STUDIES IN PERTURBATION THEORY. IV. SOLUTION OF EIGENVALUE PROBLEM BY PROJECTION OPERATOR FORMALISM

Journal Article · Sat Sep 01 04:00:00 EDT 1962 · J. Math. Phys.

DOI:https://doi.org/10.1063/1.1724312· OSTI ID:4760197

Loewdin, P

The partitioning technique for solving secular equations is reviewed. It is formulated in terms of an operator language in order to permit a discussion of the various methods of solving the Schrodinger equation. The total space is divided into two parts by means of a self-adjoint projection operator O. Introducing the symbolic inverse T = (1 - O)/ (E -- H), it can be shown that there exists an operator OMEGA = O + THO, which is an idempotent eigenoperator to H and satisfies the relations H OMEGA = E OMEGA and OMEGA /sup 2/ = OMEGA . This operator is not normal but has a form that directly corresponds to infinite- order perturbation theory. Both the Brillouin- and Schrodinger-type formulas may be derived by power series expansion of T, even if other forms are perhaps more natural. The concept of the reaction operator is discussed, and upper and lower bounds for the true eigenvalues are derived. (auth)

• Research Organization: Univ. of Uppsala and Univ. of Florida, Gainesville
• Sponsoring Organization: USDOE
• NSA Number: NSA-17-002446
• OSTI ID: 4760197
• Journal Information: J. Math. Phys., Journal Name: J. Math. Phys. Vol. Vol: 3
• Country of Publication: Country unknown/Code not available
• Language: English

Related Subjects

DIFFERENTIAL EQUATIONS
EIGENVALUES
HAMILTONIAN
HAMILTONIAN FUNCTION
MATHEMATICS
PERTURBATION THEORY
PHYSICS
POWER SERIES
QUANTUM MECHANICS
SCHROEDINGER EQUATION