QUANTUM EVOLUTION EQUATIONS

Stochastic Quantum Evolutions (SQE)
were first studied in connection with generalized Schroedinger Equation with singular (stochastic) potentials. Only recently it was realized that the deterministic unitarity condition in such evolutions should be weakened and replaced by the stochastic unitarity of wave propagators (unitary in the mean square sense). The linear stochastic wave equations of this type first appeared as renormalizations of the nonlinear quantum filtering equations in [1,2]. They were derived in full generality from the quantum stochastic equations both for quantum jump and diffusion unitary evolutions in the series of my papers on quantum stochastic filtering during the 80's. Gisen and Pearl, and then Diosi introduced a nonlinear stochastic wave equation of the particular diffusive type into quantum physics in the end of 80's. They wanted to describe the notorious wave reduction continuously in time as spontaneous jumps and localization, but could find only the simplest continuous diffusive analog for such jumps, based on the classical standard Wiener process. The jump stochastic models for continuous reductions based on the classical Poisson process were rediscovered in quantum optics in the 90's, but the corresponding filtering wave equations for quantum spontaneous jumps both in linear and nonlinear form, as well as their quantum stochastic derivation [3] and applications [4-6] remain largely unknown in physics. As it has been shown in [7], the quantum jump models allow to describe also the unstable quantum systems by a contractive stochastic wave equation.
Quantum Boundary Problems (QBP)
play important role in the singular scattering theory, and for the dynamical interpretation of QSE. In the second quantization they give Hamiltonian models for Quantum Continuous Measurements, and their solution in interaction picture represents Quantum Stochastic Cocycles resolving the SQE. This was understood in the Dynamical Measurement Theory in the beginning of 90's, and by Chebotarev in the contents of quantum stochastic equations for unitary evolutions in Fock space, but only recently was proved in full generality in [8,9]. As we proved, any SQE and quantum stochastic evolution corresponding to Hudson-Parthasarathy Schroedinger equation is equivalent to a Dirac boundary value problem in Fock space over half a line. The notorious Quantum Measurement Problem has been reduced in [8] to the Dirac boundary value problem in Fock space for the unitary dilation of the linear stochastic filtering equation. The positivity problem for the continuous spectrum of free Hamiltonian corresponding to the quantum stochastic evolution in Fock space can be solved for the ultra-relativistic initial conditions of the incoming particles interacting with the observed quantum system at the boundary by counting the scattered outgoing particles. This has been done by the WKB method for the standard relativistic free Hamiltonian evolutions in [8], and for arbitrary Hamiltonians in [9]. The generalization of WKB method to the stochastic and quantum stochastic evolution equations was first considered in [10].
My 10 Relevant Publications:
  1. V. P. Belavkin: A New Wave Equation for a Continuous Non-Demolition Measurement. Phys Letters A 140(3) 355--358 (1989). quant-ph/0512136, PDF.
  2. V. P. Belavkin: A Stochastic Posterior Schroedinger Equation for Counting Non-Demolition Measurement. Letters in Math Phys 20 85--89 (1990).
  3. V. P. Belavkin: A Posterior Schroedinger Equation for Continuous Non-Demolition Measurement. J of Math Phys 31(12) 2930--2934 (1990).
  4. V. P. Belavkin: A Posterior Stochastic Equations for Quantum Brownian Motion. In Stochastic Methods in Experimental Sciences World Scientific, Singapore 26--42 1990.
  5. V. P. Belavkin and P. Staszewski: A Quantum Particle Undergoing Continuous Observation. Phys Letters A 140(3) 359--362 (1989). quant-ph/0512137, PDF.
  6. V. P. Belavkin and P. Staszewski: A Continuous Observation of Photon Emission. Reports in Mathematical Physics 31(29) 213--225 (1990).
  7. V. P. Belavkin & P. Staszewski: Quantum Stochastic Differential Equation for Unstable Systems. Journal Math. Phys. 41 (11) 7220-7233 (2000). math-ph/0512078, PDF.
  8. V. P. Belavkin: On Stochastic Schroedinger Equation as a Dirac Boundary-Value Problem, and an Inductive Stochastic Limit. In Evolution Equations and their Applications 311--334. Lecture Notes in Pure and Applied Mathematics, Marcel Dekker, Inc., New York 2000. math-ph/0512075, PDF.
  9. V. P. Belavkin and V. Kolokoltsov: Stochastic Evolution as a quasiclassical limit of a Boundary Value problem for Schroedinger Equations. Inf. Di. Anal. Quant. Prob. & Rel. Top. 5 (1) 61-91 (2002).
  10. V. P. Belavkin and V. Kolokoltsov: Quasiclassic Solutions of Quantum Stochastic Equations.. Theoret. Math. Phys. 11 (2) 20-33 (1991).