The Library
Stochastic evolution as a quasiclassical limit of a boundary value problem for Schrödinger equations
Tools
Belavkin, V. P. and Kolokoltsov, V. N. (Vasiliĭ Nikitich). (2002) Stochastic evolution as a quasiclassical limit of a boundary value problem for Schrödinger equations. Infinite Dimensional Analysis, Quantum Probability and Related Topics, Vol.5 (No.1). pp. 6191. ISSN 0219 0257

PDF
WRAP_Kolokoltsov_bel.pdf  Submitted Version  Requires a PDF viewer such as GSview, Xpdf or Adobe Acrobat Reader Download (332Kb) 
Official URL: http://dx.doi.org/10.1142/S0219025702000717
Abstract
We develop systematically a new unifying approach to the analysis of linear stochastic, quantum stochastic and even deterministic equations in Banach spaces. Solutions to a wide class of these equations (in particular those decribing the processes of continuous quantum measurements) are proved to coincide with the interaction representations of the solutions to certain Dirac type equations with boundary conditions in pseudo Fock spaces. The latter are presented as the semiclassical limit of an appropriately dressed unitary evolutions corresponding to a boundaryvalue problem for rather general Schrödinger equations with bounded below Hamiltonians.
[error in script] [error in script]Item Type:  Journal Article 

Subjects:  Q Science > QA Mathematics 
Divisions:  Faculty of Science > Statistics 
Library of Congress Subject Headings (LCSH):  Stochastic processes, Banach spaces, Schrödinger equation 
Journal or Publication Title:  Infinite Dimensional Analysis, Quantum Probability and Related Topics 
Publisher:  World Scientific Publishing 
ISSN:  0219 0257 
Date:  2002 
Volume:  Vol.5 
Number:  No.1 
Page Range:  pp. 6191 
Identification Number:  10.1142/S0219025702000717 
Status:  Peer Reviewed 
Publication Status:  Published 
Access rights to Published version:  Restricted or Subscription Access 
References:  [1] L. Accardi, R. Alicki, A. Frigerio, and Y.G. Lu. An invitation to the weak coupling and low density limits. Quantum Probability and Related Topics VI (1991), 361. [2] l. Accardi, Y.G. Lu and I. Volovich. Quantum Theory and its Stochastic Limit. SpringerVerlag, Texts and Mopnographs in Physics, 2000. [3] S. Albeverio, F. Gesztesy, R. HoeghKrohn, and H. Holden. Solvable models in quantum mechanics. Springer 1988. [4] S.Albeverio, V.N. Kolokoltsov, O.G. Smolyanov. Continuous Quantum Measurement: Local and Global Approaches. Reviews in Math. Phys. 9:8 (1997), 907920. [5] H. Araki. Factorisable representations of current algebras. Proc. R.I.M.S., Kyoto 5 (1970/71), 361422. [6] A. Barchielli, A.S. Holevo. Constructing quantum measurement processes via classical stochastic calculus. Stochastic Processes Appl. 58:2 (1995), 293317. [7] V.P. Belavkin. A continuous counting observation and posterior quantum dynamics. J. Phys. A Math. Gen. 22 (1989), L1109L1114. [8] V.P. Belavkin. Nondemolition Principle of Quantum Measurement Theory. Found. of Physics 24:5 (1994), 685714. [9] V.P. Belavkin. A dynamical Theory of Quantum Measurement and Spontaneous Localization. Russian Journal of Mathematical Physics 3:1 (1995), 323. [10] V.P. Belavkin. On Quantum Stochastics as a Dirac Boundaryvalue Problem and an Inductive Stochastic Limit. In: Evolution Equations and their Applications, Lect. Notes Pure Appl. Math., Marcel Dekker, Inc., New York 2000, 311334. [11] V.P. Belavkin. Quantum Stochastic Dirac Boundary Value Problem and the Ultra Relativistic Limit. Rep. Math. Phys. 46:3 (2000), 359386. [12] V.P. Belavkin. Chaotic States and Stochastic Integration in Quantum Systems. Russian Math. Surveys 47:1 (1992), 47106. [13] V.P. Belavkin. A Quantum Nonadapted Ito Formula and Stochastic Analysis in Fock Scale. J. Funct. Anal. 102:2 (1991), 414447. [14] V.P. Belavkin, R. Hudson, R. Hirota (Eds.). Quantum Communications and Measurements. Proc. Intern. Workshop held in Nottingham, 1994. Plenum Press 1995. [15] A.M. Chebotarev. The quantum stochastic equation is equivalent to a boundary value problem for the Schrödinger equation. Mathematical Notes 61:4 (1997), 510518. [16] A.M. Chebotarev. The quantum stochastic equation is equivalent to a symmetric boundary value problem in Fock space. Infinite Dimensional Analysis, Quantum Probability and Related Topics, 1:2 (1998): 175199. [17] L. Diosi. Continuous quantum measurement and Ito formalism. Phys. Let. A 129 (1988), 419423. [18] J. Gough. The Stratonovich Interpretation of Quantum Stochastic Approximations. Potential Analysis 11 (1999), 213233. [19] M.Gregoratti. On the Hamiltonian Operator Associated to Some Quantum Stochastic Differential Equations. Infinite Dimensional Analysis, Quantum Probability and Related Topics, 3:4 (2000), 483504. [20] R.L. Hudson, K.R. Parthasarathy. Quantum Ito's formula and stochastic evolutions. Comm. Math. Phys. 93:3 (1984), 301323. [21] V.N. Kolokoltsov. Localisation and Analytic Properties of the Simplest Quantum Filtering equation. Reviews in Math. Phys. 10:6 (1998), 801 828. [22] V.N. Kolokoltsov. Short deduction and mathematical properties of the main equation of the theory of continuous quantum measurements. In: GROUP21 (Eds. H.D.Doebner, P.Nattermann, W.Scherer), Proc. XXI Intern. Colloq. on Group Theoret. Methods in Physics July 1996, World Scientific 1997, v.1, 326330. [23] V.N. Kolokoltsov. The stochastic HJB equation and WKB method. In: Idempotency (Ed. J. Gunawardena), Cambridge Univ. Press 1998, 285 302. [24] V.N. Kolokoltsov. Semiclassical Analysis for Diffusions and Stochastic Processes Springer Lecture Notes Math. 1724, 2000. [25] V. Koshmanenko. Singular Quadratic Forms in Perturbation Theory. Kluwer Academic 1999. [26] J. von Neumann. Foundations of Quantum Mechanics. Princeton Univ. Press 1955. [27] K.R. Parthasarathy. An Introduction to Quantum Stochastic Calculus. Birkhauser Verlag, Basel, 1992. [28] P. Protter. Stochastic Integration and Differential Equations. Applications of Mathematics 21, SpringerVerlag, 1990. [29] Quantum and Semiclassical Optics 8:1 (1996). Special Issue on Stochastic Quantum Optics. [30] R.F. Streater. Current commutation relations, continuous tensor products, and infinitely divisible group representations. In: Local Quantum Theory (Ed. R. Jost), Academic Press, 1969, 247263. [31] A. Truman, Z. Zhao. The stochastic HJ equations, stochastic heat equations and Schrödinger equations. In: (Eds. A Truman et al) Stochastic Analysis and Application. World Scientific Press (1996), 441464. 
URI:  http://wrap.warwick.ac.uk/id/eprint/39366 
Actions (login required)
View Item 