On the finite-rank approximation of wave operators and the S matrix using Hermite expansion functions

1986 ◽  
Vol 64 (8) ◽  
pp. 872-875 ◽  
Author(s):  
Helmut Kröger
Keyword(s):  
Finite Rank ◽  
Hermite Functions ◽  
Quantum Mechanical ◽  
Weak Approximation ◽  
Body System ◽  
Hermite Expansion ◽  
Wave Operators ◽  
Finite Dimensional ◽  
S Matrix ◽  
Rank Approximation

For the quantum mechanical, nonrelativistic two-body system, wave operators can be approximated by exponentials of finite-rank operators obtained by finite-dimensional projections of the full and the asymptotic Hamiltonian onto a set of Hermite functions. This yields a weak approximation of the S matrix.

A Multichannel “Potential Curves Hopping” Model for Inelastic Collisions

1986 ◽  
Vol 41 (5) ◽  
pp. 704-714
Author(s):  
D. Campos ◽  
J. M. Tejeiro ◽  
F. Cristancho
Keyword(s):  
Matrix Element ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Inelastic Collisions ◽  
Interaction Matrix ◽  
Quantum Mechanical ◽  
Interaction Matrix Element ◽  
S Matrix ◽  
Hopping Model ◽  
Potential Curves

We introduce a multichannel “potential curves hopping” model and obtain the exact quantum mechanical S-matrix by solving the associated set of coupled second-order ordinary differential equations that describes the inelastic collisions between atomic particles. The only assumption is that the interaction matrix element between each pair of channels (say, γ and β) is of the form Uγβ(r) = Uβγ(r) =: Uγβ δ( r - rγβ), where δ (c) is the Dirac deltafunction, and rγβ and Uγβ are parameters which can be chosen freely.Semiclassical techniques can be incorporated directly in the theory if the Schrödinger equations for the uncoupled channels allow this treatment. The formulation is particularized to the two-channel problem and illustrated with a semiclassical example the He+ + Ne problem at 70.9 eV.

Finite-dimensional approximation of the scattering matrix

1986 ◽  
Vol 64 (5) ◽  
pp. 611-616 ◽  
Author(s):  
Helmut Kröger ◽  
Anais Smailagic ◽  
Ralph Girard
Keyword(s):  
Weak Coupling ◽  
Weak Coupling Regime ◽  
S Wave ◽  
Proton Proton ◽  
Time Evolution Operator ◽  
Finite Dimensional Approximation ◽  
Finite Dimensional ◽  
Dimensional Approximation ◽  
S Matrix ◽  
Computer Calculations

A finite-dimensional nonperturbative approximation scheme of the time-evolution operator and the S matrix for relativistic field theories is discussed. It is amenable to computer calculations. Parallels with lattice-field theory are drawn. The method is outlined for the ϕ4 theory. Equivalence to standard perturbation theory in the weak-coupling regime is obtained in the limit of the approximation parameters. The method is tested numerically for nonrelativistic proton–proton s-wave scattering and the the ϕ4 model in the weak-coupling regime in 1 + 1 dimensions. In both examples, convergence to the reference solution is found.

scholarly journals Scattering on periodic metric graphs

2020 ◽  
Vol 32 (08) ◽  
pp. 2050024
Author(s):  
Evgeny Korotyaev ◽  
Natalia Saburova
Keyword(s):  
Dirichlet Problem ◽  
Half Plane ◽  
Fredholm Determinant ◽  
Unit Interval ◽  
Discrete Laplacian ◽  
Metric Graphs ◽  
Wave Operators ◽  
S Matrix ◽  
Upper Half Plane ◽  
Uniformly Bounded

We consider the Laplacian on a periodic metric graph and obtain its decomposition into a direct fiber integral in terms of the corresponding discrete Laplacian. Eigenfunctions and eigenvalues of the fiber metric Laplacian are expressed explicitly in terms of eigenfunctions and eigenvalues of the corresponding fiber discrete Laplacian and eigenfunctions of the Dirichlet problem on the unit interval. We show that all these eigenfunctions are uniformly bounded. We apply these results to the periodic metric Laplacian perturbed by real integrable potentials. We prove the following: (a) the wave operators exist and are complete, (b) the standard Fredholm determinant is well-defined and is analytic in the upper half-plane without any modification for any dimension, (c) the determinant and the corresponding S-matrix satisfy the Birman–Krein identity.

A quantum-mechanical central limit theorem for anti-commuting observables

1973 ◽  
Vol 10 (03) ◽  
pp. 502-509 ◽  
Author(s):  
R. L. Hudson
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Dimensional Space ◽  
Quantum Mechanical ◽  
Finite Dimensional Space ◽  
Commutation Relations ◽  
Finite Dimensional

A quantum-mechanical central limit theorem for sums of pairwise anti-commuting representations of the canonical anti-commutation relations over a finite-dimensional space is formulated and proved.

scholarly journals Groups which satisfy a weak form of Poincaré duality

1991 ◽  
Vol 34 (3) ◽  
pp. 463-486
Author(s):  
J. E. Roberts
Keyword(s):  
Finite Rank ◽  
Weak Form ◽  
Homological Dimension ◽  
Poincaré Duality ◽  
Poincare Duality ◽  
Soluble Groups ◽  
Finite Dimensional ◽  
Duality Property ◽  
The Given ◽  
Torsion Free

Our main result is that a “restricted Poincaré duality” property with respect to finite dimensional coefficient modules over a field holds for a certain class of groups which includes all soluble groups of finite Hirsch length. This relies on a generalisation to the given class of a module construction by Stammbach; an extension of his result on homological dimension to these groups is given. We also generalise the well-known result that torsion-free soluble groups of finite rank are countable.

scholarly journals On the Structure of Finite Groupoids and Their Representations

Symmetry ◽  
2019 ◽  
Vol 11 (3) ◽  
pp. 414 ◽  
Author(s):  
Alberto Ibort ◽  
Miguel Rodríguez
Keyword(s):  
Irreducible Representation ◽  
Regular Representation ◽  
Irreducible Representations ◽  
Quantum Mechanical ◽  
Linear Representations ◽  
Finite Dimensional ◽  
Completely Reducible ◽  
Quantum Mechanical Systems ◽  
Maschke’S Theorem ◽  
Totally Disconnected

In this paper, both the structure and the theory of representations of finite groupoids are discussed. A finite connected groupoid turns out to be an extension of the groupoids of pairs of its set of units by its canonical totally disconnected isotropy subgroupoid. An extension of Maschke’s theorem for groups is proved showing that the algebra of a finite groupoid is semisimple and all finite-dimensional linear representations of finite groupoids are completely reducible. The theory of characters for finite-dimensional representations of finite groupoids is developed and it is shown that irreducible representations of the groupoid are in one-to-one correspondence with irreducible representation of its isotropy groups, with an extension of Burnside’s theorem describing the decomposition of the regular representation of a finite groupoid. Some simple examples illustrating these results are exhibited with emphasis on the groupoids interpretation of Schwinger’s description of quantum mechanical systems.

S-matrix pole trajectory of the three-body system

1991 ◽  
Vol 534 (3-4) ◽  
pp. 620-636 ◽  
Author(s):  
A. Matsuyama ◽  
K. Yazaki
Keyword(s):  
Body System ◽  
S Matrix ◽  
Three Body

Orthogonality spaces of finite rank and the complex Hilbert spaces

2019 ◽  
Vol 16 (05) ◽  
pp. 1950080 ◽  
Author(s):  
Thomas Vetterlein
Keyword(s):  
Binary Relation ◽  
Hilbert Spaces ◽  
Finite Rank ◽  
Representation Theorem ◽  
Quantum Physics ◽  
Orthogonality Relation ◽  
Quadratic Space ◽  
Orthogonality Space ◽  
Finite Dimensional

An orthogonality space is a set endowed with a symmetric, irreflexive binary relation. By means of the usual orthogonality relation, each anisotropic quadratic space gives rise to such a structure. We investigate in this paper the question to which extent this strong abstraction suffices to characterize complex Hilbert spaces, which play a central role in quantum physics. To this end, we consider postulates concerning the nature and existence of symmetries. Together with a further postulate excluding the existence of nontrivial quotients, we establish a representation theorem for finite-dimensional orthomodular spaces over a dense subfield of [Formula: see text].

Sign in / Sign up

Export Citation Format

Share Document