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Research Article Review to appear in Mathematical Reviews (2006)
Copyright: American Mathematical Society
Boris F. Samsonov
"Spectral Singularities of non-Hermitian Hamiltonians and SUSY transformations"
( Journal of Physics A: Math. Gen., 38 (2005) L571-L579)
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| Reviewed by T. Cheon (Tosa Yamada, Japan)
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Non-Hermitian Hamiltonians with purely real spectra have attracted much attention among those who are in quest of expanding the repertoire of solvable quantum mechanics.
Non-Hermitian Hamiltonians, however, often suffer two ``diseases'' which would make their use as physical Hamiltonians difficult.
The first is the possibility of the appearance of {\it associated functions} which are not eigenfunctions of the Hamiltonians but are necessary to complete a basis in the corresponding Hilbert space. This property is also known as {\it non-diagonalizability}.
The second is the possibility of the appearance of {\it spectral singularities} inside a continuous spectrum. Spectral singularities are the energies which correspond to the real roots of the Jost function.
Spectral singularity embedded in a continuous spectrum, which is not to be confused with an embedded bound state, is absent in Hermitian Hamiltonians.
With the appearance of spectral singularities, the spectral decomposition of normalizable operators in Hilbert space is not guaranteed.
In his previous paper, Samsonov, the author of this paper, has shown that the first difficulty is curable with suitable SUSY transformation.
Here, he demonstrates that a similar feat is possible for the second difficulty of non-Hermitian Hamiltonian.
More precisely, the author proves a theorem that states that a smooth, finite and exponentially decreasing potential $V_0(x)$ defined on $[0, \infty)$ has a complex second-order SUSY partner potential $V_2(x)$ which is again regular and exponentially decreasing on $[0, \infty)$, but yields the Hamiltonian that has a spectral singularity.
It is also proven that there exists a real first-order SUSY partner potential $V_1(x)$ of the potential $V_2(x)$, which has a singularity $V_1(x) \to 2/x^2$ at the origin $x = 0$, giving an essentially self-adjoint Hamiltonian.
In the process of the proof, it is shown that the positive energy eigenfunction at the spectral singularity, embedded in the scattering wavefunctions, is nodeless, a very peculiar characteristics indeed.
The new variety of possibly viable quantum mechanical potentials brought home by the theorem, both real and complex, is illustrated with examples derived from spectrally singular complex potential $V_2(x) = -2a^2/\cosh^2(ax+b)$ with positive real $a$ and complex $b$.
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Copyright: American Mathematical Society, 2006
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