Quasi-periodic Schroedinger operators in one dimension, absolutely continuous spectra, Bloch waves, and integrable Hamiltonian systems

Thesis/Dissertation · Wed Jan 01 00:00:00 EST 1986

OSTI ID:7242624

Chierchia, L

In the first chapter, the eigenvalue problem for a periodic Schroedinger operator, Lf = (-d/sup 2//dx/sup 2/ + v)f = Ef, is viewed as a two-dimensional Hamiltonian system which is integrable in the sense of Arnold and Liouville. With the aid of the Floquet-BLoch theory, it is shown that such a system is conjugate to two harmonic oscillators with frequencies ..cap alpha.. and omega, being the rotation number for L and 2..pi../omega the period of the potential v. This picture is generalized in the second chapter, to quasi periodic Schroedinger operators, L/sub epsilon/, with highly irrational frequencies (omega/sub 1/, ..., omega/sub d/), which are a small perturbation of periodic operators. In the last chapter, the absolutely continuous spectrum sigma/sub ac/ of a general quasi-periodic Schroedinger operators is considered. The Radon-Nikodym derivatives (with respect to Lebesgue measure) of the spectral measures are computed in terms of special independent eigensolutions existing for almost ever E in sigma/sub ac/. Finally, it is shown that weak Bloch waves always exist for almost ever E in sigma/sub ac/ and the question of the existence of genuine Bloch waves is turned into a regularity problem for a certain nonlinear partial differential equation on a d-dimensional torus.

• Research Organization: New York Univ., NY (USA)
• OSTI ID: 7242624
• Resource Relation: Other Information: Thesis (Ph. D.)
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
QUANTUM OPERATORS
SCHROEDINGER EQUATION
BLOCH THEORY
HAMILTONIANS
NONLINEAR PROBLEMS
PARTIAL DIFFERENTIAL EQUATIONS
DIFFERENTIAL EQUATIONS
EQUATIONS
MATHEMATICAL OPERATORS
WAVE EQUATIONS
657002* - Theoretical & Mathematical Physics- Classical & Quantum Mechanics