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What Is a Quantum Stochastic Differential Equation from the Point of View of Functional Analysis?

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Abstract

We prove that a quantum stochastic differential equation is the interaction representation of the Cauchy problem for the Schrödinger equation with Hamiltonian given by a certain operator restricted by a boundary condition. If the deficiency index of the boundary-value problem is trivial, then the corresponding quantum stochastic differential equation has a unique unitary solution. Therefore, by the deficiency index of a quantum stochastic differential equation we mean the deficiency index of the related symmetric boundary-value problem.

In this paper, conditions sufficient for the essential self-adjointness of the symmetric boundary-value problem are obtained. These conditions are closely related to nonexplosion conditions for the pair of master Markov equations that we canonically assign to the quantum stochastic differential equation.

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  1. M. V. Lomonosov Moscow State University, Russia

    A. M. Chebotarev

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Chebotarev, A.M. What Is a Quantum Stochastic Differential Equation from the Point of View of Functional Analysis?. Mathematical Notes 71, 408–427 (2002). https://doi.org/10.1023/A:1014994726667

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