Resonance poles and Gamow vectors in the rigged Hilbert space formulation of quantum mechanics

Abstract

After a summary of the Jigged Hilbert space formulation of quantum mechanics and a brief statement of its advantages over von Neumann's formulation, a mathematically correct definition of Gamow's exponentially decaying vectors as generalized energy eigenvectors is suggested. It is shown that exponentially decaying vectors are obtained from the S-matrix poles in the lower half of the second sheet and exponentially growing vectors from the S-matrix poles in the upper half of the second sheet. Decaying "state" vectors are defined as functionals over half of the space of physical states and growing "state" vectors are defined as functionals over the other half. On functionals over these subs paces, the dynamical group of time development splits into two semigroups, one for t > 0 and the other for t < O. The generalized basis system connected with the spectrum of the Hamiltonian is transformed into a new basis system in which the exponentially decaying component of the density matrix is separated.

Higher order Gamow states with exponential decay

“…We insert this into (2.1) and deform the contour of integration C − through the cut along the spectrum of H into the second sheet (Bohm, 1979(Bohm, , 1980(Bohm, , 1981(Bohm, , 1993. Then one obtains…”

“…In analogy to von Neumann's definition of a pure stationary state using dyadic products |f f | of the energy eigenvectors |f in Hilbert space, microphysical Gamow states connected with first order poles can be defined as dyadic products of zeroth order Gamow vectors (Bohm, 1979(Bohm, , 1980(Bohm, , 1981(Bohm, , 1993Bohm et al, 1997),…”

“…Resonances in quantum mechanics can be described by poles of the analytically continued S-matrix on the second sheet of a two-sheeted Riemann surface (Newton, 1982;Bohm 1979Bohm , 1980Bohm , 1981Bohm , 1993. These poles appear in conjugate pairs below (at z R = E R − iΓ/2 ) and above (at z * R = E R + iΓ/2) the positive real axis, where the pole at z R corresponds to a decaying state at times t ≥ 0 and the pole at z * R to the growing state at t < 0.…”

From Poincare's Divergences to Quantum Mechanics with Broken Time Symmetry

“…Also the space in which these states live cannot be the usual Hilbert space as in this space H, which is a Hermitian operator, cannot have complex eigenvalues. Let us mention that we deal here with the so called "rigged Hilbert space" [27,28] (which admits states with zero norm). We see that LPS may have more than one spectral representation.…”

Mathematical Foundations of Time Asymmetric Quantum Mechanics

“…ii) Give a precise meaning to Gamow vectors or vector states for resonances [17][18][19]. iii) With the use of rigged Hilbert space of Hardy functions, provide a mathematical support for the time asymmetric quantum mechanics [20][21][22][23].…”