94-243 Stovicek P.

Scattering on a finite chain of vortices (83K, LaTeX) Jul 25, 94

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Abstract. This problem is related to the Aharonov--Bohm effect. The Hamiltonian in L^2(R^2) is defined as a self--adjoint extension of the symmetric operator X=-Laplacian, with the domain D(X)= smooth functions in R^2 compactly supported outside of the first coordinate axis and the extension is determined by boundary conditions on this axis. There is proven a perturbative formula for the inverted operator [\sqrt{1+P^2}+\exp(-2\pi i\nu(Q))\sqrt{1+P^2}\exp(2\pi i\nu(Q))]^{-1} where P and Q are canonically conjugated operators, [Q,P]=i, and with \nu(u) being a piecewise constant function related to the boundary conditions. This result jointly with the Krein's formula enables one to construct generalized eigen--functions of the Hamiltonian starting from the explicit form of the unitary mapping between the deficiency subspaces of X defining the self-adjoint extension. These formulae also make it possible to prove existence and completeness of the wave operators using the trace class methods.

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