Comportement semi-classique pour l’opérateur de Schrödinger à potentiel périodique. (Semi-classical behaviour of the Schrödinger operator with periodic potential). (French) Zbl 0662.35023

Outassourt, A.

J. Funct. Anal. 72, 65-93 (1987).

The Schrödinger operator with a periodic potential is considered. Let V be a smooth periodic function, we study the semi-classical behavior for a continuum spectrum of \(-h^ 2\Delta +V\) (h\(\to 0)\). We are interested in localization and width of bands. We give the interaction matrix up to an exponentially small error, measured by Agmon’s distance between the wells. A detailed investigation of the spectrum is made for the case where V has one nondegenerate minima per unit cell. We also investigate the spectral properties of \(-h^ 2\Delta +V+\Delta V,\) where \(\Delta\) V is a smooth positive perturbation with compact support.

MSC:

35J10	Schrödinger operator, Schrödinger equation
35P05	General topics in linear spectral theory for PDEs

Keywords:

Schrödinger operator; periodic potential; semi-classical behavior; continuum spectrum; localization; width of bands; interaction matrix; Agmon’s distance; spectral properties; smooth positive perturbation

References:

[1]	Berthier, A. M., Ann. Sci. Ecole Norm. Sup, 15, 1-15 (1982), (4) · Zbl 0487.35071
[2]	Callaway, J., Energy Band Theory (1964), Academic Press: Academic Press New York/London · Zbl 0121.23303
[3]	Carmona, R.; Simon, B., Pointwise bounds on eigenfunctions and wave packets in N. Body Quantum Systems, Comm. Mah. Phys., 80, 59-98 (1981) · Zbl 0464.35085
[4]	Eastham, M. S.P, The Spectral Theory of Periodic Differential Equation (1974), Scottish Academic: Scottish Academic Edinburgh/London · Zbl 0288.34012
[5]	Harrell, E. M., On the rate of asymptotic eigenvalue degeneracy, Comm. Math. Phys., 60, 73-95 (1978) · Zbl 0395.34023
[6]	Harrell, E. M., Double wells, Comm. Math. Phys., 75, 239-261 (1980) · Zbl 0445.35036
[7]	Harrell, E. M., The band-structure of a one-dimensional, periodic system in a scaling limit, Ann. Physics, 119, 351-369 (1979) · Zbl 0412.34013
[8]	Helffer, B.; Sjöstrand, J., Multiple wells in the semiclassical limit I, Comm. Partial Differential Equations, 9, 4, 337-408 (1984) · Zbl 0546.35053
[9]	Helffer, B.; Sjöstrand, J., Puits multiples en limite semi-classique. II. Interaction moléculaire-symétries-perturbation, Ann. Inst. H. Poincaré, 42, 2, 127-212 (1985) · Zbl 0595.35031
[10]	Helffer, B.; Sjöstrand, J., Multiple wells in the semiclassical limit. III. Interaction through non-resonant wells, Math. Nachr., 124, 263-313 (1985) · Zbl 0597.35023
[12]	Kittel, C., Introduction to Solid State Physics (1976), Wiley
[13]	Outassourt, A., Deuxième note: Perturbation d’un potentiel périodique en limite semi-classique, C. R. Acad. Sci. Paris Ser. I Math., 307, No. 3 (1985) · Zbl 0601.35087
[14]	Rajaraman, R., Solitons and Instantons (1982), North-Holland: North-Holland Amsterdam/New York/Oxford · Zbl 0493.35074
[15]	Reed, M.; Simon, B., (Methods of Modern Mathematical Physics Functional Analysis, Vol. 4 (1972), Academic Press: Academic Press New York) · Zbl 0242.46001
[16]	Simon, B., Instantons, double wells and large deviations, Bull. Amer. Math. Soc. (N.S), 8, No. 2 (1983) · Zbl 0529.35059
[17]	Simon, B., Semiclassical analysis of low lying eigenvalues. I. Non-degenerate minima: Asymptotic expansions, Ann. Inst. H. Poincaré, 38, 295-307 (1983) · Zbl 0526.35027
[18]	Simon, B., Semiclassical analysis of low lying eigenvalues. II. tunneling, Ann. of Math., 120, 89-118 (1984) · Zbl 0626.35070
[19]	Simon, B., Semiclassical analysis of low lying eigenvalues. III. Width of the ground state band in strongly coupled solid, Ann. Physics., 158, No. 2, 415-420 (1984) · Zbl 0596.35028
[20]	Simon, B., Semiclassical analysis of low lying eigenvalues. IV. the flea on the elephant, J. Funct. Anal., 63, 123-136 (1985) · Zbl 0652.35090
[21]	Jona-Lasinio, G.; Martinelli, F.; Scoppola, E., Multiple tunnelings in \(d\)-dimensions: A quantum particle in a hierarchical potential, Inst. H. Poincaré, Ann., 42, No. 1, 73-108 (1985) · Zbl 0586.35030