1983)
[8] Graffi, S., Grecchi, V., Harrell II, E. M., Silverstone, H. J.: The 1/R expansion forH 2 + : analyticity, summability and asymptotics. Ann. Phys.165, 441 (1985) · Zbl 0614.46068
[9] Graffi, S., Grecchi, V., Simon, B.: Borel summability: Application to the anharmonic oscillator. Phys. Lett.32B, 631 (1970)
[10] Herbst, I. W., Simon, B.: Some remarkable examples in eigenvalue perturbation theory. Phys. Lett.78B, 304 (1978)
[11] Reed, M., Simon, B.: Methods of modern mathematical physics, Vol. IV. New York: Academic Press 1978 · Zbl 0401.47001
[12] Sibuya, Y.: Global theory of a second order linear ordinary differential equation with a polynomial coefficient. Amsterdam: North-Holland 1975 · Zbl 0322.34006
[13] Simon, B.: Coupling constant analyticity for the anharmonic oscillator. Ann. Phys. (NY)58, 76 (1970)
[14] Simon, B.: Large orders and summability of eigenvalue perturbation theory: A mathematical overview. Int. J. Quant. Chem.21, 3 (1982)
[15] Vock, E., Hunziker, W.: Stability of Schrödinger eigenvalue problems. Commun. Math. Phys.83, 281 (1982) · Zbl 0528.35023
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