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Sep 23, 2013
09/13

by
Sergio Albeverio; Alexander K. Motovilov

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Let A be a self-adjoint operator on a separable Hilbert space H. Assume that the spectrum of A consists of two disjoint components s_0 and s_1 such that the set s_0 lies in a finite gap of the set s_1. Let V be a bounded self-adjoint operator on H off-diagonal with respect to the partition spec(A)=s_0 \cup s_1. It is known that if ||V||

Source: http://arxiv.org/abs/1012.1569v3

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Sep 21, 2013
09/13

by
Sergio Albeverio; Yuri Kozitsky; Yuri Kondratiev; Michael Roeckner

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The most important recent results in the theory of phase transitions and quantum effects in quantum anharmonic crystals are presented and discussed. In particular, necessary and sufficient conditions for a phase transition to occur at some temperature are given in the form of simple inequalities involving the interaction strength and the parameters describing a single oscillator. The main characteristic feature of the theory is that both mentioned phenomena are described in one and the same...

Source: http://arxiv.org/abs/1204.6279v1

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Jun 30, 2018
06/18

by
Sergio Albeverio; Sonia Mazzucchi

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An approach to infinite dimensional integration which unifies the case of oscillatory integrals and the case of probabilistic type integrals is presented. It provides a truly infinite dimensional construction of integrals as linear functionals, as much as possible independent of the underlying topological and measure theoretical structure. Various applications are given, including, next to Schr\"odinger and diffusion equations, also higher order hyperbolic and parabolic equations.

Topics: Probability, Mathematics

Source: http://arxiv.org/abs/1411.2853

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Sep 24, 2013
09/13

by
Sergio Albeverio; Sergii Kuzhel

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Generalized PT-symmetric operators acting an a Hilbert space $\mathfrak{H}$ are defined and investigated. The case of PT-symmetric extensions of a symmetric operator $S$ is investigated in detail. The possible application of the Lax-Phillips scattering methods to the investigation of PT-symmetric operators is illustrated by considering the case of 0-perturbed operators.

Source: http://arxiv.org/abs/1202.1537v1

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Jun 28, 2018
06/18

by
Sergio Albeverio; Francesco C. De Vecchi; Stefania Ugolini

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We prove the entropy-chaos property for the system of N undistinguishable interacting diffusions rigorously associated with the ground state of N trapped Bose particles in the Gross-Pitaevskii scaling limit of infinite particles. On the path-space we show that the sequence of probability measures of the one-particle interacting diffusion weakly converges to a limit probability measure, uniquely associated with the minimizer of the Gross-Pitaevskii functional.

Topics: Probability, Mathematics, Mathematical Physics

Source: http://arxiv.org/abs/1512.04729

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Jun 29, 2018
06/18

by
Sergio Albeverio; Alexander K. Motovilov

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We consider a J-self-adjoint 2x2 block operator matrix L in the Feshbach spectral case, that is, in the case where the spectrum of one main-diagonal entry is embedded into the absolutely continuous spectrum of the other main-diagonal entry. We work with the analytic continuation of one of the Schur complements of L to the unphysical sheets of the spectral parameter plane. We present the conditions under which the continued Schur complement has operator roots, in the sense of Markus-Matsaev. The...

Topics: Spectral Theory, Mathematical Physics, Mathematics

Source: http://arxiv.org/abs/1601.06299

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Sep 23, 2013
09/13

by
Sergio Albeverio; Shao-Ming Fei

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A general way to construct ladder models with certain Lie algebraic or quantum Lie algebraic symmetries is presented. These symmetric models give rise to series of integrable systems. It is shown that corresponding to these SU(2) symmetric integrable ladder models there are exactly solvable stationary discrete-time (resp. continuous-time) Markov processes with transition matrices (resp. intensity matrices) having spectra which coincide with the ones of the corresponding integrable models.

Source: http://arxiv.org/abs/cond-mat/0109327v1

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Jun 30, 2018
06/18

by
Sergio Albeverio; Luca Di Persio; Elisa Mastrogiacomo; Boubaker Smii

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We describe a class of explicit invariant measures for both finite and infinite dimensional Stochastic Differential Equations (SDE) driven by L\'evy noise. We first discuss in details the finite dimensional case with a linear, resp. non linear, drift. In particular, we exhibit a class of such SDEs for which the invariant measures are given in explicit form, coherently in all dimensions. We then indicate how to relate them to invariant measures for SDEs on separable Hilbert spaces.

Topics: Probability, Mathematics

Source: http://arxiv.org/abs/1407.3943

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Sep 22, 2013
09/13

by
Sergio Albeverio; Claudio Cacciapuoti

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We give a representation of the classical Riemann $\zeta$-function in the half plane $\Re s>0$ in terms of a Mellin transform involving the real part of the dilogarithm function with an argument on the unit circle (associated Clausen $Gl_2$-function). We also derive corresponding representations involving the derivatives of the $Gl_2$-function. A generalized symmetrized M\"untz-type formula is also derived. For a special choice of test functions it connects to our integral...

Source: http://arxiv.org/abs/1101.4786v2

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Sep 21, 2013
09/13

by
Sergio Albeverio; Saidakhmat N. Lakaev; Zahriddin I. Muminov

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A model operator $H$ associated to a system of three-particles on the three dimensional lattice $\Z^3$ and interacting via pair non-local potentials is studied. The following results are proven: (i) the operator $H$ has infinitely many eigenvalues lying below the bottom of the essential spectrum and accumulating at this point, in the case, where both Friedrichs model operators $h_{\mu_\alpha}(0),\alpha=1,2,$ have threshold resonances. (ii) the operator $H$ has a finite number of eigenvalues...

Source: http://arxiv.org/abs/math-ph/0508029v2

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Sep 18, 2013
09/13

by
Sergio Albeverio; Alexandre Kosyak

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We construct a [(n+1)/2]+1 parameters family of irreducible representations of the Braid group B_3 in arbitrary dimension n\in N, using a q-deformation of the Pascal triangle. This construction extends in particular results by S.P.Humphries [8], who constructed representations of the braid group B_3 in arbitrary dimension using the classical Pascal triangle. E.Ferrand [7] obtained an equivalent representation of B_3 by considering two special operators in the space C^n[X]. Slightly more general...

Source: http://arxiv.org/abs/0803.2778v2

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Sep 18, 2013
09/13

by
Sergio Albeverio; Shao-Ming Fei

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A general way to construct chain models with certain Lie algebraic or quantum Lie algebraic symmetries is presented. These symmetric models give rise to series of integrable systems. As an example the chain models with $A_n$ symmetry and the related Temperley-Lieb algebraic structures and representations are discussed. It is shown that corresponding to these $A_n$ symmetric integrable chain models there are exactly solvable stationary discrete-time (resp. continuous-time) Markov chains whose...

Source: http://arxiv.org/abs/hep-th/9605130v1

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Sep 23, 2013
09/13

by
Sergio Albeverio; Elisa Mastrogiacomo; Boubaker Smii

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We study a reaction-diffusion evolution equation perturbed by a space-time L\'evy noise. The associated Kolmogorov operator is the sum of the infinitesimal generator of a $C_0$-semigroup of strictly negative type acting in a Hilbert space and a nonlinear term which has at most polynomial growth, is non necessarily Lipschitz and is such that the whole system is dissipative. The corresponding It\^o stochastic equation describes a process on a Hilbert space with dissi- pative nonlinear, non...

Source: http://arxiv.org/abs/1212.5804v1

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Sep 22, 2013
09/13

by
Sergio Albeverio; Shao-Ming Fei

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Symmetry properties of stochastic dynamical systems described by stochastic differential equation of Stratonovich type and related conserved quantities are discussed, extending previous results by Misawa. New conserved quantities are given by applying symmetry operators to known conserved quantities. Some detailed examples are presented.

Source: http://arxiv.org/abs/hep-th/9503050v1

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Sep 18, 2013
09/13

by
Sergio Albeverio; Shao-Ming Fei

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A study of symplectic forms associated with two dimensional quantum planes and the quantum sphere in a three dimensional orthogonal quantum plane is provided. The associated Hamiltonian vector fields and Poissonian algebraic relations are made explicit.

Source: http://arxiv.org/abs/math/9903016v1

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Sep 19, 2013
09/13

by
Sergio Albeverio; Shao-Ming Fei; Wen-Li Yang

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We study optimal teleportation based on the Bell measurements. An explicit expression for the quantum channel associated with the optimal teleportation with an arbitrary mixed state resource is presented. The optimal transmission fidelity of the corresponding quantum channel is calculated and shown to be related to the fully entangled fraction of the quantum resource, rather than the singlet fraction as in the standard teleportation protocol.

Source: http://arxiv.org/abs/quant-ph/0208015v2

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Sep 22, 2013
09/13

by
Sergio Albeverio; Saidakhmat N. Lakaev; Konstantin A. Makarov; Zahriddin I. Muminov

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For a wide class of two-body energy operators $h(k)$ on the three-dimensional lattice $\bbZ^3$, $k$ being the two-particle quasi-momentum, we prove that if the following two assumptions (i) and (ii) are satisfied, then for all nontrivial values $k$, $k\ne 0$, the discrete spectrum of $h(k)$ below its threshold is non-empty. The assumptions are: (i) the two-particle Hamiltonian $h(0)$ corresponding to the zero value of the quasi-momentum has either an eigenvalue or a virtual level at the bottom...

Source: http://arxiv.org/abs/math-ph/0501013v1

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4.0

Jun 30, 2018
06/18

by
Sergio Albeverio; Alexander K. Motovilov

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We overview the recent results on the shift of the spectrum and norm bounds for variation of spectral subspaces of a Hermitian operator under an additive Hermitian perturbation. Along with the known results, we present a new subspace variation bound for the generic off-diagonal subspace perturbation problem. We also demonstrate how some of the abstract results may work for few-body Hamiltonians.

Topics: Quantum Physics, Mathematics, Mathematical Physics

Source: http://arxiv.org/abs/1410.3231

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Sep 19, 2013
09/13

by
Sergio Albeverio; Shao-Ming Fei

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By investigating the symplectic geometry and geometric quantization on a class of supermanifolds, we exhibit BRST structures for a certain kind of algebras. We discuss the undeformed and q-deformed cases in the classical as well as in the quantum cases.

Source: http://arxiv.org/abs/hep-th/9405097v1

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Sep 23, 2013
09/13

by
Zong-Guo Li; Shao-Ming Fei; Sergio Albeverio; W. M. Liu

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We investigate the entanglement of assistance which quantifies capabilities of producing pure bipartite entangled states from a pure tripartite state. The lower bound and upper bound of entanglement of assistance are obtained. In the light of the upper bound, monogamy constraints are proved for arbitrary n-qubit states.

Source: http://arxiv.org/abs/0904.2809v1

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Sep 19, 2013
09/13

by
Sergio Albeverio; Laura Cattaneo; Shao-Ming Fei; Xiao-Hong Wang

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In previous work the authors introduced a notion of generic states and obtained criteria for local equivalence of them. Here they introduce the concept of CHG states maintaining the criteria of local equivalence. This fact allows the authors to halve the number of invariants necessary to characterize the equivalence classes under local unitary transformations for the set of tripartite states whose partial trace with respect to one of the subsystems belongs to the class of CHG mixed states.

Source: http://arxiv.org/abs/quant-ph/0703026v1

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Jul 20, 2013
07/13

by
Sergio Albeverio; Saidakhmat N. Lakaev; Ramiza Kh. Djumanova

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A family of Friedrichs models with rank one perturbations $h_\mu(p),$ $p \in (-\pi,\pi]^3,\mu>0$ associated to a system of two particles on the lattice $\Z^3$ is considered. The existence of a unique strictly positive eigenvalue below the bottom of the essential spectrum of $h_\mu(p)$ for all nontrivial values $p \in (-\pi,\pi]^3$ under the assumption that $h_\mu(0)$ has either a zero energy resonance (virtual level) or a threshold eigenvalue is proved. Low energy asymptotic expansion for...

Source: http://arxiv.org/abs/math/0604282v1

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Sep 22, 2013
09/13

by
Sergio Albeverio; Saidakhmat N. Lakaev; Janikul I. Abdullaev

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Two-particle discrete Schr\"{o}dinger operators $H(k)=H_{0}(k)-V$ on the three-dimensional lattice $\Z^3,$ $k$ being the two-particle quasi-momentum, are considered. An estimate for the number of the eigenvalues lying outside of the band of $H_{0}(k)$ via the number of eigenvalues of the potential operator $V$ bigger than the width of the band of $H_{0}(k)$ is obtained. The existence of non negative eigenvalues below the band of $H_{0}(k)$ is proven for nontrivial values of the...

Source: http://arxiv.org/abs/math-ph/0501036v1

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35

Sep 18, 2013
09/13

by
Sergio Albeverio; Shao-Ming Fei

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Teleportation of finite dimensional quantum states by a non-local entangled state is studied. For a generally given entangled state, an explicit equation that governs the teleportation is presented. Detailed examples and the roles played by the dimensions of the Hilbert spaces related to the sender, receiver and the auxiliary space are discussed.

Source: http://arxiv.org/abs/quant-ph/0012035v1

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Sep 18, 2013
09/13

by
Sergio Albeverio; Shao-Ming Fei

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An integral representation of the partition function for general $n$-dimensional Ising models with nearest or non-nearest neighbours interactions is given. The representation is used to derive some properties of the partition function. An exact solution is given in a particular case.

Source: http://arxiv.org/abs/cond-mat/9605112v1

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Sep 18, 2013
09/13

by
Sergio Albeverio; Aleksey Kostenko; Mark Malamud; Hagen Neidhardt

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We investigate the spectral properties of the Schr\"odinger operators in $L^2(\mathbb{R}^n)$ with a singular interaction supported by an infinite family of concentric spheres $$ \mathbf{H}_{R,\alpha}=-\Delta+\sum_{k=1}^\infty\alpha_k\delta(|x|-r_k). $$ We obtain necessary and sufficient conditions for the operator $\mathbf{H}_{R,\alpha}$ to be self-adjoint, lower-semibounded. Also we investigate the spectral types of $\mathbf{H}_{R,\alpha}$.

Source: http://arxiv.org/abs/1211.4048v1

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Sep 23, 2013
09/13

by
Sergio Albeverio; Carlo Marinelli

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The problem of reconstructing the drift of a diffusion in $\erre^d$, $d\geq 2$, from the transition probability density observed outside a domain is considered. The solution of this problem also solves a new inverse problem for a class of parabolic partial differential equations. This work considerably extends \cite{jsp} in terms of generality, both concerning assumptions on the drift coefficient, and allowing for non-constant diffusion coefficient. Sufficient conditions for solvability of this...

Source: http://arxiv.org/abs/math/0411008v1

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Jun 27, 2018
06/18

by
Sergio Albeverio; Hanno Gottschalk

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It is proven that the relativistic quantum fields obtained from analytic continuation of convoluted generalized (L\'evy type) noise fields have positive metric, if and only if the noise is Gaussian. This follows as an easy observation from a criterion by K. Baumann, based on the Dell'Antonio-Robinson-Greenberg theorem, for a relativistic quantum field in positive metric to be a free field.

Topics: Mathematical Physics, Mathematics

Source: http://arxiv.org/abs/1506.01329

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Jun 28, 2018
06/18

by
Sergio Albeverio; Bruce K. Driver; Maria Gordina; A. M. Vershik

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We consider two unitary representations of the infinite-dimensional groups of smooth paths with values in a compact Lie group. The first representation is induced by quasi-invariance of the Wiener measure, and the second representation is the energy representation. We define these representations and their basic properties, and then we prove that these representations are unitarily equivalent.

Topics: Representation Theory, Probability, Mathematics, Mathematical Physics

Source: http://arxiv.org/abs/1511.07378

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Jun 30, 2018
06/18

by
Sergio Albeverio; Zhi Ming Ma; Michael Röckner

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After recalling basic features of the theory of symmetric quasi regular Dirichlet forms we show how by applying it to the stochastic quantization equation, with Gaussian space-time noise, one obtains weak solutions in a large invariant set. Subsequently, we discuss non symmetric quasi regular Dirichlet forms and show in particular by two simple examples in infinite dimensions that infinitesimal invariance, does not imply global invariance. We also present a simple example of non-Markov...

Topics: Probability, Mathematics

Source: http://arxiv.org/abs/1404.2757

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Sep 21, 2013
09/13

by
Sergio Albeverio; Eugene Lytvynov; Andrea Mahnig

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An extension of the Heath--Jarrow--Morton model for the development of instantaneous forward interest rates with deterministic coefficients and Gaussian as well as L\'evy field noise terms is given. In the special case where the L\'evy field is absent, one recovers a model discussed by D.P.~Kennedy.

Source: http://arxiv.org/abs/math/0311144v1

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Jun 28, 2018
06/18

by
Sergio Albeverio; Iryna Pratsiovyta; Grygoriy Torbin

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We consider the second Ostrogradsky expansion from the number theory, probability theory, dynamical systems and fractal geometry points of view, and establish several new phenomena connected with this expansion. First of all we prove the singularity of the random second Ostrogradsky expansion. Secondly we study properties of the symbolic dynamical system generated by the natural one-sided shift-transformation $T$ on the second Ostrogradsky expansion. It is shown, in particular, that there are...

Topics: Mathematics, Probability

Source: http://arxiv.org/abs/1506.04353

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55

Sep 19, 2013
09/13

by
Sergio Albeverio; Olga Rozanova

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It is well known that the solutions to the non-viscous Burgers equation develop a gradient catastrophe at a critical time provided the initial data have a negative derivative in certain points. We consider this equation assuming that the particle paths in the medium are governed by a random process with a variance which depends in a polynomial way on the velocity. Given an initial distribution of the particles which is uniform in space and with the initial velocity linearly depending on the...

Source: http://arxiv.org/abs/0708.2320v3

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Sep 21, 2013
09/13

by
Sergio Albeverio; Zhi-Ming Ma

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The theory of Dirichlet forms as originated by Beurling-Deny and developed particularly by Fukushima and Silverstein, is a natural functional analytic extension of classical (and axiomatic) potential theory. Although some parts of it have abstract measure theoretic versions, the basic general construction of a Hunt process properly associated with the form, obtained by Fukushima and Silverstein, requires the form to be defined on a locally compact separable space with a Radon measure $m$ and...

Source: http://arxiv.org/abs/math/9204224v1

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Jul 20, 2013
07/13

by
Sergio Albeverio; Saidakhmat N. Lakaev; Zahriddin I. Muminov

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A family of Friedrichs models under rank one perturbations $h_{\mu}(p),$ $p \in (-\pi,\pi]^3$, $\mu>0,$ associated to a system of two particles on the three dimensional lattice $\Z^3$ is considered. We prove the existence of a unique eigenvalue below the bottom of the essential spectrum of $h_\mu(p)$ for all nontrivial values of $p$ under the assumption that $h_\mu(0)$ has either a threshold energy resonance (virtual level) or a threshold eigenvalue. The threshold energy expansion for the...

Source: http://arxiv.org/abs/math/0604277v2

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75

Sep 18, 2013
09/13

by
Sergio Albeverio; Shao-Ming Fei

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We study the highest weight and continuous tensor product representations of q-deformed Lie algebras through the mappings of a manifold into a locally compact group. As an example the highest weight representation of the q-deformed algebra $sl_q(2,\Cb)$ is calculated in detail.

Source: http://arxiv.org/abs/q-alg/9412012v1

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Sep 18, 2013
09/13

by
Sergio Albeverio; Shao-Ming Fei; Debashish Goswami

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We investigate the equivalence of quantum states under local unitary transformations. A complete set of invariants under local unitary transformations is presented for a class of mixed states. It is shown that two states in this class are locally equivalent if and only if all these invariants have equal values for them.

Source: http://arxiv.org/abs/quant-ph/0505156v1

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Sep 22, 2013
09/13

by
Kai Chen; Sergio Albeverio; Shao-Ming Fei

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We give explicit expressions for entanglement measures of Werner states in arbitrary dimensions in terms of concurrence and tangle. We show that an optimal ensemble decomposition for a joint density matrix of a Werner state can achieve the minimum average concurrence and tangle simultaneously. Furthermore, the same decomposition also attains entanglement of formation for Werner states.

Source: http://arxiv.org/abs/quant-ph/0702017v1

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Sep 20, 2013
09/13

by
Sergio Albeverio; Alexander K. Motovilov

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We introduce the concept of Stieltjes integral of an operator-valued function with respect to the spectral measure associated with a normal operator. We give sufficient conditions for the existence of this integral and find bounds on its norm. The results obtained are applied to the Sylvester and Riccati operator equations. Assuming that the entry C is a normal operator, the spectrum of the entry A is separated from the spectrum of C, and D is a bounded operator, we obtain a representation for...

Source: http://arxiv.org/abs/math/0410577v2

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Sep 20, 2013
09/13

by
Sergio Albeverio; Shao-Ming Fei

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The relations between integrable Poisson algebras with three generators and two-dimensional manifolds are investigated. Poisson algebraic maps are also discussed.

Source: http://arxiv.org/abs/hep-th/9603125v1

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Sep 17, 2013
09/13

by
Sergio Albeverio; Debashish Goswami

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We study the structure of the generator of a symmetric, conservative quantum dynamical semigroup with norm-bounded generator on a von Neumann algebra equipped with a faithful semifinite trace. For von Neumann algebras with abelian commutant (i.e. type I von Neumann algebras), we give a necessary and sufficient algebraic condition for the generator of such a semigroup to be written as a sum of square of self-adjoint derivations of the von Neumann algebra. This generalizes some of the results...

Source: http://arxiv.org/abs/math-ph/0207047v1

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Sep 21, 2013
09/13

by
Sergio Albeverio; Matthias Gundlach; Andrei Khrennikov; Karl-Olof Lindahl

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We study Markovian and non-Markovian behaviour of stochastic processes generated by $p$-adic random dynamical systems. Given a family of $p$-adic monomial random mappings generating a random dynamical system. Under which conditions do the orbits under such a random dynamical system form Markov chains? It is necessary that the mappings are Markov dependent. We show, however, that this is in general not sufficient. In fact, in many cases we have to require that the mappings are independent....

Source: http://arxiv.org/abs/nlin/0402043v1

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Jul 20, 2013
07/13

by
Sergio Albeverio; Olga Rozanova

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We consider the Langevin equation describing a stochastically perturbed by uniform noise non-viscous Burgers fluid and introduce a deterministic function that corresponds to the mean of the velocity when we keep the value of position fixed. We study interrelations between this function and the solution of the non-perturbed Burgers equation. Especially we are interested in the property of the solution of the latter equation to develop unbounded gradients within a finite time. We study the...

Source: http://arxiv.org/abs/0804.1553v3

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Sep 24, 2013
09/13

by
Sergio Albeverio; Alexander K. Motovilov

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Let A be a self-adjoint operator on a Hilbert space H. Assume that {\sigma} is an isolated component of the spectrum of A, i.e. dist({\sigma},{\Sigma})=d>0 where {\Sigma}=spec(A)\{\sigma}. Suppose that V is a bounded self-adjoint operator on H such that ||V||R^+, that is essentially stronger than the previously known estimates for ||P-Q||. In particular, the bound obtained ensures that ||P-Q||

Source: http://arxiv.org/abs/1112.0149v3

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Sep 22, 2013
09/13

by
Sergio Albeverio; Volodymyr Koshmanenko; Mykola Pratsiovytyi; Grygoriy Torbin

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A $\widetilde{Q}-$representation of real numbers is introduced as a generalization of the $p-$adic and $Q-$representations. It is shown that the $\widetilde{Q}-$representation may be used as a convenient tool for the construction and study of fractals and sets with complicated local structure. Distributions of random variables $\xi$ with independent $\widetilde{Q}-$symbols are studied in details. Necessary and sufficient conditions for the probability measures $\mu_\xi $ associated with $\xi$...

Source: http://arxiv.org/abs/math/0308007v2

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Sep 23, 2013
09/13

by
Sergio Albeverio; Mark Malamud; Vadim Mogilevskii

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We study general (not necessarily Hamiltonian) first-order symmetric systems $J y'(t)-B(t)y(t)=\D(t) f(t)$ on an interval $\cI=[a,b> $ with the regular endpoint $a$. It is assumed that the deficiency indices $n_\pm(\Tmi)$ of the minimal relation $\Tmi$ in $\LI$ satisfy $n_-(\Tmi)\leq n_+(\Tmi)$. By using a Nevanlinna boundary parameter $\tau=\tau(\l)$ at the singular endpoint $b$ we define self-adjoint and $\l$-depending Nevanlinna boundary conditions which are analogs of separated...

Source: http://arxiv.org/abs/1303.6153v1

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Jun 28, 2018
06/18

by
Sergio Albeverio; Yuri Kondratiev; Roman Nikiforov; Grygoriy Torbin

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We establish several new fractal and number theoretical phenomena connected with expansions which are generated by infinite linear iterated function systems. First of all we show that the systems $\Phi$ of cylinders of generalized L\"uroth expansions are, generally speaking, not faithful for the Hausdorff dimension calculation. Using Yuval Peres' approach, we prove sufficient conditions for the non-faithfulness of such families of cylinders. On the other hand, rather general sufficient...

Topics: Metric Geometry, Functional Analysis, Number Theory, Mathematics, Probability

Source: http://arxiv.org/abs/1507.05672

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Sep 19, 2013
09/13

by
Sergio Albeverio; Saidakhmat N. Lakaev; Axmad M. Xalxo'jaev

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We consider the Hamiltonian of a system of three quantum mechanical particles on the three-dimensional lattice $\Z^3$ interacting via short-range pair potentials. We prove for the two-particle energy operator $h(k),$ $k\in \T^3$ the two-particle quasi-momentum, the existence of a unique positive eigenvalue $z(k)$ lying below the essential spectrum under assumption that the operator $h(0)$ corresponding to the zero value of $k$ has a zero energy resonance. We describe the location of the...

Source: http://arxiv.org/abs/math/0703301v1

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Sep 22, 2013
09/13

by
Sergio Albeverio; Shao-Ming Fei; Preeti Parashar; Wen-Li Yang

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The nonlocal properties for a kind of generic N-dimensional bipartite quantum systems are investigated. A complete set of invariants under local unitary transformations is presented. It is shown that two generic density matrices are locally equivalent if and only if all these invariants have equal values in these density matrices.

Source: http://arxiv.org/abs/quant-ph/0307164v1

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Sep 18, 2013
09/13

by
Sergio Albeverio; Claudio Cacciapuoti; Domenico Finco

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We analyze the problem of approximating a smooth quantum waveguide with a quantum graph. We consider a planar curve with compactly supported curvature and a strip of constant width around the curve. We rescale the curvature and the width in such a way that the strip can be approximated by a singular limit curve, consisting of one vertex and two infinite, straight edges, i.e. a broken line. We discuss the convergence of the Laplacian, with Dirichlet boundary conditions on the strip, in a...

Source: http://arxiv.org/abs/math-ph/0611059v2