 02126 R. de la Madrid, A. Bohm, M. Gadella
 Rigged Hilbert Space Treatment of Continuous Spectrum
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Mar 16, 02

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Abstract. The ability of the Rigged Hilbert Space formalism to deal with
continuous spectrum is demonstrated within the example of the square
barrier potential. The nonsquare integrable solutions of the
timeindependent Schrodinger equation are used to define Dirac kets,
which are (generalized) eigenvectors of the Hamiltonian. These Dirac
kets are antilinear functionals over the space of physical wave
functions. They are also basis vectors that expand any physical wave
function in a Dirac basis vector expansion. It is shown that an
acceptable physical wave function must fulfill stronger conditions
than just square integrabilitythe space of physical wave functions
is not the whole Hilbert space but rather a dense subspace of the
Hilbert space. We construct the position and energy representations
of the Rigged Hilbert Space generated by the square barrier potential
Hamiltonian. We shall also construct the unitary operator that
transforms from the position into the energy representation. We shall
see that in the energy representation the Dirac kets act as the
antilinear Schwartz delta functional. In constructing the Rigged
Hilbert Space of the square barrier potential, we will find a
systematic procedure to construct the Rigged Hilbert Space of a large
class of spherically symmetric potentials. The example of the square
barrier potential will also make apparent that the natural framework
for the solutions of a Schrodinger operator with continuous
spectrum is the Rigged Hilbert Space rather than just the Hilbert space.
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