What is a quantum stochastic equation from the point of view of functional analysis? (Q1810090)

The author is concerned with a quantum stochastic differential equation in a Boson-Fock space \({\mathcal H}\otimes\Gamma(L^2({\mathbb R}))\) (\({\mathcal H}\) Hilbert space) of the form \[ dU_t = \left((W-I)d\Lambda + LdA^\dagger - L^*WdA - Gdt\right)U_t, \quad U_0=I \] where \(W,L\) and \(G=iH+(1/2)L^*L\) are operators on \({\mathcal H}\), acting on the bigger space by standard amplication with \(W\) unitary, \(H\) self-adjoint, \(I\) is the identity operator and \(\Lambda,A^\dagger, A\) are the fundamental noises of Boson-Fock quantum stochastic calculus. Under suitable conditions on \(W,L,G\) the solution is a family of unitary operators \((U_t)_{t\in{\mathbb R}}\) on \({\mathcal H}\otimes\Gamma(L^2({\mathbb R}))\). Since its introduction in the seminal paper by \textit{R. L. Hudson} and \textit{K. R. Parthasarathy} [Commun. Math. Phys. 93, No. 3, 301--323 (1984; Zbl 0546.60058)] several efforts have been done to understand the relationship of this ``quantum stochastic'' evolution equation with the usual Schrödinger equation. In this paper the author shows that, under some natural conditions, \(U_t=\text{e}^{-t\nabla}\text{e}^{-it\widehat{C}}\) where \(\widehat{C}\) is a selfadjoint operator on \({\mathcal H}\otimes\Gamma(L^2({\mathbb R}))\) as well as \(i\nabla\), the generator of the time shift. The operator \(\widehat{C}\) is essentially equal to \(i(\nabla + G^* -L^*A_{-})\) where \(A_{-}\) is a suitable creation-annihilation like operator and the domain of \(\widehat{C}\) is restricted by boundary conditions, involving also \(W\), describing the jumps in phase and amplitude of the evolution on the coordinate hyperplanes in Fock space. The relationship between conditions for the solution of the quantum stochastic differential equation to be unitary and defect indices of symmetric operators is also discussed. For related work see also [Zbl 0919.60093, Zbl 1010.60093, Zbl 1042.81055, Zbl 0017.81029 and \textit{A. M. Chebotarev}, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 1, No. 2, 175--199 (1998; Zbl 0942.60097)].