The Quantum Stochastic Differential Equation Is Unitarily Equivalent to a Symmetric Boundary Value Problem for the Schr\"odinger Equation

Abstract

We prove that the solution of the Hudson-Parthasarathy quantum stochastic differential equation in the Fock space coincides with the solution of a symmetric boundary value problem for the Schr\"odinger equation in the interaction representation generated by the energy operator of the environment. The boundary conditions describe the jumps in the phase and the amplitude of the Fourier transforms of the Fock vector components as any of its arguments changes the sign. The corresponding Markov evolution equation (the Lindblad equation or the ``master equation'') is derived from the boundary value problem for the Schr\"odinger equation.