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Journal of Mathematical Physics

Analytic Properties of Resolvents and Completeness

Luciano Fonda, G. C. Ghirardi, T. Weber, A. Rimini

Published Sep 1, 1966• 15 citations
Publisher

Abstract

The main purpose of this paper is to prove, by using a simple formal procedure, the completeness of the set of eigenstates of a non-Hermitian operator whose resolvent satisfies certain physically plausible analytic properties. It is also shown how the obtained completeness relation can be related to the scattering solutions of the eigenvalue equation with extension to the multichannel case. All proofs are heuristic only.

DOI
10.1063/1.1705077

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The main purpose of this paper is to prove, by using a simple formal procedure, the completeness of the set of eigenstates of a non-Hermitian operator whose resolvent satisfies certain physically plausible analytic properties. It is also shown how the obtained completeness relation can be related to the scattering solutions of the eigenvalue equation with extension to the multichannel case. All proofs are heuristic only.
The mathematical theory of the scattering operator is developed for the general scattering systems involving an arbitrary number of channels. It includes as a special case the theory for 'simple scattering systems' given in an earlier paper. The scattering system is defined as a quantum mechanical system which satisfies certain asymptotic and completeness conditions given in Section 4. The existence of the S-operator as well as its unitary property is then a rigorous mathematical consequence of this property. A crucial step in these deductions is the orthogonality theorem for the left projections of the wave operators which is proved in Section 5. In the last Section 7, we discuss the various ways of introducing the 'in' and 'out'-operators and their relation to the S-operator.
The spectral analysis of a non-Hermitian unbounded operator appearing in quantum physics is our main concern. The properties of such an operator are essentially different from those of Hermitian Hamiltonians, namely due to spectral instabilities. We demonstrate that the considered operator and its adjoint can be diagonalized when expressed in terms of certain conveniently constructed operators. We show that their eigenfunctions constitute complete systems, but do not form Riesz bases. Attempts to overcome this difficulty in the quantum mechanical set up are pointed out.
We delineate the scope of research on the completeness of eigenstates in quantum mechanics. Based on the limit of the potential function at infinity, the proof of completeness is divided into eight cases, and theoretical proofs or numerical simulations are provided for each case. We present the definition of orthonormalization for general free states and the solution to the normalization coefficients, as well as a general set of initial states, which simplifies and concretizes the proof of completeness. Additionally, we define the spectral function for continuous energy eigenvalues. By taking the spectral function as the original integral variable of the expansion function, the relationship between the measured probability amplitude and the expansion function is endowed with the physical meaning of coordinate-momentum transformation.
A basic fact, having fundamental significance in quantum mechanics, is that hermitian (or self-adjoint) operators have only real eigenvalues. However, in certain applications in molecular physics, one deals with non-hermitian operators. We discuss a condition for non-hermitian operators to have real eigenvalues, proving that it is the case if and only if it can be decomposed as a product of two, generally non-commuting hermitian operators, one of which is positive definite. The theorem is illustrated on the example of non-hermitian effective Hamiltonians occurring in the non-perturbative form of the Bloch equation.
The Fredholm theory of integral equations is used to give a rigorous proof of the analyticity and boundedness of the ordinary nonrelativistic scattering amplitude for a fixed momentum transfer. The results follow from ordinary quantum mechanics and certain conditions on the potentials. These conditions are stated explicitly, and the bound states are treated with rigor. It is shown that the amplitude vanishes in the limit of large momenta, and thus simple dispersion relations are derived. Finally, it is proved that the partial-wave expansion is convergent in the unphysical region, provided the potentials satisfy the same conditions as above.