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Journal of Mathematical Physics

On the equality of S operators corresponding to unitarily equivalent Hamiltonians in single channel scattering

A. W. Saénz, W. W. Zachary

Published Mar 1, 1976• 6 citations
Publisher

Abstract

Some years ago, conditions in order that unitarily equivalent Hamiltonians H and W*HW yield the same S operator were investigated by Ekstein and by Coester and his collaborators in the framework of nonrelativistic time-dependent single-channel quantum scattering theory. This subject has turned out to be of considerable importance for nuclear physics, since it constitutes the foundation of a widely used method for constructing phase-equivalent nucleon–nucleon potentials. In the present paper we derive a rigorous and concise necessary and sufficient condition for a pair of unitarily equivalent Hamiltonians, governing the dynamics of a pair of nonrelativistic particles in the center-of-mass system, to yield the same S operator. Our theory, based on a time-dependent approach, applies to very general types of short-range potentials, with or without hard cores, and to an extensive class of long-range potentials. Our necessary and sufficient condition simplifies when certain strong limits W±, related to W, are unitary and when W+=W−. Requirements sufficient for each of these properties to hold are determined. Various examples of operators W such that H and W*HW have the same S operator are discussed.

DOI
10.1063/1.522908

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Some years ago, conditions in order that unitarily equivalent Hamiltonians H and WHW yield the same S operator were investigated by Ekstein and by Coester and his collaborators in the framework of nonrelativistic time-dependent single-channel quantum scattering theory. This subject has turned out to be of considerable importance for nuclear physics, since it constitutes the foundation of a widely used method for constructing phase-equivalent nucleon–nucleon potentials. In the present paper we derive a rigorous and concise necessary and sufficient condition for a pair of unitarily equivalent Hamiltonians, governing the dynamics of a pair of nonrelativistic particles in the center-of-mass system, to yield the same S operator. Our theory, based on a time-dependent approach, applies to very general types of short-range potentials, with or without hard cores, and to an extensive class of long-range potentials. Our necessary and sufficient condition simplifies when certain strong limits W±, related to W, are unitary and when W+=W−. Requirements sufficient for each of these properties to hold are determined. Various examples of operators W such that H and WHW have the same S operator are discussed.
Consider a pair of nonrelativistic N-particle (N?2) systems with unitarily equivalent Hamiltonians H and W*HW. Typically, W*HW involves nonlocal multiparticle interactions even when only local pair interactions are present in H. For the respective cases when H involves short-range interactions alone and also when it involves long-range ones, sufficient conditions are established in order for a scattering amplitude of the first system pertaining to any states in given entry and exit channels to equal the amplitude of the second system pertaining to corresponding states. This is accomplished by a time-dependent approach. The correspondence in question assigns to each channel state of the first system a channel state of the second system in a bijective and intuitively natural manner. Nontrivial examples are given of unitary operators W for which the above equality holds for all channels of these systems. This applies to a large class of interactions in H, including interactions with suitable long-range parts. The present work is the theoretical foundation of a new method of the authors, discussed elsewhere, for investigating many-body nuclear forces phenomenologically.
A time-dependent multichannel nonrelativistic quantum scattering theory is established for systems which contain identical particles. The symmetry properties of the theory are rooted in the abstract algebraic relationships between permutations. The formalism of second quantization is thus avoided. Appropriate wave and scattering operators are defined, including those for systems with long range (e.g., Coulomb) interactions. A correctly symmetrized dynamical equation for the symmetric form of the transition operator is derived and used to develop an optical potential formalism.NUCLEAR REACTIONS Nonrelativistic multichannel scattering theory for identical particles, wave and scattering operators, dynamical equations, optical potentials.
This paper is a contribution to the discussion of the question: to what extent does the scattering matrix determine the Hamiltonian? The Hamiltonians considered are nonrelativistic, but in extension of previous studies, "nonlocal" potentials and many-body potentials are allowed. A large class of unitary transformations is found which produce Hamiltonians leading to the same SSS matrix. In the last section, it is shown that this equivalence is only a special consequence of the general axiomatic formulation of scattering in field theory.
We analyze the issue of unitary equivalence within Generalized Uncertainty Principle (GUP) theories in the one-dimensional case. For a deformed Heisenberg algebra, its representation in terms of Hilbert space and conjugate operators is not uniquely determined, raising the question of whether different realizations of the same algebra are equivalent and describe the same physics. After proposing a definition of a quantum GUP theory, we establish conditions for unitary equivalence. Using this framework, we rigorously prove that two commonly used representations are unitarily equivalent, specifying the conditions under which this equivalence holds. We demonstrate this equivalence explicitly by providing a unitary map and showing how both GUP formulations yield the same physical results in two examples: the quantum harmonic oscillator and a free-falling particle. Finally, we discuss a case where equivalence fails, suggesting that a generalization of the Stone-von Neumann theorem may not be possible within the GUP framework under our definition of unitary equivalence.