2,284 research outputs found
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
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
Traditions of Research in Architecture: Finnish Trend
Finnish trends in research in architecture are described and related to three traditions of research: the subjective idealistic, the empirical positivistic, and the hermeneutic interpretative traditions. The influence of the contexts on research are discussed
Quasi-periodic attractors, Borel summability and the Bryuno condition for strongly dissipative systems
We consider a class of ordinary differential equations describing
one-dimensional analytic systems with a quasi-periodic forcing term and in the
presence of damping. In the limit of large damping, under some generic
non-degeneracy condition on the force, there are quasi-periodic solutions which
have the same frequency vector as the forcing term. We prove that such
solutions are Borel summable at the origin when the frequency vector is either
any one-dimensional number or a two-dimensional vector such that the ratio of
its components is an irrational number of constant type. In the first case the
proof given simplifies that provided in a previous work of ours. We also show
that in any dimension , for the existence of a quasi-periodic solution with
the same frequency vector as the forcing term, the standard Diophantine
condition can be weakened into the Bryuno condition. In all cases, under a
suitable positivity condition, the quasi-periodic solution is proved to
describe a local attractor.Comment: 10 page
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