scholarly journals Quantum white noises—White noise approach to quantum stochastic calculus

1993 ◽  
Vol 129 ◽  
pp. 23-42 ◽  
Author(s):  
Zhiyuan Huang
Keyword(s):  
Hilbert Space ◽  
White Noise ◽  
Fock Space ◽  
Stochastic Calculus ◽  
Quantum Stochastic Calculus ◽  
Integrable Functions ◽  
Boson Fock Space ◽  
Square Integrable Functions ◽  
Complex Valued ◽  
White Noises

Let H = L2 (R) be the Hilbert space of all complex-valued square integrable functions defined on R, Ф = Γ(H) be the Boson Fock space over H. For each h ∈ H, denote by ε(h) the corresponding exponential vector:in particular ε(0) is the Fock vacuum.

scholarly journals A generalization of the Itô formula

2002 ◽  
Vol 31 (8) ◽  
pp. 477-496
Author(s):  
Said Ngobi
Keyword(s):  
Sobolev Space ◽  
White Noise ◽  
Itô Formula ◽  
Ito Formula ◽  
Integrable Functions ◽  
Square Integrable Functions ◽  
White Noise Space

The classical Itô formula is generalized to some anticipating processes. The processes we consider are in a Sobolev space which is a subset of the space of square integrable functions over a white noise space. The proof of the result uses white noise techniques.

scholarly journals Resolution of the Identity of the Classical Hardy Space by Means of Barut-Girardello Coherent States

2012 ◽  
Vol 2012 ◽  
pp. 1-12 ◽  
Author(s):  
Zouhaïr Mouayn
Keyword(s):  
Fourier Transform ◽  
Hardy Space ◽  
Real Line ◽  
Coherent States ◽  
Positive Real ◽  
Integrable Functions ◽  
Resolution Of The Identity ◽  
Square Integrable Functions ◽  
Complex Valued ◽  
Classical Hardy Space

We construct a one-parameter family of coherent states of Barut-Girdrardello type performing a resolution of the identity of the classical Hardy space of complex-valued square integrable functions on the real line, whose Fourier transform is supported by the positive real semiaxis.

scholarly journals On the Parity Under Metapletic Operators and an Extension of a Result of Lyubarskii and Nes

2019 ◽  
Vol 75 (1) ◽  
Author(s):  
Markus Faulhuber
Keyword(s):  
Hilbert Space ◽  
Real Line ◽  
Gabor Frame ◽  
Window Function ◽  
Gabor System ◽  
The Real ◽  
Metaplectic Group ◽  
Integrable Functions ◽  
Square Integrable Functions

AbstractIn this work we show that if the frame property of a Gabor frame with window in Feichtinger’s algebra and a fixed lattice only depends on the parity of the window, then the lattice can be replaced by any other lattice of the same density without losing the frame property. As a byproduct we derive a generalization of a result of Lyubarskii and Nes, who could show that any Gabor system consisting of an odd window function from Feichtinger’s algebra and any separable lattice of density $$\tfrac{n+1}{n}$$n+1n, $$n \in \mathbb {N}_+$$n∈N+, cannot be a Gabor frame for the Hilbert space of square-integrable functions on the real line. We extend this result by removing the assumption that the lattice has to be separable. This is achieved by exploiting the interplay between the symplectic and the metaplectic group.

QUANTUM STOCHASTIC CALCULUS ON BOOLEAN FOCK SPACE

2004 ◽  
Vol 07 (04) ◽  
pp. 631-650 ◽  
Author(s):  
ANIS BEN GHORBAL ◽  
MICHAEL SCHÜRMANN
Keyword(s):  
Fock Space ◽  
Field Operator ◽  
Stochastic Calculus ◽  
Product Formula ◽  
Stochastic Integration ◽  
Quantum Stochastic Calculus ◽  
Basic Field ◽  
Quantum Dynamical ◽  
Quantum Dynamical Semigroups

In this paper we establish a theory of stochastic integration with respect to the basic field operator processes in the Boolean case. This leads to a Boolean version of quantum Itô's product formula and has applications to the theory of dilations of quantum dynamical semigroups.

scholarly journals The qq-bit (III): Symmetric q-Jordan–Wigner embeddings

2019 ◽  
Vol 22 (01) ◽  
pp. 1850023
Author(s):  
Luigi Accardi ◽  
Yun-Gang Lu
Keyword(s):  
Limit Theorem ◽  
Fock Space ◽  
Random Variable ◽  
Stochastic Calculus ◽  
General Context ◽  
Quantum Stochastic Calculus ◽  
Continuous Version ◽  
Text Mode ◽  
Interacting Fock Space ◽  
Canonical Quantum

We prove that, replacing the left Jordan–Wigner [Formula: see text]-embedding by the symmetric [Formula: see text]-embedding described in Sec. 2, the result of the corresponding central limit theorem changes drastically with respect to those obtained in Ref. 5. In fact, in the former case, for any [Formula: see text], the limit space is precisely the [Formula: see text]-mode Interacting Fock Space (IFS) that realizes the canonical quantum decomposition of the limit classical random variable. In the latter case, this happens if and only if [Formula: see text]. Furthermore, as shown in Sec. 4, the limit classical random variable turns out to coincide with the [Formula: see text]-mode version of the [Formula: see text]-deformed quantum Brownian introduced by Parthasarathy[Formula: see text], and extended to the general context of bi-algebras by Schürman[Formula: see text]. The last section of the paper (Appendix) describes this continuous version in white noise language, leading to a simplification of the original proofs, based on quantum stochastic calculus.

scholarly journals On an angle between two rings

1991 ◽  
Vol 43 (1) ◽  
pp. 79-87
Author(s):  
Grzegorz Rzadkowski
Keyword(s):  
Complex Plane ◽  
Fock Space ◽  
Integrable Functions ◽  
Square Integrable Functions

Jakóbczak and Mazur found the L2-angle between two concentric rings on the complex plane. In this note we investigate the same case but for spaces of square integrable functions with various weights. Moreover the continuity of the L2-angle for the Fock space is examined.

QUANTUM STOCHASTIC ANALYSIS VIA WHITE NOISE OPERATORS IN WEIGHTED FOCK SPACE

2002 ◽  
Vol 14 (03) ◽  
pp. 241-272 ◽  
Author(s):  
DONG MYUNG CHUNG ◽  
UN CIG JI ◽  
NOBUAKI OBATA
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
White Noise ◽  
Fock Space ◽  
Existence Of A Solution ◽  
Quantum Stochastic Differential Equation ◽  
Unique Existence ◽  
Regularity Properties ◽  
Weighted Fock Space ◽  
White Noises

White noise theory allows to formulate quantum white noises explicitly as elemental quantum stochastic processes. A traditional quantum stochastic differential equation of Itô type is brought into a normal-ordered white noise differential equation driven by lower powers of quantum white noises. The class of normal-ordered white noise differential equations covers quantum stochastic differential equations with highly singular noises such as higher powers or higher order derivatives of quantum white noises, which are far beyond the traditional Itô theory. For a general normal-ordered white noise differential equation unique existence of a solution is proved in the sense of white noise distribution. Its regularity properties are investigated by means of weighted Fock spaces interpolating spaces of white noise distributions and associated characterization theorems for S-transform and for operator symbols.

Eisenstein Series for Reductive Groups over Global Function Fields I.

1982 ◽  
Vol 34 (1) ◽  
pp. 91-168 ◽  
Author(s):  
L. E. Morris
Keyword(s):  
Hilbert Space ◽  
Continuous Spectrum ◽  
Eisenstein Series ◽  
Half Plane ◽  
Discrete Subgroup ◽  
Reductive Groups ◽  
Global Function Fields ◽  
Integrable Functions ◽  
Generalized Eigenfunctions ◽  
Square Integrable Functions

Let G be the Lie group SL(2, R) and Γ a discrete subgroup of arithmetic type. The homogeneous space Γ\G can be equipped with an invariant measure so that there is a Hilbert space of square integrable functions, denoted L2(Γ\G), on which G acts by right translations. If Γ\G is compact then this Hilbert space breaks up into a countable direct sum of irreducible representations of G, each occurring with finite multiplicity. Quite often however Γ\G is not compact, but of finite volume; in this case L2(Γ\G) splits into a discrete spectrum Ld2 which behaves as if Γ\G were compact, and a continuous spectrum Lc2 which is described by the so called theory of Eisenstein series. These are generalized eigenfunctions of the Casimir operator of G, which are parametrized by a right half plane in C, and as such are analytic functions on this half-plane; in the course of describing the continuous spectrum Lc2 however, one analytically continues them to meromorphic functions over all of C, and shows them to satisfy functional equations.

EXOTIC LAPLACIANS AND ASSOCIATED STOCHASTIC PROCESSES

2009 ◽  
Vol 12 (01) ◽  
pp. 1-19 ◽  
Author(s):  
LUIGI ACCARDI ◽  
UN CIG JI ◽  
KIMIAKI SAITÔ
Keyword(s):  
Hilbert Space ◽  
Brownian Motion ◽  
Stochastic Processes ◽  
Explicit Expression ◽  
Separable Hilbert Space ◽  
Fock Space ◽  
Heat Semigroup ◽  
Tempered Distributions ◽  
Infinite Dimensional ◽  
Boson Fock Space

In this paper, we give a decomposition of the space of tempered distributions by the Cesàro norm, and for any [Formula: see text] we construct directly from the exotic trace an infinite dimensional separable Hilbert space Hc,2a-1 on which the exotic trace plays the role as the usual trace. This implies that the Exotic Laplacian coincides with the Volterra–Gross Laplacian in the Boson Fock space Γ(Hc,2a-1) over the Hilbert space Hc,2a-1. Finally we construct the Brownian motion naturally associated to the Exotic Laplacian of order 2a-1 and we find an explicit expression for the associated heat semigroup.

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