515 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
Generalized Faddeev equations in the AGS form for deuteron stripping with explicit inclusion of target excitations and Coulomb interaction
Theoretical description of reactions in general, and the theory for
reactions, in particular, needs to advance into the new century. Here deuteron
stripping processes off a target nucleus consisting of nucleons are
treated within the framework of the few-body integral equations theory. The
generalized Faddeev equations in the AGS form, which take into account the
target excitations, with realistic optical potentials provide the most advanced
and complete description of the deuteron stripping. The main problem in
practical application of such equations is the screening of the Coulomb
potential, which works only for light nuclei. In this paper we present a new
formulation of the Faddeev equations in the AGS form taking into account the
target excitations with explicit inclusion of the Coulomb interaction. By
projecting the -body operators onto target states, matrix three-body
integral equations are derived which allow for the incorporation of the excited
states of the target nucleons. Using the explicit equations for the partial
Coulomb scattering wave functions in the momentum space we present the AGS
equations in the Coulomb distorted wave representation without screening
procedure. We also use the explicit expression for the off-shell two-body
Coulomb scattering -matrix which is needed to calculate the effective
potentials in the AGS equations. The integrals containing the off-shell Coulomb
T-matrix are regularized to make the obtained equations suitable for
calculations. For and nucleon-target nuclear interactions we assume the
separable potentials what significantly simplifies solution of the AGS
equations.Comment: 34 pages, 13 figure
Meromorphic traveling wave solutions of the complex cubic-quintic Ginzburg-Landau equation
We look for singlevalued solutions of the squared modulus M of the traveling
wave reduction of the complex cubic-quintic Ginzburg-Landau equation. Using
Clunie's lemma, we first prove that any meromorphic solution M is necessarily
elliptic or degenerate elliptic. We then give the two canonical decompositions
of the new elliptic solution recently obtained by the subequation method.Comment: 14 pages, no figure, to appear, Acta Applicandae Mathematica
Influence of low energy scattering on loosely bound states
Compact algebraic equations are derived, which connect the binding energy and
the asymptotic normalization constant (ANC) of a subthreshold bound state with
the effective-range expansion of the corresponding partial wave. These
relations are established for positively-charged and neutral particles, using
the analytic continuation of the scattering (S) matrix in the complex
wave-number plane. Their accuracy is checked on simple local potential models
for the 16O+n, 16O+p and 12C+alpha nuclear systems, with exotic nuclei and
nuclear astrophysics applications in mind
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
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