| 1. |
- Albeverino, S., et al.
(författare)
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Pseudo-Differential Operators with Point Interactions
- 1997
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Ingår i: Letters in Mathematical Physics. - 0377-9017 .- 1573-0530. ; 41, s. 79-92
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Tidskriftsartikel (refereegranskat)abstract
- Point interactions for pseudo-differential operators are studied. Necessary and sufficient conditions for a pseudo-differential operator to have nontrivial point perturbations are given. The results are applied to the construction of relativistic spin zero Hamiltonians with point interactions.
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| 2. |
- Albeverio, S., et al.
(författare)
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Finite rank perturbations and distribution theory
- 1999
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Ingår i: Proceedings of the American Mathematical Society. - 0002-9939 .- 1088-6826. ; 127:4, s. 1151-1161
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Tidskriftsartikel (refereegranskat)abstract
- Perturbations AT of a selfadjoint operator A by symmetric finite rank operators T from H2A) to H-2(A) are studied. The finite dimensional family of selfadjoint extensions determined by AT is given explicitly.
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| 3. |
- Albeverio, S., et al.
(författare)
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Many body problems with "spin"-related contact interactions
- 2001
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Ingår i: Reports on mathematical physics. - 0034-4877 .- 1879-0674. ; 47:2, s. 157-166
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Tidskriftsartikel (refereegranskat)abstract
- We study quantum mechanical systems with "spin"-related contact interactions in one dimension. The boundary conditions describing the contact interactions are dependent on the spin states of the particles. In particular we investigate the integrability of N-body systems with δ-interactions and point spin couplings. Bethe ansatz solutions, bound states and scattering matrices are explicitly given. The cases of generalized separated boundary condition and some Hamiltonian operators corresponding to special spin related boundary conditions are also discussed.
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| 4. |
- Albeverio, S., et al.
(författare)
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Rank one perturbations of not semibounded operators
- 1997
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Ingår i: Integral equations and operator theory. - 0378-620X .- 1420-8989. ; 27:4, s. 379-400
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Tidskriftsartikel (refereegranskat)abstract
- Rank one perturbations of selfadjoint operators which are not necessarily semibounded are studied in the present paper. It is proven that such perturbations are uniquely defined, if they are bounded in the sense of forms. We also show that form unbounded rank one perturbations can be uniquely defined if the original operator and the perturbation are homogeneous with respect to a certain one parameter semigroup. The perturbed operator is defined using the extension theory for symmetric operators. The resolvent of the perturbed operator is calculated using Krein's formula. It is proven that every rank one perturbation can be approximated in the operator norm. We prove that some form unbounded perturbations can be approximated in the strong resolvent sense without renormalization of the coupling constant only if the original operator is not semibounded. The present approach is applied to study first derivative and Dirac operators with point interaction, in one dimension
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| 5. |
- Albeverio, S., et al.
(författare)
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Symmetries of Schrödinger operator with point interactions
- 1998
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Ingår i: Letters in Mathematical Physics. - 0377-9017 .- 1573-0530. ; 45, s. 33-47
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Tidskriftsartikel (refereegranskat)abstract
- The transformations of all the Schrödinger operators with point interactions in dimension one under space reflection P, time reversal T and (Weyl) scaling Wλ are presented. In particular, those operators which are invariant (possibly up to a scale) are selected. Some recent papers on related topics are commented upon
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| 6. |
- Kurasov, Pavel
(författare)
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Distribution theory for discontinuous test functions and differential operators with generalized coefficients
- 1996
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Ingår i: Banach Journal of Mathematical Analysis. - : Elsevier BV. - 1735-8787 .- 0022-247X. ; 201:1, s. 297-323
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Tidskriftsartikel (refereegranskat)abstract
- Investigation of the differential operators with the generalized coefficients having singular support on a disjoint set of points requires the consideration of the distribution theory with the set of discontinuous test functions. Such a distribution theory for test functions having discontinuity at one point is developed. A four-parameter family of Schrödinger operators, formed by the operators with singular potential, singular metrics and singular gauge field, is considered. It is proved that this family of singular interactions describes all possible selfadjoint extensions of the second derivative operator defined on the functions vanishing in a neighbourhood of the point. Approximation by operators with smooth coefficients is discussed.
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| 7. |
- Kurasov, Pavel
(författare)
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Energy dependent boundary conditions and few-body scattering problem
- 1997
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Ingår i: Reviews in Mathematical Physics. - 0129-055X. ; 9:7, s. 853-906
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Tidskriftsartikel (refereegranskat)abstract
- An exactly solvable problem with energy dependent interaction is investigated in the present paper. The selfadjoint model operator describes the scattering problem for three one-dimensional particles. It is shown that this problem is equivalent to the diffraction problem in the sector with energy dependent boundary conditions. The problem is solved with the help of the Sommerfeld-Maluzhinetz representation, which transforms the partial differential equation for the eigenfunctions to a functional equation on the integral densities. The solution of the functional equation can be constructed explicitly in the case of identical particles. The three-body scattering matrix describing rearrangement and excitation processes is represented in terms of analytic functions.
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| 8. |
- Kurasov, Pavel
(författare)
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On the Coulomb potential in one dimension
- 1996
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Ingår i: Journal of Physics A. - : IOP Publishing. - 0305-4470 .- 1361-6447. ; 29:8, s. 1767-1771
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Tidskriftsartikel (refereegranskat)abstract
- A mathematically rigorous definition of the one-dimensional Schrödinger operator -d2/dx2 - γ/x is given. It is proven that the domain of the operator is defined by the boundary conditions connecting the values of the function on the left and right half-axes. The investigated operator is compared with the Schrödinger operator containing the Coulomb potential -γ/|x
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| 9. |
- Kurasov, Pavel
(författare)
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Scattering matrices with finite phase shift and the inverse scattering problem
- 1996
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Ingår i: Inverse Problems. - : IOP Publishing. - 0266-5611 .- 1361-6420. ; 12:3, s. 295-307
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Tidskriftsartikel (refereegranskat)abstract
- The inverse scattering problem for the Schrödinger operator on the half-axis is studied. it is shown that this problem can be solved for the scattering matrices with arbitrary finite phase shift on the real axis if one admits potentials with long-range oscillating tails at infinity. The solution of the problem is constructed with the help of the Gelfand-Levitan-Marchenko procedure. The inverse problem has no unique solution for the standared set of scattering data which includes the scattering matrix, energies of the bound states and corresponding normalizing constants. This fact is related to zeros of the spectral density on the real axis. It is proven that the inverse problem has a unique solution in the defined class of potentials if the zeros of the spectral density are added to the set of scattering data.
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| 10. |
- Avdonin, Sergei, et al.
(författare)
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Inverse problems for quantum trees
- 2008
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Ingår i: INVERSE PROBLEMS AND IMAGING. - 1930-8337 .- 1930-8345. ; 2:1, s. 1-21
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Tidskriftsartikel (refereegranskat)abstract
- Three different inverse problems for the Schrodinger operator on a metric tree are considered, so far with standard boundary conditions at the vertices. These inverse problems are connected with the matrix Titchmarsh-Weyl function, response operator ( dynamic Dirichlet-to-Neumann map) and scattering matrix. Our approach is based on the boundary control ( BC) method and in particular on the study of the response operator. It is proven that the response operator determines the quantum tree completely, i.e. its connectivity, lengths of the edges and potentials on them. The same holds if the response operator is known for all but one boundary points, as well as for the Titchmarsh-Weyl function and scattering matrix. If the connectivity of the graph is known, then the lengths of the edges and the corresponding potentials are determined by just the diagonal terms of the data.
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