403 research outputs found
Anharmonic Oscillators with Infinitely Many Real Eigenvalues and PT-Symmetry
We study the eigenvalue problem in the complex plane
with the boundary condition that decays to zero as tends to infinity
along the two rays , where
for complex-valued polynomials of degree at most
. We provide an asymptotic formula for eigenvalues and a necessary
and sufficient condition for the anharmonic oscillator to have infinitely many
real eigenvalues
Trace Formulas for Non-Self-Adjoint Periodic Schr\"odinger Operators and some Applications
Recently, a trace formula for non-self-adjoint periodic Schr\"odinger
operators in associated with Dirichlet eigenvalues was proved
in [9]. Here we prove a corresponding trace formula associated with Neumann
eigenvalues.
In addition we investigate Dirichlet and Neumann eigenvalues of such
operators. In particular, using the Dirichlet and Neumann trace formulas we
provide detailed information on location of the Dirichlet and Neumann
eigenvalues for the model operator with the potential , where
.Comment: 26 pages, no figure
The potential (iz)^m generates real eigenvalues only, under symmetric rapid decay conditions
We consider the eigenvalue problems -u"(z) +/- (iz)^m u(z) = lambda u(z), m
>= 3, under every rapid decay boundary condition that is symmetric with respect
to the imaginary axis in the complex z-plane. We prove that the eigenvalues
lambda are all positive real.Comment: 23 pages and 1 figur
On the shape of spectra for non-self-adjoint periodic Schr\"odinger operators
The spectra of the Schr\"odinger operators with periodic potentials are
studied. When the potential is real and periodic, the spectrum consists of at
most countably many line segments (energy bands) on the real line, while when
the potential is complex and periodic, the spectrum consists of at most
countably many analytic arcs in the complex plane.
In some recent papers, such operators with complex -symmetric
periodic potentials are studied. In particular, the authors argued that some
energy bands would appear and disappear under perturbations. Here, we show that
appearance and disappearance of such energy bands imply existence of nonreal
spectra. This is a consequence of a more general result, describing the local
shape of the spectrum.Comment: 5 pages, 2 figure
Eigenvalues of PT-symmetric oscillators with polynomial potentials
We study the eigenvalue problem
with the boundary
conditions that decays to zero as tends to infinity along the rays
, where is a polynomial and integers . We provide an
asymptotic expansion of the eigenvalues as , and prove
that for each {\it real} polynomial , the eigenvalues are all real and
positive, with only finitely many exceptions.Comment: 23 pages, 1 figure. v2: equation (14) as well as a few subsequent
equations has been changed. v3: typos correcte
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