Singular Quadratic Forms in Perturbation Theory

Koshmanenko, V. D.

doi:10.1007/978-94-011-4619-7

Cited by 50 publications

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“…(where we used notations introduced in (3)) which solves the Cauchy problem for equations (25), (26). Moreover,…”

Section: Proposition 6 (I) the Operator (28) Generates A Continuous Smentioning

confidence: 99%

“…Therefore, the solutions to these stochastic equations can thus be obtained as the interaction representation of the solutions of problem (25), (26) with respect to the "free" shift T n . In Section 7 it is shown that the model (25), (26) in its turn can be obtained as a semiclassical limit of Schrödinger evolutions with a bounded below Hamiltonian. Moreover, as linear stochastic equations driven by Wiener process can be obtained as a limit of equations driven by Poisson process (see Appendix A for a concrete case of quantum filtering equation), the solutions to these equations can be obtained as a limit of the solutions of the boundary value problems considered above.…”

Section: Theorem 2 Solution (24) To the Cauchy Problem Of Equation (2mentioning

confidence: 99%

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Stochastic Evolution as a Quasiclassical Limit of a Boundary Value Problem for Schrödinger Equations

Belavkin

Kolokoltsov

2002

Infin. Dimens. Anal. Quantum. Probab. Relat. Top.

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We develop systematically a new unifying approach to the analysis of linear stochastic, quantum stochastic and even deterministic equations in Banach spaces. Solutions to a wide class of these equations (in particular those describing the processes of continuous quantum measurements) are proved to coincide with the interaction representations of the solutions to certain Dirac type equations with boundary conditions in pseudo-Fock spaces. The latter are presented as the semiclassical limit of an appropriately dressed unitary evolutions corresponding to a boundary-value problem for rather general Schrödinger equations with bounded below Hamiltonians.

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“…(where we used notations introduced in (3)) which solves the Cauchy problem for equations (25), (26). Moreover,…”

Section: Proposition 6 (I) the Operator (28) Generates A Continuous Smentioning

confidence: 99%

Section: Theorem 2 Solution (24) To the Cauchy Problem Of Equation (2mentioning

confidence: 99%

Stochastic Evolution as a Quasiclassical Limit of a Boundary Value Problem for Schrödinger Equations

Belavkin

Kolokoltsov

2002

Infin. Dimens. Anal. Quantum. Probab. Relat. Top.

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“…The theory of finite rank perturbations of self-adjoint operators attracted much attention of physicists and mathematicians, in particular it leads to exactly solvable models [1,4,5,17,18,19,20,21,22,29]. This theory has been presented in several review papers and summarized in research monographs [3,9,14,22,31,32,35].…”

Section: Introductionmentioning

confidence: 99%

“…This theory has been presented in several review papers and summarized in research monographs [3,9,14,22,31,32,35]. The aim of the current paper is to clarify a few additional properties of such perturbations, namely the possibility to construct approximations by operators with regular perturbations.…”

Section: Introductionmentioning

confidence: 99%

On approximations of rank one ℋ₋₂-perturbations

Albeverio

Koshmanenko

Kurasov

et al. 2002

Proc. Amer. Math. Soc.

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Abstract. Approximations of rank one H −2 -perturbations of self-adjoint operators by operators with regular rank one perturbations are discussed. It is proven that in the case of arbitrary not semibounded operators such approximations in the norm resolvent sense can be constructed without any renormalization of the coupling constant. Approximations of semibounded operators are constructed using rank one non-symmetric regular perturbations.

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“…The books [2,4,37] and the papers [6,7,11,26,30,39,42,44,46,48] provide a detailed discussion and an extensive list of references on this subject. In the present work we will address the problem, that emerges naturally in this context, of constructing a boundary potential formulation of the operators L that is entirely defined in terms of standard Schwartz distributions.…”

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confidence: 99%

One-dimensional Schrödinger operators with singular potentials: A Schwartz distributional formulation

Dias

Jorge

Prata

2016

Journal of Differential Equations

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Abstract. Using an extension of the Hörmander product of distributions, we obtain an intrinsic formulation of one-dimensional Schrödinger operators with singular potentials. This formulation is entirely defined in terms of standard Schwartz distributions and does not require (as some previous approaches) the use of more general distributions or generalized functions. We determine, in the new formulation, the action and domain of the Schrödinger operators with arbitrary singular boundary potentials. We also consider the inverse problem, and obtain a general procedure for constructing the singular (pseudo) potential that imposes a specific set of (local) boundary conditions. This procedure is used to determine the boundary operators for the complete four-parameter family of one-dimensional Schrödinger operators with a point interaction. Finally, the δ and δ ′ potentials are studied in detail, and the corresponding Schrödinger operators are shown to coincide with the norm resolvent limit of specific sequences of Schrödinger operators with regular potentials.

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Singular Quadratic Forms in Perturbation Theory

Cited by 50 publications

References 0 publications

Stochastic Evolution as a Quasiclassical Limit of a Boundary Value Problem for Schrödinger Equations

Stochastic Evolution as a Quasiclassical Limit of a Boundary Value Problem for Schrödinger Equations

On approximations of rank one ℋ₋₂-perturbations

One-dimensional Schrödinger operators with singular potentials: A Schwartz distributional formulation

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