Bulletin of the American Physical Society
APS March Meeting 2011
Volume 56, Number 1
Monday–Friday, March 21–25, 2011; Dallas, Texas
Abstract: K1.00048 : Anharmonic Energy Spectrum for $V(x)=\pm x^{4},\pm x^{6},\pm x^{8}$
Author:
In this work we wish to revisit the energy spectrum for the
anharmonic
potentials%
\[
H=\frac{p^{2}}{2m}\pm x^{N},
\]
where $N=4,6,8$. Using the second quantized operator formalism of
Dirac, we
have evaluated matrix truncations of up to $100\times100$. Our
results for the
energy spectrum are in disagreement with the work of Bender and
Boettcher (PRL
80, 5243). They studied a \emph{PT} symmetric Hamiltonian whose
potential is
given by $V(x)=-(ix)^{N}$ and who maintain that \textquotedblleft
when
$N\geq2$, the spectrum is infinite, discrete and entirely real and
positive\textquotedblright. We find, for the potentials with
$N=4,6,8$ that
the spectrum is not completely positive and in fact has no lower
bound.
To cite this abstract, use the following reference: http://meetings.aps.org/link/BAPS.2011.MAR.K1.48
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