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International Journal of Quantum Chemistry

Some properties of inner projections

Per‐Olov Löwdin

Published Jun 18, 2009• 29 citations
Publisher

Abstract

During the last decade, inner projections of positive-definite operators have turned out to be of essential importance for constructing lower bounds to self-adjoint operators which are bounded from below. They have been utilized to construct lower bounds to energy eigenvalues, and to study the wave and reaction operators in modern perturbation theory. In this paper, a brief review is given of some of the basic properties of the inner projections.

DOI
10.1002/qua.560050725

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During the last decade, inner projections of positive-definite operators have turned out to be of essential importance for constructing lower bounds to self-adjoint operators which are bounded from below. They have been utilized to construct lower bounds to energy eigenvalues, and to study the wave and reaction operators in modern perturbation theory. In this paper, a brief review is given of some of the basic properties of the inner projections.
This chapter develops a basic working knowledge of operators — objects central to quantum mechanics. The fact that quantum operators are linear determines how they may be manipulated, so linearity forms the chapter's first topic. The adjoint of an operator is then introduced. This provides the basis for Hermitian or self-adjoint operators, which are discussed for both discrete eigenvalue systems and wavefunctions. Projection operators, which ‘project out’ a certain part of a quantum state, are then introduced. These form the basis for the identity operator — a sum over projection operators — which changes the representation in which a state is expressed without changing the state itself. Finally, unitary operators are briefly discussed, including the defining unitarity condition, the fact that such operators preserve inner products, and their physical meaning as effecting transformations in space and time. Unitary operators are discussed in greater detail in Appendix D.
The set of bounded observables for a quantum system is represented by the set of bounded self-adjoint operators S(H) on a complex Hilbert space H, and the quantum effects for a physical system can be described by the set E(H) of positive contractive operators on a complex Hilbert space H. In this note, by the techniques of operator block and spectral, we give the simpler representation of A∧P and obtained the new necessary and sufficient conditions for A∨P, for A∊S(H) and P∊P(H), where P(H) is the set of all orthogonal projection operators on H. In particular, we get that if A∨P exists, then A∨P∊E(H) for A∊E(H) and P∊P(H). In addition, we consider the relations between the existence of A∨B, A−∨B−, and A+∨B+, where A+, B+, A−, and B− are the positive and negative parts of A,B∊S(H).
We explore the Hamiltonian operator , where is the Dirac delta function and z is an arbitrary complex coupling constant. For a purely imaginary z , H has a spectral singularity at . For Re( z ) < 0, H has an eigenvalue at E = − z 2 /4. For the case that Re( z ) > 0, H has a real, positive, continuous spectrum that is free from spectral singularities. For this latter case, we construct an associated biorthonormal system and use it to perform a perturbative calculation of a positive-definite inner product that renders H self-adjoint. This allows us to address the intriguing question of the nonlocal aspects of the equivalent Hermitian Hamiltonian for the system. In particular, we compute the energy expectation values for various Gaussian wave packets to show that the non-Hermiticity effect diminishes rapidly outside an effective interaction region.