127 research outputs found
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 eigenvalues of the Schr\"odinger operator with a complex-valued polynomial potential
In this paper, we generalize a recent result of A. Eremenko and A. Gabrielov
on irreducibility of the spectral discriminant for the Schr\"odinger equation
with quartic potentials. We consider the eigenvalue problem with a
complex-valued polynomial potential of arbitrary degree d and show that the
spectral determinant of this problem is connected and irreducible. In other
words, every eigenvalue can be reached from any other by analytic continuation.
We also prove connectedness of the parameter spaces of the potentials that
admit eigenfunctions satisfying k>2 boundary conditions, except for the case d
is even and k=d/2. In the latter case, connected components of the parameter
space are distinguished by the number of zeros of the eigenfunctions.Comment: 23 page
Quasi-exactly solvable quartic: real algebraic spectral locus
We describe the real quasi-exactly solvable spectral locus of the
PT-symmetric quartic using the Nevanlinna parametrization.Comment: 17 pages, 11 figure
On eigenvalues of the Schr\"odinger operator with an even complex-valued polynomial potential
In this paper, we generalize several results of the article "Analytic
continuation of eigenvalues of a quartic oscillator" of A. Eremenko and A.
Gabrielov.
We consider a family of eigenvalue problems for a Schr\"odinger equation with
even polynomial potentials of arbitrary degree d with complex coefficients, and
k<(d+2)/2 boundary conditions. We show that the spectral determinant in this
case consists of two components, containing even and odd eigenvalues
respectively.
In the case with k=(d+2)/2 boundary conditions, we show that the
corresponding parameter space consists of infinitely many connected components
PT-Symmetric Quantum Theory Defined in a Krein Space
We provide a mathematical framework for PT-symmetric quantum theory, which is
applicable irrespective of whether a system is defined on R or a complex
contour, whether PT symmetry is unbroken, and so on. The linear space in which
PT-symmetric quantum theory is naturally defined is a Krein space constructed
by introducing an indefinite metric into a Hilbert space composed of square
integrable complex functions in a complex contour. We show that in this Krein
space every PT-symmetric operator is P-Hermitian if and only if it has
transposition symmetry as well, from which the characteristic properties of the
PT-symmetric Hamiltonians found in the literature follow. Some possible ways to
construct physical theories are discussed within the restriction to the class
K(H).Comment: 8 pages, no figures; Refs. added, minor revisio
Quantum toboggans: models exhibiting a multisheeted PT symmetry
A generalization of the concept of PT-symmetric Hamiltonians H=p^2+V(x) is
described. It uses analytic potentials V(x) (with singularities) and a
generalized concept of PT-symmetric asymptotic boundary conditions. Nontrivial
toboggans are defined as integrated along topologically nontrivial paths of
coordinates running over several Riemann sheets of wave functions.Comment: 16 pp, 5 figs. Written version of the talk given during 5th
International Symposium on Quantum Theory and Symmetries, University of
Valladolid, Spain, July 22 - 28 2007, webpage http://tristan.fam.uva.es/~qts
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
Joint distribution of the first and second eigenvalues at the soft edge of unitary ensembles
The density function for the joint distribution of the first and second
eigenvalues at the soft edge of unitary ensembles is found in terms of a
Painlev\'e II transcendent and its associated isomonodromic system. As a
corollary, the density function for the spacing between these two eigenvalues
is similarly characterized.The particular solution of Painlev\'e II that arises
is a double shifted B\"acklund transformation of the Hasting-McLeod solution,
which applies in the case of the distribution of the largest eigenvalue at the
soft edge. Our deductions are made by employing the hard-to-soft edge
transitions to existing results for the joint distribution of the first and
second eigenvalue at the hard edge \cite{FW_2007}. In addition recursions under
of quantities specifying the latter are obtained. A Fredholm
determinant type characterisation is used to provide accurate numerics for the
distribution of the spacing between the two largest eigenvalues.Comment: 26 pages, 1 Figure, 2 Table
Y-System and Deformed Thermodynamic Bethe Ansatz
We introduce a new tool, the Deformed TBA (Deformed Thermodynamic Bethe
Ansatz), to analyze the monodromy problem of the cubic oscillator. The Deformed
TBA is a system of five coupled nonlinear integral equations, which in a
particular case reduces to the Zamolodchikov TBA equation for the 3-state Potts
model. Our method generalizes the Dorey-Tateo analysis of the (monomial) cubic
oscillator. We introduce a Y-system corresponding to the Deformed TBA and give
it an elegant geometric interpretation.Comment: 12 pages. Minor corrections in Section
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