Invariance principle and asymptotic completeness for a quantum mechanical system

Thesis/Dissertation · Fri Jan 01 04:00:00 EST 1982

OSTI ID:6087803

Herzberg, M

Operators H and H/sub 0/ acting on the Hilbert Space L/sup 2/(R) are studied. The free or unperturbed operator H/sub 0/ is the multiplication (also known as the position) operator x. H = H/sub 0/ + A is considered, where the perturbation A is an integral operator which may be unbounded. H/sub 0/ is known to be self-adjoint. Conditions are given on the pertubation A for H to be self-adjoint as well. The primary objective of this dissertation is to prove the main conclusions of scattering theory for operators of the type just described. Both time-dependent and stationary methods are utilized. A factored perturbation technique to prove existence and completeness of the wave operators W +- (H,H/sub 0/). Conditions are obtained for the existence of the wave operators for a generalized integral perturbation where A is not factorable. This is accomplished by first finding operators L/sub 0/ and B in momentum (Fourier-transform) space which are equivalent to the operators H and H/sub 0/ in configuration space. Then, by showing that the two pairs of Hamiltonians are equilavent, it has been proved that all theorems proved for H and H/sub 0/ are true for L(=L/sub 0/ + B) and L/sub 0/, and vice versa. Then the operators L and L/sub 0/ can be examined and the results can be transferred to H and H/sub 0/.

• Research Organization: Yeshiva Univ., New York (USA)
• OSTI ID: 6087803
• Country of Publication: United States
• Language: English

Related Subjects

657002* -- Theoretical & Mathematical Physics-- Classical & Quantum Mechanics
71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
BANACH SPACE
HAMILTONIANS
HILBERT SPACE
INVARIANCE PRINCIPLES
MATHEMATICAL OPERATORS
MATHEMATICAL SPACE
MECHANICS
QUANTUM MECHANICS
QUANTUM OPERATORS
SPACE