 94243 Stovicek P.
 Scattering on a finite chain of vortices
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Jul 25, 94

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Abstract. This problem is related to the AharonovBohm effect. The Hamiltonian
in L^2(R^2) is defined as a selfadjoint 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 eigenfunctions of the Hamiltonian starting from
the explicit form of the unitary mapping between the deficiency subspaces
of X defining the selfadjoint 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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