## On the rate of asymptotic eigenvalue degeneracy.(English)Zbl 0395.34023

### MSC:

 34L99 Ordinary differential operators 34E20 Singular perturbations, turning point theory, WKB methods for ordinary differential equations 34C11 Growth and boundedness of solutions to ordinary differential equations 34E05 Asymptotic expansions of solutions to ordinary differential equations
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### References:

 [1] Thompson, C.J., Kac, M.: Phase transitions and eigenvalue degeneracy of a one-dimensional anharmonic oscillator. Studies Appl. Math.48, 257–264 (1969) [2] Isaacson, D.: Singular perturbations and asymptotic eigenvalue degeneracy. Commun. Pure Appl. Math.29, 531–551 (1976) · Zbl 0336.34055 [3] Reed, M., Simon, B.: Methods of modern mathematical physics, Vol. 4. New York: Academic Press 1978 · Zbl 0401.47001 [4] Kato, T.: Perturbation theory for linear operators. Die Grundlehren der mathematischen Wissenschaften, Vol. 132. Berlin-Heidelberg-New York: Springer-Verlag 1966 [5] Reed, M., Simon, B.: Methods of modern mathematical physics, Vol. III. New York: Academic Press 1978 · Zbl 0401.47001 [6] Kac, M.: Mathematical mechanisms of phase transitions. In: Brandeis University Summer Institute in Theoretical Physics 1966, Vol. 1 (eds. M. Chrétien, E. P. Gross, S. Deser). New York: Gordon and Breach 1968 [7] Glimm, J., Jaffe, A., Spencer, T.: Phase transitions for {$$\phi$$} 2 4 quantum fields. Commun. math. Phys.45, 203–216 (1975) · Zbl 0956.82501 [8] Simon, B.: Coupling constant analyticity for the anharmonic oscillator. Ann. Phys.58, 76–136 (1970) [9] Loeffel, J.J., Martin, A.: Propriétés analytiques des niveaux de l’oscillateur anharmonique et convergence des approximants de Padé. CERN Ref. TH. 1167 (1970) [10] Hsieh, P.-F., Sibuya, Y.: On the asymptotic integration of second order linear ordinary differential equations with polynomial coefficients. J. Math. Anal. Appl.16, 84–103 (1966) · Zbl 0161.05803 [11] Whittaker, E.T., Watson, G.N.: A course of modern analysis. Cambridge: Cambridge University Press 1969 · JFM 45.0433.02 [12] Abramowitz, M., Stegun, I.A., (eds.): Handbook of mathematical functions. Washington: National Bureau of Standards 1964 · Zbl 0171.38503 [13] Coddington, E.A., Levinson, N.: The theory of differential equations. New York: McGraw-Hill 1955 · Zbl 0064.33002 [14] Gildener, E., Patrascioiu, A.: Pseudoparticle contributions to the energy spectrum of a one-dimensional system. Phys. Rev. D16, 423–430 (1977)
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