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- Albeverio, Sergio, et al.
(författare)
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Estimates uniform in time for the transition probability ofdiusions with small drift and for stochastically perturbed Newton equations
- 1999
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Ingår i: Journal of theoretical probability. - 0894-9840 .- 1572-9230. ; 12:2, s. 293-300
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Tidskriftsartikel (refereegranskat)abstract
- An estimate uniform in time for the transition probability of di®usion processes with small drift is given. This also covers the case of a degenerate di®usion describing a stochastic perturbation of a particle moving according to the Newton's law. Moreover the random wave operator for such a particle is described and the analogue of asymptotic completeness is proven, the latter in the case of a su±ciently small drift.
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- Albeverio, Sergio, et al.
(författare)
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Gauge fields, point interactions and few-body problems in one dimension
- 2004
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Ingår i: Reports on Mathematical Physics. - 0034-4877. ; 53:3, s. 363-370
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Tidskriftsartikel (refereegranskat)abstract
- Point interactions for the second derivative operator in one dimension are studied. Every operator from this family is described by the boundary conditions which include a 2 x 2 real matrix with the unit determinant and a phase. The role of the phase parameter leading to unitarily equivalent operators is discussed in the present paper. In particular, it is shown that the phase parameter is not redundant (contrary to previous studios) if nonstationary problems are concerned. It is proven that the phase parameter can be interpreted as the amplitude of a singular gauge field. Considering the few-body problem we extend the range of parameters for which the exact solution can be found using the Bethe Ansatz.
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- Albeverio, Sergio, et al.
(författare)
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On *-representations of a deformation of a Wick analogue of the CAR algebra
- 2005
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Ingår i: Reports on Mathematical Physics. - 0034-4877. ; 56:2, s. 175-196-196
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Tidskriftsartikel (refereegranskat)abstract
- The classification of irreducible *-representations of deformations of canonical anti-commutation relations is given. A description of the corresponding enveloping C*-algebra is presented in terms of continuous matrix-functions satisfying certain boundary conditions.
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- Albeverio, Sergio, et al.
(författare)
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p-Adic valued quantization
- 2009
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Ingår i: P-Adic Numbers, Ultrametric Analysis, and Applications. - Berlin : Springer. - 2070-0466 .- 2070-0474. ; 1:2, s. 91-104
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Forskningsöversikt (refereegranskat)abstract
- This review covers an important domain of p-adic mathematical physics — quantum mechanics with p-adic valued wave functions. We start with basic mathematical constructions of this quantum model: Hilbert spaces over quadratic extensions of the field of p-adic numbers ℚ p , operators — symmetric, unitary, isometric, one-parameter groups of unitary isometric operators, the p-adic version of Schrödinger’s quantization, representation of canonical commutation relations in Heisenberg andWeyl forms, spectral properties of the operator of p-adic coordinate.We also present postulates of p-adic valued quantization. Here observables as well as probabilities take values in ℚ p . A physical interpretation of p-adic quantities is provided through approximation by rational numbers.
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- Albeverio, Sergio, et al.
(författare)
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Singular Perturbations of Differential Operators
- 2000
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Bok (övrigt vetenskapligt/konstnärligt)abstract
- Singular perturbations of Schrödinger type operators are of interest in mathematics, e.g. to study spectral phenomena, and in applications of mathematics in various sciences, e.g. in physics, chemistry, biology, and in technology. They also often lead to models in quantum theory which are solvable in the sense that the spectral characteristics (eigenvalues, eigenfunctions, and scattering matrix) can be computed. Such models then allow us to grasp the essential features of interesting and complicated phenomena and serve as an orientation in handling more realistic situations. In the last ten years two books have appeared on solvable models in quantum theory built using special singular perturbations of Schrödinger operators. The book by S. Albeverio, F. Gesztesy, R. Hoegh-Krohn and H. Holden "Solvable Models in Quantum Mechanics" describes the models in rigorous mathematical terms. It gives a detailed analysis of perturbations of the Laplacian in R^d, d=1,2,3, by potentials with support on a discrete finite or infinite set of point sources (chosen in a deterministic, respectively, stochastic manner). Physically these operators describe the motion of a quantum mechanical particle moving under the action of a potential supported, e.g., by the points of a crystal lattice or a random solid. Such systems and models are also described in physical terms in the book by Yu.N.Demkov and V.N.Ostrovsky "Zero-range Potentials in Atomic Physics", which also contains a description of applications in other areas such as in optics and electromagnetism. Let us also remark that a translation of the book by S.Albeverio, F.Gesztesy, R.Hoegh-Krohn and H.Holden in Russian has been published with additional comments and literature. Since the appearance of these books several important new developments have taken place. It is the main aim of the present book to present some of these new developments in a unified formalism which also puts some of the basic results of the preceding books into a new light. The new developments concern in particular a systematic study of finite rank perturbations of (self--adjoint) operators (in particular differential operators), of generalized (singular) perturbations, of the corresponding scattering theory as well as infinite rank perturbations and multiple particles (many--body) problems in quantum theory. We also present the theory of point interaction Hamiltonians, as a particular case of a general theory of singular perturbations of differential operators. This theory has received steadily increasing attention over the years also for its many applications in physics (solid state physics, nuclear physics), electromagnetism (antennas), and technology (metallurgy, nanophysics). We hope this monograph can serve as a basis for orientation in a rapidly developing area of analysis, mathematical physics and their applications.
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