zbMATH — the first resource for mathematics

Trapping and cascading of eigenvalues in the large coupling limit. (English) Zbl 0669.35085
Authors’ abstract: We consider eigenvalues \(E_{\lambda}\) of the Hamiltonian \(H_{\lambda}=-\Delta +V+\lambda W,\) W compactly supported, in the \(\lambda\) \(\to \infty\) limit. For \(W>0\) we find monotonic convergence of \(E_{\lambda}\) to the eigenvalues of a limiting operator \(H_{\infty}\) (associated with an exterior Dirichlet problem), and we estimate the rate of convergence for 1-dimensional systems. In 1- dimensional systems with \(W\leq 0\), or with W changing sign, we do not find convergence. Instead, we find a cascade phenomenon, in which, as \(\lambda\) \(\to \infty\), each eigenvalue \(E_{\lambda}\) stays near a Dirichlet eigenvalue for a long interval (of length O(\(\sqrt{\lambda}))\) of the scaling range, quickly drops to the next lower Dirichlet eigenvalue, stays there for a long interval, drops again, and so on. As a result, for most large values of \(\lambda\) the discrete spectrum of \(H_{\lambda}\) is close to that of \(H_{\infty}\), but when \(\lambda\) reaches a transition region, the entire spectrum quickly shifts down by one. We also explore the behaviour of several explicit models, as \(\lambda\) \(\to \infty\).
Reviewer: W.D.Frans

35P05 General topics in linear spectral theory for PDEs
35J25 Boundary value problems for second-order elliptic equations
47A55 Perturbation theory of linear operators
35Q99 Partial differential equations of mathematical physics and other areas of application
81Q15 Perturbation theories for operators and differential equations in quantum theory
Full Text: DOI
[1] Abramowitz, M., Stegun, I. A.: Handbook of mathematical functions. New York: Dover 1972 · Zbl 0543.33001
[2] Alama, S., Deift, P., Hempel, R.: Eigenvalue problems arising in the theory of the color of crystals (to appear) · Zbl 0676.47032
[3] Albeverio, S., Gesztesy, F., Høegh-Krohn, R., Holden, H.: Solvable models in quantum mechanics. Texts and Monographs in Physics. Berlin, Heidelberg, New York: Springer 1988 · Zbl 0679.46057
[4] Ashbaugh, M. S., Harrell, E. M.: Perturbation theory for shape resonances and large barrier potentials. Commun. Math. Phys.83, 151-170 (1982) · Zbl 0494.34044 · doi:10.1007/BF01976039
[5] Baumgärtel, H., Demuth, M.: Decoupling by a projection, Rep. Math. Phys.15, 173-186 (1979) · Zbl 0426.47007 · doi:10.1016/0034-4877(79)90017-X
[6] Bulla, W., Gesztesy, F.: Deficiency indices and singular boundary conditions in quantum mechanics. J. Math. Phys.26, 2520-2528 (1985) · Zbl 0583.35029 · doi:10.1063/1.526768
[7] Deift, P. A.: Applications of a communication formula, Duke Math. J.45, 267-310 (1978) · Zbl 0392.47013 · doi:10.1215/S0012-7094-78-04516-7
[8] Deift, P. A., Hempel, R.: On the existence of eigenvalues of the Schrödinger operatorH??W in a gap of?(H). Commun. Math. Phys.103, 461-490 (1986) · Zbl 0594.34022 · doi:10.1007/BF01211761
[9] Dunford, N., Schwartz, J. T.: Linear Operators II. New York: Interscience 1963 · Zbl 0128.34803
[10] Gesztesy, F., Simon, B.: On a theorem of Deift and Hempel. Commun. Math. Phys.116, 503-505 (1988) · Zbl 0647.35063 · doi:10.1007/BF01229205
[11] Grinberg, A. A.: Energy spectrum of an electron placed in the fields of a small-radius potential well and an attractive coulomb potential. Sov. Phys. Semicond.11, 1118-1120 (1977)
[12] Harrell, E. M.: The band structure of a one-dimensional periodic system in a scaling limit. Ann. Phys.119, 351-369 (1979) · Zbl 0412.34013 · doi:10.1016/0003-4916(79)90191-X
[13] Hempel, R.: A left-indefinite generalized eigenvalue problem for Schrödinger operators. Habilitationsschrift, University of Munich, FRG, 1987 · Zbl 0754.35026
[14] Hempel, R., Hinz, A. M., Kalf, H.: On the essential spectrum of Schrödinger operators with spherically symmetric potentials. Math. Ann.277, 197-208 (1987) · Zbl 0629.35028 · doi:10.1007/BF01457359
[15] Hille, E., Phillips, R. S.: Functional analysis and semigroups. Rev. ed., Providence, RI: Am. Math. Soc. Colloq. Publ.31, (1957) · Zbl 0078.10004
[16] Kato, T.: On the Trotter-Lie product formula. Proc. Jpn. Acad.50, 694-698 (1974) · Zbl 0336.47018 · doi:10.3792/pja/1195518790
[17] Kato, T.: Trotter’s product formula for an arbitrary pair of self-adjoint contraction semi-groups. In: Topics in functional analysis. Adv. Math. Suppl. Stud.3, 185-195 (1978)
[18] Kato, T.: Monotonicity theorems in scattering theory. Hadronic J.1, 134-154 (1978) · Zbl 0426.47004
[19] Kato, T.: Perturbation theory for linear operators. 2nd corr. (ed.), Berlin, Heidelberg, New York: Springer 1980 · Zbl 0435.47001
[20] Klaus, M.: Some applications of the Birman-Schwinger principle. Helv. Phys. Acta55, 49-68 (1982)
[21] Kudryavtsev, A. E., Markushin, V. E., Shapiro, I. S.: Nuclear level shift in the (p21-1) Atom. Sov. Phys. JETP47, 225-232 (1978)
[22] Olver, F. W. J.: Asymptotics and special functions. New York: Academic Press 1974 · Zbl 0303.41035
[23] Pötz, W., Vogl, P.: High magnetic field effects on shallow and deep impurities in semiconductors. Solid State Comm.48, 249-252 (1983) · doi:10.1016/0038-1098(83)90280-6
[24] Popov, V. S.: On the properties of the discrete spectrum for Z close to 137. Sov. Phys. JETP33, 665-673 (1971)
[25] Reed, M., Simon, B.: Methods of modern mathematical physics I: Functional analysis, rev. and enlarged ed., New York: Academic Press 1980 · Zbl 0459.46001
[26] Reed, M., Simon, B.: Methods of modern mathematical physics IV: Analysis of operators. New York: Academic Press 1978 · Zbl 0401.47001
[27] Saxon, D. S., Hutner, R. A.: Some electronic properties of a one-dimensional crystal model. Phillips Res. Rep.4, 81-122 (1949)
[28] Simon, B.: Coupling constant analyticity for the anharmonic oscillator. Ann. Phys.58, 76-136 (1970) · doi:10.1016/0003-4916(70)90240-X
[29] Simon, B.: Lower semicontinuity of positive quadratic forms. Proc. Roy. Soc. Edinburgh79, 267-273 (1977) · Zbl 0442.47017
[30] Simon, B.: A canonical decomposition for quadratic forms with applications to monotone convergence theorems. J. Funct. Anal.28, 377-385 (1978) · Zbl 0413.47029 · doi:10.1016/0022-1236(78)90094-0
[31] Thirring, W.: A course in mathematical physics Vol. 3: Quantum mechanics of atoms and molecules. Berlin, Heidelberg, New York: Springer 1981 · Zbl 0462.46046
[32] Veseli?, K.: Perturbation of pseudoresolvents and analyticity in 1/c in relativistic quantum mechanics. Commun. Math. Phys.22, 27-43 (1971) · Zbl 0212.15701 · doi:10.1007/BF01651582
[33] Vogl, P.: Chemical trends of deep impurity levels in covalent semiconductors. In: Festkörperprobleme XXI (1981), pp. 191-219. Wiesbaden, Vieweg
[34] Weidmann, J.: Oszillationsmethoden für systeme Gewöhnlicher Differentialgleichungen. Math. Z.119, 349-373 (1971) · Zbl 0206.10002 · doi:10.1007/BF01109887
[35] Weidmann, J.: Linear operators in Hilbert spaces. Graduate texts in mathematics Vol.68, Berlin, Heidelberg, New York: Springer 1980 · Zbl 0434.47001
[36] Zel’dovich, Ya.B.: Energy levels in a distorted Coulomb field. Sov. Phys. Solid State1, 1497-1501 (1960)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.