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1

On the eigenvalues of a 2 x 2 block operator matrix

Muminov M. I., Rasulov T. H.

Opuscula Mathematica

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2015

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Vol. 35, no. 3

371--395

EN

A 2 x 2 block operator matrix H acting in the direct sum of one- and two-particle subspaces of a Fock space is considered. The existence of infinitely many negative eigenvalues of H22 (the second diagonal entry of H) is proved for the case where both of the associated Friedrichs models have a zero energy resonance. For the number N(z) of eigenvalues of H22 lying below z < 0, the following asymptotics is found [formula]. Under some natural conditions the infiniteness of the number of eigenvalues located respectively inside, in the gap, and below the bottom of the essential spectrum of H is proved.

2

Quantum Laplacians on generalized operators on boson fock space

Accardi L., Barhoumi A., Ji U. C.

Probability and Mathematical Statistics

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2011

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Vol. 31, Fasc. 2

203--225

EN

By adapting the white noise theory, the quantum analogues of the (classical) Gross Laplacian and Lévy Laplacian, so called the quantum Gross Laplacian and quantum Lévy Laplacian, respectively, are introduced as the Laplacians acting on the spaces of generalized operators. Then the integral representations of the quantum Laplacians in terms of quantum white noise derivatives are studied. Correspondences of the classical Laplacians and quantum Laplacians are studied. The solutions of heat equations associated with the quantum Laplacians are obtained from a normal-ordered white noise differential equation.

3

Matrices related to some Fock space operators

Rudol K.

Opuscula Mathematica

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2011

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Vol. 31, no. 2

289-296

EN

Matrices of operators with respect to frames are sometimes more natural and easier to compute than the ones related to bases. The present work investigates such operators on the Segal-Bargmann space, known also as the Fock space. We consider in particular some properties of matrices related to Toeplitz and Hankel operators. The underlying frame is provided by normalised reproducing kernel functions at some lattice points.

4

Optimal holomorphic hypercontractivity for CAR algebras

Królak I.

Bulletin of the Polish Academy of Sciences. Mathematics

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2010

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Vol. 58, no 1

79-90

EN

We present a new proof of Janson's strong hypercontractivity inequality for the Ornstein-Uhlenbeck semigroup in holomorphic algebras associated with CAR (canonical anticommutation relations) algebras. In the one generator case we calculate optimal bounds for t such that Ut is a contraction as a map L2(H) → Lp(H) for arbitrary p ≥ 2. We also prove a logarithmic Sobolev inequality.

5

A Levy-Ciesielski expansion for quantum Brownian motion and the construction of quantum Brownian bridges

Applebaum D.

Journal of Applied Analysis

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2007

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Vol. 13, nr 2

275-290

EN

We introduce "probabilistic" and "stochastic Hilbertian structures". These seem to be a suitable context for developing a theory of "quantum Gaussian processes". The Schauder system is utilised to give a Levy-Ciesielski representation of quantum (bosonic) Brownian motion as operators in Fock space over a space of square summable sequences. Similar results hold for non-Fock, fermion, free and monotone Brownian motions. Quantum Brownian bridges are defined and a number of representations of these are given.