SCHRODINGER-TYPE OPERATORS WITH CONTINUOUS SPECTRA-EASTHAM, MSP, KALF, H (original) (raw)

Book Review: Schrödinger-type operators with continuous spectra

Bulletin of the American Mathematical Society, 1984

What ultimately constitutes a good mathematics book? It seems to the reviewer that this is a function ƒ(e, r, c) of the variables e = effort needed to comprehend the book, r = reward in the form of valuable understanding gained, and c = cost of the book. Of these, the first two are highly dependent on the reader, and, given the first two, dependence on the third is completely individual, hence need not be discussed here. We assume ƒ is decreasing in e and increasing in r. On these assumptions, Chung's book comes out very well indeed for the present reviewer. But let us beware. The reviewer has recently written a book [Essentials of Brownian motion and diffusion, Math. Surveys, vol. 18, Amer. Math. Soc., Providence, R.I., 1981] which complements Chung's book to a considerable degree. It gets one over the hard beginning (especially §1.3, Optional Times) without appreciably encroaching on the content. For the two books together one might suggest the title From Brownian motion to Markov processes and back.

Review: M. S. P. Eastham and H. Kalf, Schr�dinger-type operators with continuous spectra

Bull Amer Math Soc, 1984

What ultimately constitutes a good mathematics book? It seems to the reviewer that this is a function ƒ(e, r, c) of the variables e = effort needed to comprehend the book, r = reward in the form of valuable understanding gained, and c = cost of the book. Of these, the first two are highly dependent on the reader, and, given the first two, dependence on the third is completely individual, hence need not be discussed here. We assume ƒ is decreasing in e and increasing in r. On these assumptions, Chung's book comes out very well indeed for the present reviewer. But let us beware. The reviewer has recently written a book [Essentials of Brownian motion and diffusion, Math. Surveys, vol. 18, Amer. Math. Soc., Providence, R.I., 1981] which complements Chung's book to a considerable degree. It gets one over the hard beginning (especially §1.3, Optional Times) without appreciably encroaching on the content. For the two books together one might suggest the title From Brownian motion to Markov processes and back.

Graduate Texts in Mathematics Brownian Motion, Martingales, and Stochastic Calculus

Graduate Texts in Mathematics bridge the gap between passive study and creative understanding, offering graduate-level introductions to advanced topics in mathematics. The volumes are carefully written as teaching aids and highlight characteristic features of the theory. Although these books are frequently used as textbooks in graduate courses, they are also suitable for individual study.

Stochastic Schroedinger equations and applications to Ehrenfest-typetheorems

Latin American Journal of Probability and Mathematical Statistics, 2012

We study stochastic evolution equations describing the dynamics of open quantum systems. First, using resolvent approximations, we obtain a sufficient condition for regularity of solutions to linear stochastic Schrödinger equations driven by cylindrical Brownian motions applying to many physical systems. Then, we establish well-posedness and norm conservation property of a wide class of open quantum systems described in position representation. Moreover, we prove Ehrenfest-type theorems that describe the evolution of the mean value of quantum observables in open systems. Finally, we give a new criterion for the existence and uniqueness of weak solutions to non-linear stochastic Schrödinger equations. We apply our results to physical systems such as fluctuating ion traps and quantum measurement processes of position.

Functional-calculus approach to stochastic differential equations

Physical Review A, 1986

The connection between stochastic differential equations and associated Fokker-Planck equations is elucidated by the full functional calculus. One-variable equations with either additive or multiplicative noise are considered. The central focus is on approximate Fokker-Planck equations which describe the consequences of using "colored" noise, which has an exponential correlation function and a correlation time~. To leading order in v. , the functional-calculus approach generalizes the~expansion result and produces an approximate Fokker-Planck equation free from certain difficulties which have plagued the less general approximations. Mean first-passage-tine behavior for bistable potentials, an additive case, is discussed in detail. The new result presented here leads to a mean first-passage-time formula in quantitative agreement with the results of numerical simulation and in contrast with earlier theoretical conclusions. The theory provides new results for the multiplicative case as well.

Markov processes and generalized Schroedinger equations

2011

Starting from the forward and backward infinitesimal generators of bilateral, time-homogeneous Markov processes, the self-adjoint Hamiltonians of the generalized Schroedinger equations are first introduced by means of suitable Doob transformations. Then, by broadening with the aid of the Dirichlet forms the results of the Nelson stochastic mechanics, we prove that it is possible to associate bilateral, and time-homogeneous Markov processes to the wave functions stationary solutions of our generalized Schroedinger equations. Particular attention is then paid to the special case of the Levy-Schroedinger equations and to their associated Levy-type Markov processes, and to a few examples of Cauchy background noise.

Markov processes and generalized Schrödinger equations

Journal of Mathematical Physics, 2011

Starting from the forward and backward infinitesimal generators of bilateral, time-homogeneous Markov processes, the self-adjoint Hamiltonians of the generalized Schrödinger equations are first introduced by means of suitable Doob transformations. Then, by broadening with the aid of the Dirichlet forms the results of the Nelson stochastic mechanics, we prove that it is possible to associate bilateral, and time-homogeneous Markov processes to the wave functions stationary solutions of our generalized Schrödinger equations. Particular attention is then paid to the special case of the Lévy-Schrödinger (L-S ) equations and to their associated Lévy-type Markov processes, and to a few examples of Cauchy background noise.