Abstract
Spectral properties of − Δ +V(x), whereV(x) lies in a ‘neighbourhood’ of the periodic case and describes various ‘models of disorder’, are studied. We prove the exponential decay of generalized eigenfunctions corresponding to energies in the resolvent set of the ‘unperturbed’ periodic Hamiltonian, as well as the stability of the essential spectrum for the ‘dislocation disorder’ in two dimensions.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Berthier-(Boutet de Monvel), A.,Spectral Theory and Wave Operators for the Schrödinger Equation, Research Notes in Mathematics 71, Pitman, London, 1982, and Nauka, Moscow, to appear.
Reed, M. and Simon, B.,Methods of Modern Mathematical Physics IV: Analysis of Operators, Academic Press, New York, 1978.
Skriganov, M. M.,Invent. Math. 10, 107–129 (1985).
Carmona, R. and Lacroix, J.,Spectral Theory of Random Schrödinger Operators, Birkhäuser, Boston, 1990.
Pastur, L., Figotin, A.,Spectra of Random and Almost Periodic Operators, Springer, Berlin, 1992.
Ziman, J. M.,Models of Disorder, Cambridge University Press, Cambridge, 1979.
Agmon, S.,Lectures on Exponential Decay of Solutions of Second Order Elliptic Equations, Princeton University Press, Princeton, NJ, 1982.
Herbst, I.,Comm. Math. Physics 158, 517–536 (1993).
Boutet de Monvel-Berthier, A. and Georgescu, V.,J. Funct. Anal. 71, 309–338 (1987).
Wang, X. P.,Ann. Inst. Henri Poincaré 43, 269 (1985).
Parisse, B.,Ann. Inst. Henri Poincaré 56, 235 (1992).
Perry, P. A., A remark on continuous eigenfunctions ofN-body Schrödinger operators, in I. W. Knowles and R. T. Lewis (eds),Differential Equations, North-Holland, Amsterdam, 1984.
Combes, J.-M. and Thomas, L.,Comm. Math. Physics 34, 251 (1973).
Cycon, H. L., Froese, R. G., Kirsch, W., and Simon, B.,Schrödinger Operators, Springer, Berlin 1987.
Froese, R. G., Herbst, I., Hoffmann-Ostenhof, M., and Hoffmann-Ostenhof, T.,Comm. Math. Phys. 87, 265 (1982).
Nenciu, Gh.,Rev. Mod. Phys. 63, 91–127 (1991).
Helffer, B. and Sjöstrand, J., Equation de Schrödinger avec champ magnétique et équation de Harper, in H. Holden and A. Jensen (eds),Schrödinger Operators, Lecture Notes in Physics 345, Springer, Berlin, 1989, p. 118.
Shubin, M. A.,Astérisque 207, 37 (1992).
Schechter, M.,Spectra of Differential Operators, 2nd edn, North-Holland, Amsterdam, 1986.
Pantelides, S. T.,Rev. Modern Phys. 50, 797–858 (1978).
Bentosela, F.,Comm. Math. Phys. 46, 153 (1976).
Agmon, S., On positive solutions of elliptic equations with periodic coefficients in ℝn; spectral results and extensions to elliptic operators on Riemannian manifolds, in I. W. Knowles and R. T. Lewis (eds)Differential Equations, North-Holland, Amsterdam, 1984, pp. 7-17.
Gubanov, A. I.,JETP 23, 484 (1955).
Nenciu, A. and Nenciu, Gh.,J. Phys. A. 14, L.109-L.112 (1981).
Nenciu, A., PhD. Thesis, ICEFIZ, Bucharest 1984 (published inStudii si cercetan de Fizica 39, 494 (1987).
Kato, T.,Perturbation Theory for Linear Operators, 2nd edn, Springer, Berlin 1970.
Simon, B.,Quantum Mechanics for Hamiltonians Defined as Quadratic Forms, Princeton University Press, Princeton, NJ, 1971.
Simon, B., Spectrum and continuous eigenfunctions of Schrödinger operators,J. Funct. Anal. 42, 347–355 (1981).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
De Monvel-Berthier, A.B., Nenciu, A. & Nenciu, G. Perturbed periodic Hamiltonians: Essential Spectrum and exponential decay of eigenfunctions. Lett Math Phys 34, 119–133 (1995). https://doi.org/10.1007/BF00739091
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00739091