Coupling Constant Thresholds Research Articles

Convergent Expansions of Eigenvalues of the Generalized Friedrichs Model with a Rank-One Perturbation

We study analytic behavior of eigenvalues of the generalized Friedrichs model \(H_\mu (p)\), with a rank-one perturbation, depending on parameters \(\mu >0\) and \(p\in {\mathbb {T}}^2\). Under certain conditions, the existence of a unique eigenvalue lying below the essential spectrum has been shown in Lakaev (Abstract Appl Anal 2012, 2012). Here, we obtain an absolutely convergent expansion for that eigenvalue at \(\mu (p)\), the coupling constant threshold. The expansion is dependent to a large extent on whether the lower bound of the essential spectrum is a threshold resonance, a threshold eigenvalue or neither of them.

On coupling constant thresholds in one dimension

The threshold behaviour of negative eigenvalues for Schr\"{o}dinger operators of the type \[ H_\lambda=-\frac{d^2}{dx^2}+U+ \lambda\alpha_\lambda V(\alpha_\lambda \cdot) \] is considered. The potentials $U$ and $V$ are real-valued bounded functions of compact support, $\lambda$ is a positive parameter, and positive sequence $\alpha_\lambda$ has a finite or infinite limit as $\lambda\to 0$. Under certain conditions on the potentials there exists a bound state of $H_\lambda$ which is absorbed at the bottom of the continuous spectrum. For several cases of the limiting behaviour of sequence $\alpha_\lambda$, asymptotic formulas for the bound states are proved and the first order terms are computed explicitly.

ON THE SPECTRUM OF THE ONE-DIMENSIONAL SCHRÖDINGER HAMILTONIAN PERTURBED BY AN ATTRACTIVE GAUSSIAN POTENTIAL

<p>We propose a new approach to the problem of finding the eigenvalues (energy levels) in the discrete spectrum of the one-dimensional Hamiltonian with an attractive Gaussian potential by using the well-known Birman-Schwinger technique. However, in place of the Birman-Schwinger integral operator we consider an isospectral operator in momentum space, taking advantage of the unique feature of this potential, that is to say its invariance under Fourier transform. <br />Given that such integral operators are trace class, it is possible to determine the energy levels in the discrete spectrum of the Hamiltonian as functions of <span>the coupling constant with great accuracy by solving a finite number of transcendental equations. We also address the important issue of the coupling constant thresholds of the Hamiltonian, that is to say the critical values of λ for which we have the emergence of an additional bound state out of the absolutely continuous spectrum. </span></p>

Ground-state counterpropagating solitons in photorefractive media with saturable nonlinearity

We investigate the existence and form of (2+1)-dimensional ground-state counterpropagating solitons in photorefractive media with saturable nonlinearity. General conditions for the existence of fundamental solitons in a local isotropic model that includes an intensity-dependent saturable nonlinearity are identified. We confirm our theoretical findings numerically and determine the ground-state profiles. We check their stability in propagation and identify the coupling constant threshold for their existence. Critical exponents of the power and beam width are determined as functions of the propagation constant at the threshold. We finally formulate a variational approach to the same problem, introduce an approximate fundamental Gaussian solution, and verify that this method leads to the same threshold and similar critical exponents as the theoretical and numerical methods.

Coupling constant thresholds of perturbed periodic Hamiltonians

We consider Schrödinger operators of the form Hλ=−Δ+V+λW on L2(Rν) (ν=1, 2, or 3) with V periodic, W short range, and λ a real non-negative parameter. Then the continuous spectrum of Hλ has the typical band structure consisting of intervals, separated by gaps. In the gaps there may be discrete eigenvalues of Hλ that are functions of the parameter λ. Let (a,b) be a gap and E(λ)∈(a,b) an eigenvalue of Hλ. We study the asymptotic behavior of E(λ) as λ approaches a critical value λ0, called a coupling constant threshold, at which the eigenvalue either emerges from or is absorbed into the continuous spectrum. A typical question is the following: Assuming E(λ)↓a as λ↓λ0, is E(λ)−a∼c(λ−λ0)α for some α>0 and c≠0, or is there an expansion in some other quantity? As one expects from previous work in the case V=0, the answer strongly depends on ν.

DUALITY SYMMETRIES IN GENERALIZED COXETER ORBIFOLDS

We derive the duality symmetries relevant to moduli-dependent gauge coupling constant threshold corrections, in generalized Coxeter ZN orbifolds. The underlying sixdimensional lattice of the heterotic string theory is not assumed to decompose into a direct sum of a four-dimensional and a two-dimensional sublattice and the point group leaves fixed a two-dimensional sublattice.

Duality symmetries in orbifold models

We derive the duality symmetries relevant to moduli dependent gauge coupling constant threshold corrections, in Coreter Z N orbifolds. We consider those orbifolds for which the point group leaves fixed a two-dimensional sublattice Λ 2, of the six-dimensional torus lattice Λ 6, where Λ 6 cannot be decomposed as Λ 2 ⊗ Λ 4.

A unified approach to eigenvalues and resonances of Schrödinger operators using Fredholm determinants

We study in detail the threshold behaviour of eigenvalues and resonances of the Schrödinger operator H( λ) = - Δ + λV in L 2( R 3) as they are absorbed at the bottom of the continuous spectrum when λ → λ 0, λ 0 a so called coupling constant threshold. Using Fredholm determinants we give their analytic structure in terms of λ − λ 0 and compute the first order terms explicitly.

Hydrogen atom with a Yukawa potential: Perturbation theory and continued-fractions-Padé approximants at large order.

A simple power-series method is developed to calculate to large order the Rayleigh-Schr\odinger perturbation expansions for energy levels of a hydrogen atom with a Yukawa-type screened Coulomb potential. Perturbation series for the 1s, 2s, and 2p levels, shown not to be of the Stieltjes type, are calculated to 100th order. Nevertheless, the poles of the Pad\'e approximants to these series generally avoid the region of the positive real axis 0${\ensuremath{\lambda}}^{\mathrm{*}}$, where ${\ensuremath{\lambda}}^{\mathrm{*}}$ represents the coupling constant threshold. As a result, the Pad\'e sums afford accurate approximations to E(\ensuremath{\lambda}) in this domain. The continued-fraction representations to these perturbation series have been accurately calculated to large (100th) order and demonstrate a curious ``quasioscillatory,'' but non-Stieltjes, behavior. Accurate values of E(\ensuremath{\lambda}) as well as ${\ensuremath{\lambda}}^{\mathrm{*}}$ for the 1s, 2s, and 2p levels are reported.

On coupling constant thresholds and related eigenvalue properties of Dirac operators.

Article On coupling constant thresholds and related eigenvalue properties of Dirac operators. was published on January 1, 1985 in the journal Journal für die reine und angewandte Mathematik (volume 1985, issue 362).

Coupling constant thresholds in nonrelativistic quantum mechanics. I. Short-range two-body case

Consider − Δ + λV with V short range at a value λ 0 where some eigenvalue e( λ) → 0 as λ ↓ λ 0. We analyze two questions: (i) What is the leading order of e( λ), i.e., for what α does e( λ) ∼ c( λ − λ 0) α ? (ii) Is e( λ) analytic at λ = λ 0 and, if not, what is the natural expansion parameter? The results are highly dimension dependent.

Coupling constant thresholds in nonrelativistic quantum mechanics

We extend the analysis of absorbtion of eigenvalues for the two body case to situations where absorbtion occurs at a two cluster threshold in anN-body system. The result depends on a Birman-Schwinger kernel for such anN-body system, an object which we apply in other ways. In particular, we control the number of discrete eigenvalues in the ħ→0 limit.

Coupling constant thresholds in nonrelativistic quantum mechanics. I. Short-range two-body case

The problem on mapping between two Lagrangian descriptions (using a commuting c-number spinor ψα or anticommuting pseudovector ξμ and pseudoscalar ξ5 variables) of the spin degrees of freedom of a color spinning massive particle interacting with background non-Abelian gauge field, is considered. A general analysis of the mapping between a pair of Majorana spinors (ψα,θα) (θα is some auxiliary anticommuting spinor) and a real anticommuting tensor aggregate (S,Vμ,Tμν⁎,Aμ,P), is presented. A complete system of bilinear relations between the tensor quantities, is obtained. The analysis we have given is used for the above problem of the equivalence of two different ways of describing the spin degrees of freedom of the relativistic particle. The mapping of the kinetic term (iħ/2)(θ¯θ)(ψ¯˙ψ−ψ¯ψ˙), the term (1/e)(θ¯θ)x˙μ(ψ¯γμψ) that provides a couple of the spinning variable ψ and the particle velocity x˙μ, and the interaction term ħ(θ¯θ)QaFμνa(ψ¯σμνψ) with an external non-Abelian gauge field, are considered in detail. In the former case a corresponding system of bilinear identities including both the tensor variables and their derivatives (S˙,V˙μ,T˙μν⁎,A˙μ,P˙), is defined. A detailed analysis of the local bosonic symmetry of the Lagrangian with the commuting spinor ψα, is carried out. A connection of this symmetry with the local SUSY transformation of the Lagrangian containing anticommuting pseudovector and pseudoscalar variables, is considered. The approach of obtaining a supersymmetric Lagrangian in terms of the even ψα and odd θα spinors, is offered.