Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Cambridge University Press eBooks

Propagator and Resolvent

Marcos Mariño

Published Dec 2, 2021• 0 citations
Publisher

Abstract

It presents the unitary time evolution operator and its integral kernel in the space representation, also known as the quantum-mechanical propagator. It introduces the resolvent operator and its analytical properties, and it discusses in some detail scattering theory in one dimension.
DOI
10.1017/9781108863384.002

Related papers

Causality and propagation in the Wigner, Husimi, Glauber, and Kirkwood phase-space representations

Bilha Segev

Time evolution is considered in phase space in terms of evolution kernels for various phase-space quasi-distributions. The propagators for the Wigner function, the standard-ordered function, the Kirkwood (antistandard-ordered) function, the Glauber P and Q functions, and the Husimi function are explicitly written as bilinear transforms of the evolution operator. Free propagation, propagation in dispersive media, and scattering, are studied, and manifestations of causality and interference are analyzed. It is shown that free propagation and scattering in the Husimi, Glauber, and Kirkwood representations with the underlying dynamics of the Schr\"odinger equation involve divergent evolution kernels connecting distant phase-space points at all times. The time evolution is much simpler in the Wigner representation where (i) free propagation is a simple classical translation involving no interference, and (ii) analytical properties of the scattering matrix restrict the velocities of propagation so that no information can travel due to scattering faster than free motion. As an example, a correlation is found between the coordinate and momentum of particles detected after they are released from a box. Propagators with relativistic dispersion relations of free photons or Klein-Gordon particles are briefly discussed in an Appendix.

Physical Review A•Apr 19, 2001•8 citations

A NOTE ON THE EXISTENCE OF UNITARY PROPAGATOR OF EQUATIONS IN QUANTUM MECHANICS(Spectral and Scattering Theory and Its Related Topics)

Wataru Ichinose

数理解析研究所講究録•Apr 1, 1995•0 citations

A NOTE ON THE EXISTENCE OF UNITARY PROPAGATOR OF EQUATIONS IN QUANTUM MECHANICS(Spectral and Scattering Theory and Its Related Topics)

弥 一ノ瀬

数理解析研究所講究録•Apr 1, 1995•0 citations

Quantum walk on orbit spaces

Satoshi Ohya

Inspired by the covering-space method in path integrals on multiply connected spaces, we here present a universal formula of time-evolution kernels for continuous- and discrete-time quantum walks on orbit spaces. In this note, we focus on the case in which walkers' configuration space is the orbit space $\mathrm{\ensuremath{\Lambda}}/\mathrm{\ensuremath{\Gamma}}$, where $\mathrm{\ensuremath{\Lambda}}$ is an arbitrary lattice and $\mathrm{\ensuremath{\Gamma}}$ is a discrete group whose action on $\mathrm{\ensuremath{\Lambda}}$ has no fixed points. We show that the time-evolution kernel on $\mathrm{\ensuremath{\Lambda}}/\mathrm{\ensuremath{\Gamma}}$ can be written as a weighted sum of time-evolution kernels on $\mathrm{\ensuremath{\Lambda}}$, where the summation is over the orbit of initial point in $\mathrm{\ensuremath{\Lambda}}$ and weight factors are given by a one-dimensional unitary representation of $\mathrm{\ensuremath{\Gamma}}$. Focusing on one dimension, we present a number of examples of the formula. We also present universal formulas of resolvent kernels, canonical density matrices, and unitary representations of arbitrary groups in quantum walks on $\mathrm{\ensuremath{\Lambda}}/\mathrm{\ensuremath{\Gamma}}$, all of which are constructed in exactly the same way as for the time-evolution kernel.

arXiv:2301.03193•Physical Review A•Jun 2, 2023•2 citations

Density Matrix Formalism. Resolvent Operator

Ole Keller

A detailed account of the density matrix formalism and the resolvent operator theory is given. The statistical mixture of quantum states is described in terms of the density matrix operator, a concept which in an elegant manner allows one to obtain the ensemble average of an observable [a Hermitian operator, which eigenspectrum is complete]. On the basis of an integral equation for the evolution operator, described in the Interaction Representation, the Liouville equation, describing the dynamical evolution of the density matrix operator, is established. A comprehensive analysis of thermal equilibrium in electron systems is presented. For a description of the dynamical time evolution of a system in contact with a heat bath, a reservoir in thermal equilibrium, the den- sity matrix operator approach is shown to be extremely convenient. The Kubo formalism for the time-dependent density matrix is discussed in its most general form. The so-called resolvent propagator formalism is set up starting from a discussion of the analytical part of a Dirac delta function. Having established integral equations for the retarded and advanced scalar propagator operators, the resolvent propagator concept is introduced and analyzed. The resolvent of the Hamiltonian is of particular importance in physics, and the singularities of this resolvent (poles and branch cuts) are identified. We finish by a study of correlated systems in terms of the reduced density matrix operator concept.

CRC Press eBooks•Aug 6, 2024•0 citations

Some remarks on the unitary time evolution operator of the quantum-mechanical forced oscillator. The Weierstrass propagator

Horacio Grinberg

Il Nuovo cimento della Società italiana di fisica. B/Il Nuovo cimento B•Sep 1, 1984•0 citations
Time evolution is considered in phase space in terms of evolution kernels for various phase-space quasi-distributions. The propagators for the Wigner function, the standard-ordered function, the Kirkwood (antistandard-ordered) function, the Glauber P and Q functions, and the Husimi function are explicitly written as bilinear transforms of the evolution operator. Free propagation, propagation in dispersive media, and scattering, are studied, and manifestations of causality and interference are analyzed. It is shown that free propagation and scattering in the Husimi, Glauber, and Kirkwood representations with the underlying dynamics of the Schr\"odinger equation involve divergent evolution kernels connecting distant phase-space points at all times. The time evolution is much simpler in the Wigner representation where (i) free propagation is a simple classical translation involving no interference, and (ii) analytical properties of the scattering matrix restrict the velocities of propagation so that no information can travel due to scattering faster than free motion. As an example, a correlation is found between the coordinate and momentum of particles detected after they are released from a box. Propagators with relativistic dispersion relations of free photons or Klein-Gordon particles are briefly discussed in an Appendix.
Inspired by the covering-space method in path integrals on multiply connected spaces, we here present a universal formula of time-evolution kernels for continuous- and discrete-time quantum walks on orbit spaces. In this note, we focus on the case in which walkers' configuration space is the orbit space $\mathrm{\ensuremath{\Lambda}}/\mathrm{\ensuremath{\Gamma}}$, where $\mathrm{\ensuremath{\Lambda}}$ is an arbitrary lattice and $\mathrm{\ensuremath{\Gamma}}$ is a discrete group whose action on $\mathrm{\ensuremath{\Lambda}}$ has no fixed points. We show that the time-evolution kernel on $\mathrm{\ensuremath{\Lambda}}/\mathrm{\ensuremath{\Gamma}}$ can be written as a weighted sum of time-evolution kernels on $\mathrm{\ensuremath{\Lambda}}$, where the summation is over the orbit of initial point in $\mathrm{\ensuremath{\Lambda}}$ and weight factors are given by a one-dimensional unitary representation of $\mathrm{\ensuremath{\Gamma}}$. Focusing on one dimension, we present a number of examples of the formula. We also present universal formulas of resolvent kernels, canonical density matrices, and unitary representations of arbitrary groups in quantum walks on $\mathrm{\ensuremath{\Lambda}}/\mathrm{\ensuremath{\Gamma}}$, all of which are constructed in exactly the same way as for the time-evolution kernel.
A detailed account of the density matrix formalism and the resolvent operator theory is given. The statistical mixture of quantum states is described in terms of the density matrix operator, a concept which in an elegant manner allows one to obtain the ensemble average of an observable [a Hermitian operator, which eigenspectrum is complete]. On the basis of an integral equation for the evolution operator, described in the Interaction Representation, the Liouville equation, describing the dynamical evolution of the density matrix operator, is established. A comprehensive analysis of thermal equilibrium in electron systems is presented. For a description of the dynamical time evolution of a system in contact with a heat bath, a reservoir in thermal equilibrium, the den- sity matrix operator approach is shown to be extremely convenient. The Kubo formalism for the time-dependent density matrix is discussed in its most general form. The so-called resolvent propagator formalism is set up starting from a discussion of the analytical part of a Dirac delta function. Having established integral equations for the retarded and advanced scalar propagator operators, the resolvent propagator concept is introduced and analyzed. The resolvent of the Hamiltonian is of particular importance in physics, and the singularities of this resolvent (poles and branch cuts) are identified. We finish by a study of correlated systems in terms of the reduced density matrix operator concept.