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

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.     

Local solvability and a priori estimates for classical solutions to an equation of Benjamin-Bona-Mahony-Bürgers type

We establish the local (in time) solvability in the classical sense for the Cauchy problem and first and second boundary-value problems on the half-line for a nonlinear equation similar to Benjamin-Bona-Mahony-Bürgers-type equation. We also derive an a priori estimate that implies sufficient blow-up conditions for the second boundary-value problem. We obtain analytically an upper bound of the blow-up time and refine it numerically using Richardson effective accuracy order technique.     

Well-posed boundary-value problems,right hyperbolicity,and exponential dichotomy

A relationship between the existence of well-posed boundary-value problems and exponential dichotomy for functional equations and linear differential-operator equations on a half-line is considered. It is shown that well-posed boundary-value problems can exist for equations without the exponential dichotomy property.     

Stochastic integral representation theorem for quantum semimartingales

The quantum stochastic integral of Itô type formulated by Hudson and Parthasarathy is extended to a wider class of adapted quantum stochastic processes on Boson Fock space. An Itô formula is established and a quantum stochastic integral representation theorem is proved for a class of unbounded semimartingales which includes polynomials and (Wick) exponentials of the basic martingales in quantum stochastic calculus.     

Transformation of a singular boundary-value problem on a half-line to an integral equation with a discontinuous operator

An investigation of a boundary-value problem on a half-line for a nonlinear ordinary second order differential equation whose free term has a discontinuity in a strip. A method is proposed for the transformation of the boundary-value problem into an integral equation with a discontinuous operator. Some results have recently been obtained concerning the existence, the comparison, and integral representations of solutions of this integral equation.Translated from Matematicheskie Zematki, Vol. 9, No. 1, pp. 77–82, January, 1971.     

Global solvability of massless Dirac–Maxwell systems

We consider the Cauchy problem for massless Dirac–Maxwell equations on an asymptotically flat background and give a global existence and uniqueness theorem for initial values small in an appropriate weighted Sobolev space. The result can be extended via analogous methods to Dirac–Higgs–Yang–Mills theories.     

Validity of the NLS approximation for periodic quantum graphs

We consider a nonlinear Schrödinger (NLS) equation on a spatially extended periodic quantum graph. With a multiple scaling expansion, an effective amplitude equation can be derived in order to describe slow modulations in time and space of an oscillating wave packet. Using Bloch wave analysis and Gronwall’s inequality, we estimate the distance between the macroscopic approximation which is obtained via the amplitude equation and true solutions of the NLS equation on the periodic quantum graph. Moreover, we prove an approximation result for the amplitude equations which occur at the Dirac points of the system.     

Solution method for the inverse boundary-value problem for the wave equation

Existence and uniqueness issues are considered for the inverse boundary-value problem of determining the variable coefficient of the one-dimensional wave equation on a half-line. A Tikhonov-regularizing algorithm is constructed to find an approximate solution using input data that are specified with an error.Translated from Metody Matematicheskogo Modelirovaniya, Avtomatizatsiya Obrabotki Nablyudenii i Ikh Primeneniya, pp. 80–88, 1986.     

Resonances for the Dirac Operator on the Half-Line

Functional Analysis and Its Applications - We consider the inverse problem for a massless Dirac operator on the half-line such that the support of its potential has fixed upper boundary and solve...     

Two-stage stochastic lot-sizing problem under cost uncertainty

For production planning problems, cost parameters can be uncertain due to marketing activities and interest rate fluctuation. In this paper, we consider a single-item two-stage stochastic lot-sizing problem under cost parameter uncertainty. Assuming cost parameters will increase or decrease after time period p each with certain probability, we minimize the total expected cost for a finite horizon problem. We develop an extended linear programming formulation in a higher dimensional space that can provide integral solutions by showing that its constraint matrix is totally unimodular. We also project this extended formulation to a lower dimensional space and obtain a corresponding extended formulation in the lower dimensional space. Final computational experiments demonstrate that the extended formulation is more efficient and performs more stable than the two-stage stochastic mixed-integer programming formulation.     

On bounded solutions of a nonlinear Schrödinger equation on the half-line

We consider the problem on nonzero solutions of the Schrödinger equation on the half-line with potential that implicitly depends on the wave function via a nonlinear ordinary differential equation of the second order under zero boundary conditions for the wave function and the condition that the potential is zero at the beginning of the interval and its derivative is zero at infinity. The problem is reduced to the analysis and investigation of solutions of the Cauchy problem for a system of two nonlinear second-order ordinary differential equations with initial conditions depending on two parameters. We show that if the solution of the Cauchy problem for some parameter values can be extended to the entire half-line, then there exists a nonzero solution of the original problem with finitely many zeros.     

On the Skorokhod mapping for equations with reflection and possible jump-like exit from a boundary

For a solution of a reflection problem on a half-line similar to the Skorokhod reflection problem but with possible jump-like exit from zero, we obtain an explicit formula and study its properties. We also construct a Wiener process on a half-line with Wentzell boundary condition as a strong solution of a certain stochastic differential equation.     

On the Strong Resolvent Convergence of the Schrödinger Evolution to Quantum Stochastics

For a class of Hamiltonians including a model of the quantum detector of gravitational waves, we prove the strong convergence of the Schrödinger evolution to quantum stochastics. We show that the strong resolvent limit of a sequence of self-adjoint Hamiltonians is a symmetric boundary-value problem in Fock space, and the limit evolution of the partial trace with respect to the mixed state cannot be described by a unique equation of Lindblad type. On the contrary, each component of the mixed state generates a proper evolution law.     

Optimal multiple quantum statistical hypothesis testing

This paper is concerned with the problem of optimal M-alternative determination of quantum statistical states. A review of newest achievement of solving this problem is given. A notion of an effective decision Hilbert space is introduced and necessary and sufficient condkions for optimality of multiple quantum hypothesis testing in this space are formulated. The general solution is found for the case of a two-dimensional decision space. Another problem solved is that of discrimination of quantum pure non-orthogonal states. The result is represented in explicit analytical form for an "equidiagonal" case, which is quite general. In particular, we find explicit solutions of optimal discrimination problem of homogeneous and equiangle sets of pure states. These results are used for the M-ary detection problem in solving for the quantum coherent non-orthogonal signals. It is proved that the simplex signals are optimal elso in quantum case. The optimal estimatesof phaseandamplitude of quantum coherent signals are found. For decision operators a notion of IT-representation is introduced to get a general quasi-classical (optimal in quasi-classical limit) M-ary detection procedure of stochastic fields and particles, which submits to Bose-Einstein statistics. An optimal solution of problem of non-coherent detection of quantum stochastic (including optical) signals are found in the extreme quantum limit (weaknoise and signals with unknown phase).     

Sesquilinear quantum stochastic analysis in Banach space

A theory of quantum stochastic processes in Banach space is initiated. The processes considered here consist of Banach space valued sesquilinear maps. We establish an existence and uniqueness theorem for quantum stochastic differential equations in Banach modules, show that solutions in unital Banach algebras yield stochastic cocycles, give sufficient conditions for a stochastic cocycle to satisfy such an equation, and prove a stochastic Lie–Trotter product formula. The theory is used to extend, unify and refine standard quantum stochastic analysis through different choices of Banach space, of which there are three paradigm classes: spaces of bounded Hilbert space operators, operator mapping spaces and duals of operator space coalgebras. Our results provide the basis for a general theory of quantum stochastic processes in operator spaces, of which Lévy processes on compact quantum groups is a special case.     

Star Product for Second-Class Constraint Systems from a BRST Theory

We propose an explicit construction of the deformation quantization of a general second-class constraint system that is covariant with respect to local coordinates on the phase space. The approach is based on constructing the effective first-class constraint (gauge) system equivalent to the original second-class constraint system and can also be understood as a far-reaching generalization of the Fedosov quantization. The effective gauge system is quantized by the BFV–BRST procedure. The star product for the Dirac bracket is explicitly constructed as the quantum multiplication of BRST observables. We introduce and explicitly construct a Dirac bracket counterpart of the symplectic connection, called the Dirac connection. We identify a particular star product associated with the Dirac connection for which the constraints are in the center of the respective star-commutator algebra. It is shown that when reduced to the constraint surface, this star product is a Fedosov star product on the constraint surface considered as a symplectic manifold.     

The Cauchy problem in Sobolev spaces for Dirac operators

In this paper we consider the Cauchy problem as a typical example of ill-posed boundary-value problems. We obtain the necessary and (separately) sufficient conditions for the solvability of the Cauchy problem for a Dirac operator A in Sobolev spaces in a bounded domain D ? ? ⁿ with a piecewise smooth boundary. Namely, we reduce the Cauchy problem for the Dirac operator to the problem of harmonic extension from a smaller domain to a larger one. Moreover, along with the solvability conditions for the problem, using bases with double orthogonality, we construct a Carleman formula for recovering a function u in a Sobolev space H ^s (D), s ∈ ?, from its values on Γ and values Au in D, where Γ is an open connected subset of the boundary ?D. It is worth pointing out that we impose no assumptions about geometric properties of the domain D, except for its connectedness.     

Anomalous waves as an object of statistical topography: Problem statement

Based on ideas of statistical topography, we analyze the boundary-value problem of the appearance of anomalous large waves (rogue waves) on the sea surface. The boundary condition for the sea surface is regarded as a closed stochastic quasilinear equation in the kinematic approximation. We obtain the stochastic Liouville equation, which underlies the derivation of an equation describing the joint probability density of fields of sea surface displacement and its gradient. We formulate the statistical problem with the stochastic topographic inhomogeneities of the sea bottom taken into account. It describes diffusion in the phase space, and its solution must answer the question whether information about the existence of anomalous large waves is contained in the quasilinear equation under consideration.     

Synthesis of averaging methods in stochastic boundary-value problems of the mechanics of microheterogeneous solids

The method of periodic components is further developed, which allows us, on unified positions, to predict the effective characteristics and structural strain fields in partially or fully disordered composites. The stochastic boundary-value problem of elasticity theory for microheterogeneous solids with a statistically homogeneous structure is treated. The heterogeneous solid is considered to be macroscopically homogeneous and macroanisotropic (or quasi-isotropic) with geometric form and properties of the structural components determined and given. The structural elements are assumed to have perfect interfaces, i.e., the displacements and tractions are continuous across the interface. The boundary-value problem was solved by the method of local approximation. Numerical results were obtained for composites with a stochastic structure.Perm' State Technical University, Russia. Translated from Mekhanika Kompozitnykh Materialov, Vol. 35, No. 1, pp. 3–12, January–February, 1999.