Abstract
Schrödinger operators with interactions symmetric about a plane (double-well potentials) occur in several branches of physics, such as chemistry and quantum field theory. They commonly exhibit asymptotic eigenvalue degeneracy, i.e., pairs of eigenvalues coalesce as the potential wells get farther apart. After a sketch of the theory of double wells, it is shown that the problem of estimating the gap between two such eigenvalues is reducible to finding asymptotics of eigenfunctions. For several examples and classes of potentials the gap is estimated or bounded above and below. The general case is fullyn-dimensional.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Coleman, S.: The uses of instantons. In: Proceedings of the International School of Physics, Erice, Italy, 1977 (to appear)
Harrell, E.M.: Commun. Math. Phys.60, 73–95 (1978)
Brézin, E., Parisi, G., Zinn-Justin, J.: Phys. Rev. D16, 408–412 (1977)
Isaacson, D.: Comm. Pure App. Math.29, 531–551 (1976)
Fröman, N., Fröman, P.-O.: JWKB approximation, contribution to the theory. Amsterdam: North-Holland, 1965;
Fröman, N.: Ark. Fys.31, 445–451 (1966);
Fröman, N., Fröman, P.-O., Myhrman, U., Paulsson, R.: Ann. Phys.74, 314–323 (1972)
Kac, M.: Mathematical mechanisms of phase transitions. In: Brandeis Univ. Summer Institute in Theoretical Physics, 1966, Vol. 1. (eds. M. Chrétien, E.P. Gross, S. Desai). New York: Gordon and Breach 1968
Ashkin, J., Lamb, W.E., Jr.: Phys. Rev.64, 159–178 (1943)
Newell, G.F., Montroll, E.W.: Rev. Mod. Phys.25, 353–389 (1953)
Thompson, C.J., Kac, M.: Studies Appl. Math.48, 257–264 (1969)
Condon, E.U., Shortley, G.H.: The theory of atomic spectra. Cambridge: Cambridge University Press 1970
Herring, C.: Rev. Mod. Phys.34, 631–645 (1962)
Jacobi, C.G.J.: Vorlesungen über Dynamik. Berlin: G. Reiner 1884
Thirring, W.: Course in mathematical physics, Vol. I. Classical dynamical systems. New York, Vienna: Springer 1978
Pauli, W.: Ann. Phys.68, 177–240 (1922)
Jaffé, G.: Z. Phys.87, 535–544 (1934)
Bates, D.R., Ledsham, K., Stewart, A.L.: Phil. Trans. Roy. Soc. London A246, 215–240 (1953)
Damburg, R.J., Propin, R.Kh.: J. Phys. B, Ser. 2,1, 681–691 (1968)
Thirring, W.: Lehrbuch der mathematischen Physik, Vol. III. Quantenmechanik von Atomen und Molekülen. Vienna: Springer 1979. English translation to appear 1980
Aventini, P., Seiler, R.: Commun. Math. Phys.41, 119–134 (1975)
Combes, J.-M., Seiler, R.: Internat. J. Quantum Chem.14, 213–229 (1978)
Morgan J.D., III, Simon, B.: Behavior of molecular potential envergy curves for large nuclear separations (to appear)
Reed, M., Simon, B.: Methods of modern mathematical physics, in four volumes. New York: Academic Press 1972, 1975, 1979, 1978
Temple, G.: Proc. Roy. Soc. London119A, 276–293 (1928)
Kato, T.: J. Phys. Soc. Japan4, 334–339 (1949)
Harrell, E.M.: Proc. Am. Math. Soc.69, 271–276 (1978)
Harrell, E.M.: Ann. Phys.119, 351–369 (1979)
Slaggie, E.L., Wichmann, E.H.: J. Math. Phys.3, 946–968 (1962)
Glazman: I.M.: Direct methods of qualitative spectral analysis of singular differential operators. Jerusalem: Israel Program for Scientific Translation 1965
Bardos, C., Mérigot, M.: Comptes Rendues Acad. Sci. Paris281A, 561–563 (1975)
Deift, P., Hunziker, W., Simon, B., Vock, E.: Comm. Math. Phys.64, 1–34 (1978)
Hoffmann-Ostenhof, M., Hoffmann-Ostenhof, T.: Phys. Rev. A16, 1782–1785 (1977)
Kato, T.: Commun. on Pure and Appl. Math.10, 151–177 (1957)
Abramowitz, M., Stegun, I. (eds.): Handbook of mathematical functions. NBS Applied Mathematics Series 55. Washington: National Bureau of Standards 1964
Klaus, M., Simon, B.: Ann. Inst. Henri Poincaré30, 83–87 (1979)
Schiff, L.I.: Quantum mechanics, 3rd ed. New York: McGraw-Hill 1968
Simon, B.: Ann. Phys.58, 76–136 (1970)
Hsieh, P.-F., Sibuya, Y.: J. Math. Anal. Appl.16, 84–103 (1966)
Lubkin, G.B.: Physics Today32, No. 5, 17–19 (1979)
Flammer, C.: Spheroidal wave functions. Stanford, California: Stanford University Press 1957
Buchholz, H.: The confluent hypergeometric function. Berlin, Heidelberg, New York: Springer 1969
Herbst, I., Sloan, A.: Trans. Am. Math. Soc.236, 325–360 (1978)
Author information
Authors and Affiliations
Additional information
Communicated by B. Simon
Much of this work was done with support of an NSF National Needs Fellowship at the Department of Mathematics, M.I.T.
Rights and permissions
About this article
Cite this article
Harrell, E.M. Double Wells. Commun.Math. Phys. 75, 239–261 (1980). https://doi.org/10.1007/BF01212711
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01212711