On Quantum Stochastic Differential Equations as Dirac Boundary-Value Problems

V. P. Belavkin

V. P. Belavkin

Alexander Mikhailovich Chebotarev

We show a new remarkable connection between the symmetric form of a quantum stochastic differential equation (QSDE) and the strong resolvent limit of Schrödinger equations in Fock space: the strong resolvent limit is unitary equivalent to QSDE in the adapted (or Ito) form, and the weak limit is unitary equivalent to the symmetric (or Stratonovich) form of QSDE. We prove that QSDE is unitary equivalent to a symmetric boundary value problem for the Schrödinger equation in Fock space. The boundary condition describes standard jumps of the phase and amplitude of components of Fock vectors belonging to the range of the resolvent. The corresponding Markov evolution equation (the Lindblad or Markov master equation) is derived from the boundary value problem for the Schrödinger equation.}

J. Martin Lindsay, Adam Skalski

Franco Fagnola

Alexander Mikhailovich Chebotarev

We show a new remarkable connection between the symmetric form of a quantum stochastic differential equation (QSDE) and the strong resolvent limit of the Schrödinger equations in Fock space: the strong resolvent limit is unitarily equivalent to QSDE in the adapted (or Ito) form, and the weak limit is unitarily equivalent to the symmetric (or Stratonovich) form of QSDE. We also prove that QSDE is unitarily equivalent to a symmetric boundary value problem for the Schrödinger equation in Fock space. The boundary condition describes standard jumps in phase and amplitude of components of Fock vectors belonging to the range of the resolvent. The corresponding Markov evolution equation (the Lindblad or Markov master equation) is derived from the boundary value problem for the Schrödinger equation.