144 research outputs found
Realizing Discontinuous Wave Functions with Renormalized Short-Range Potentials
It is shown that a potential consisting of three Dirac's delta functions on
the line with disappearing distances can give rise to the discontinuity in wave
functions with the proper renormalization of the delta function strength. This
can be used as a building block, along with the usual Dirac's delta, to
construct the most general three-parameter family of point interactions, which
allow both discontinuity and asymmetry of the wave function, as the zero-size
limit of self-adjoint local operators in one-dimensional quantum mechanics.
Experimental realization of the Neumann boundary is discussed.
KEYWORDS: point interaction, self-adjoint extension, potential,
wave function discontinuity, Neumann boundary
PACS Nos: 3.65.-w, 11.10.Gh, 68.65+gComment: 4 pages, ReVTeX double column format with an epsf figure, expanded
reference
A Spectral Mapping Theorem and Invariant Manifolds for Nonlinear Schr\"odinger Equations
A spectral mapping theorem is proved that resolves a key problem in applying
invariant manifold theorems to nonlinear Schr\" odinger type equations. The
theorem is applied to the operator that arises as the linearization of the
equation around a standing wave solution. We cast the problem in the context of
space-dependent nonlinearities that arise in optical waveguide problems. The
result is, however, more generally applicable including to equations in higher
dimensions and even systems. The consequence is that stable, unstable, and
center manifolds exist in the neighborhood of a (stable or unstable) standing
wave, such as a waveguide mode, under simple and commonly verifiable spectral
conditions.Comment: LaTeX, 16 page
Duality and Anholonomy in Quantum Mechanics of 1D Contact Interactions
We study systems with parity invariant contact interactions in one dimension.
The model analyzed is the simplest nontrivial one --- a quantum wire with a
point defect --- and yet is shown to exhibit exotic phenomena, such as strong
vs weak coupling duality and spiral anholonomy in the spectral flow. The
structure underlying these phenomena is SU(2), which arises as accidental
symmetry for a particular class of interactions.Comment: 4 pages ReVTeX with 4 epsf figures. KEK preprint 2000-3. Correction
in Eq.(14
Low-energy expansion formula for one-dimensional Fokker-Planck and Schr\"odinger equations with asymptotically periodic potentials
We consider one-dimensional Fokker-Planck and Schr\"odinger equations with a
potential which approaches a periodic function at spatial infinity. We extend
the low-energy expansion method, which was introduced in previous papers, to be
applicable to such asymptotically periodic cases. Using this method, we study
the low-energy behavior of the Green function.Comment: author-created, un-copyedited version of an article accepted for
publication in Journal of Physics A: Mathematical and Theoretica
On the Two Spectra Inverse Problem for Semi-Infinite Jacobi Matrices
We present results on the unique reconstruction of a semi-infinite Jacobi
operator from the spectra of the operator with two different boundary
conditions. This is the discrete analogue of the Borg-Marchenko theorem for
Schr{\"o}dinger operators in the half-line. Furthermore, we give necessary and
sufficient conditions for two real sequences to be the spectra of a Jacobi
operator with different boundary conditions.Comment: In this slightly revised version we have reworded some of the
theorems, and we updated two reference
The inverse resonance problem for perturbations of algebro-geometric potentials
We prove that a compactly supported perturbation of a rational or simply
periodic algebro-geometric potential of the one-dimensional Schr\"odinger
equation on the half line is uniquely determined by the location of its
Dirichlet eigenvalues and resonances.Comment: 14 page
The Two-Spectra Inverse Problem for Semi-Infinite Jacobi Matrices in The Limit-Circle Case
We present a technique for reconstructing a semi-infinite Jacobi operator in
the limit circle case from the spectra of two different self-adjoint
extensions. Moreover, we give necessary and sufficient conditions for two real
sequences to be the spectra of two different self-adjoint extensions of a
Jacobi operator in the limit circle case.Comment: 26 pages. Changes in the presentation of some result
On essential self-adjointness for magnetic Schroedinger and Pauli operators on the unit disc in R^2
We study the question of magnetic confinement of quantum particles on the
unit disk \ID in \IR^2, i.e. we wish to achieve confinement solely by means
of the growth of the magnetic field near the boundary of the disk.
In the spinless case we show that , for close to 1, insures the confinement provided we
assume that the non-radially symmetric part of the magnetic field is not very
singular near the boundary. Both constants and
are optimal. This answers, in this context, an open
question from Y. Colin de Verdi\`ere and F. Truc. We also derive growth
conditions for radially symmetric magnetic fields which lead to confinement of
spin 1/2 particles.Comment: 18 pages; the main theorem has been expanded and generalize
Equivalence of Local and Separable Realizations of the Discontinuity-Inducing Contact Interaction and Its Perturbative Renormalizability
We prove that the separable and local approximations of the
discontinuity-inducing zero-range interaction in one-dimensional quantum
mechanics are equivalent. We further show that the interaction allows the
perturbative treatment through the coupling renormalization.
Keywords: one-dimensional system, generalized contact interaction,
renormalization, perturbative expansion. PACS Nos: 3.65.-w, 11.10.Gh, 31.15.MdComment: ReVTeX 7pgs, doubl column, no figure, See also the website
http://www.mech.kochi-tech.ac.jp/cheon
The Heun equation and the Calogero-Moser-Sutherland system V: generalized Darboux transformations
We obtain isomonodromic transformations for Heun's equation by generalizing
Darboux transformation, and we find pairs and triplets of Heun's equation which
have the same monodromy structure. By composing generalized Darboux
transformations, we establish a new construction of the commuting operator
which ensures finite-gap property. As an application, we prove conjectures in
part III.Comment: 24 page
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