Levinson's Theorem Research Articles

Pion dissociation and Levinson’s theorem in hot PNJL quark matter

Pion dissociation by the Mott effect in quark plasma is described within the generalized Beth-Uhlenbeck approach on the basis of the PNJL model which allows for a unified description of bound, resonant and scattering states. As a first approximation, we utilize the Breit-Wigner ansatz for the spectral function and clarify its relation to the complex mass pole solution of the pion Bethe-Salpeter equation. Application of the Levinson theorem proves that describing the pion Mott dissociation solely by means of spectral broadening of the pion bound state beyond T_Mott leaves out a significant aspect. Thus we acknowledge the importance of the continuum of scattering states and show its role for the thermodynamics of pion dissociation.

Asymptotic Solutions of nth Order Dynamic Equation and Oscillations

We establish a new asymptotic theorem for the nth order nonautonomous dynamic equation by its transformation to the almost diagonal system and applying Levinson's asymptotic theorem. Our transformation is given in the terms of unknown phase functions and is chosen in such a way that the entries of the perturbation matrix are the weighted characteristic functions. The characteristic function is defined in the terms of the phase functions and their choice is exible. Further applying this asymptotic theorem we prove the new oscillation and nonoscillation theorems for the solutions of the nth order linear nonautonomous differential equation with complex-valued coefficients. We show that the existence of the oscillatory solutions is connected with the existence of the special pairs of phase functions.

A discrete Levinson theorem for systems with singular limit and estimates of generalized eigenvectors of some Jacobi operators

We prove a discrete Levinson type theorem on asymptotic properties of solutions of discrete d-dimensional systems with being bounded variation sequence and with the perturbation being ‘appropriately small’, for the case when has zero eigenvalue. We use this result to obtain estimates for some generalized eigenvectors of Jacobi operators. In particular we get estimates of the eigenvectors decay for Jacobi matrices with ‘shaky weights’.

New Expressions for the Wave Operators of Schrödinger Operators in $${\mathbb{R}^3}$$ R 3

We prove new and explicit formulas for the wave operators of Schroedinger operators in R^3. These formulas put into light the very special role played by the generator of dilations and validate the topological approach of Levinson's theorem introduced in a previous publication. Our results hold for general (not spherically symmetric) potentials decaying fast enough at infinity, without any assumption on the absence of eigenvalue or resonance at 0-energy.

Casimir energy for a coupled fermion-kink system and its stability

We compute the Casimir energy for a system consisting of a fermion and a pseudoscalar field in the form of a prescribed kink. This model is not exactly solvable and we use the phase shift method to compute the Casimir energy. We use the relaxation method to find the bound states and the Runge-Kutta-Fehlberg method to obtain the scattering wavefunctions of the fermion in the whole interval of $x$. The resulting phase shifts are consistent with the weak and strong forms of the Levinson theorem. Then, we compute and plot the Casimir energy as a function of the parameters of the pseudoscalar field, i.e. the slope of $\phi(x)$ at x=0 ($\mu$) and the value of $\phi(x)$ at infinity ($\theta_0$). In the graph of the Casimir energy as a function of $\mu$ there is a sharp maximum occurring when the fermion bound state energy crosses the line of E=0. Furthermore, this graph shows that the Casimir energy goes to zero for $\mu\rightarrow 0$, and also for $\mu\rightarrow \infty$ when $\theta_0$ is an integer multiple of $\pi$. Moreover, the graph of the Casimir energy as a function of $\theta_0$ shows that this energy is on the average an increasing function of $\theta_0$ and has a cusp whenever there is a zero fermionic mode. We finally compute the total energy of a system consisting of a valence fermion in the ground state. Most importantly, we show that this energy (the sum of the Casimir energy and the energy of the fermion) is minimum when the background field has winding number one, independent of the details of the background profile. Throughout the paper we compare our results with those of a simple exactly solvable model, where a piece-wise linear profile approximates the kink. We find that the kink is an almost reflectionless barrier for the fermions, within the context of our model.

High-energy analysis and Levinson's theorem for the selfadjoint matrix Schrödinger operator on the half line

The matrix Schrödinger equation with a selfadjoint matrix potential is considered on the half line with the general selfadjoint boundary condition at the origin. When the matrix potential is integrable, the high-energy asymptotics are established for the related Jost matrix, the inverse of the Jost matrix, and the scattering matrix. Under the additional assumption that the matrix potential has a first moment, Levinson's theorem is derived, relating the number of bound states to the change in the argument of the determinant of the scattering matrix.

Levinson's theorem for graphs II

We prove Levinson's theorem for scattering on an (m+n)-vertex graph with n semi-infinite paths each attached to a different vertex, generalizing a previous result for the case n = 1. This theorem counts the number of bound states in terms of the winding of the determinant of the S-matrix. We also provide a proof that the bound states and incoming scattering states of the Hamiltonian together form a complete basis for the Hilbert space, generalizing another result for the case n = 1.

SCATTERING THEORY FOR LATTICE OPERATORS IN DIMENSION d ≥ 3

This paper analyzes the scattering theory for periodic tight-binding Hamiltonians perturbed by a finite range impurity. The classical energy gradient flow is used to construct a conjugate (or dilation) operator to the unperturbed Hamiltonian. For dimension d ≥ 3, the wave operator is given by an explicit formula in terms of this dilation operator, the free resolvent and the perturbation. From this formula, the scattering and time delay operators can be read off. Using the index theorem approach, a Levinson theorem is proved which also holds in the presence of embedded eigenvalues and threshold singularities.

ON THE WAVE OPERATORS AND LEVINSON'S THEOREM FOR POTENTIAL SCATTERING IN ℝ3

The paper is a presentation of recent investigations on potential scattering in ℝ3. We advocate a new formula for the wave operators and deduce the various outcomes that follow from this formula. A topological version of Levinson's theorem is proposed by interpreting it as an index theorem.

Parametric sensitivity analysis for the helium dimers on a model potential

Potential parameters sensitivity analysis for helium unlike molecules, HeNe, HeAr, HeKr and HeXe is the subject of this work. Number of bound states these rare gas dimers can support, for different angular momentum, will be presented and discussed. The variable phase method, together with the Levinson's theorem, is used to explore the quantum scattering process at very low collision energy using the Tang and Toennies potential. These diatomic dimers can support a bound state even for relative angular momentum equal to five, as in HeXe. Vibrational excited states, with zero angular momentum, are also possible for HeKr and HeXe. Results from sensitive analysis will give acceptable order of magnitude on potentials parameters.

Asymptotic behaviour of solutions of the difference Schrödinger equation

We use the method of averaging and the discrete analogue of Levinson's theorem to construct the asymptotics for solutions of the difference Schrödinger equation. Moreover, we present the general form for the averaging change of variable.

Spectral analysis of higher order differential operators with unbounded coefficients

AbstractHigher even order linear differential operators with unbounded coefficients are studied. For these operators the eigenvalues of the characteristic polynomials fall into distinct classes or clusters. Consequently the spectral properties, deficiency indices and spectra, of the underlying differential operators are superpositions of the contributions from the individual clusters. These results are based on a quantitative improvement of Levinson's Theorem. Our methods will also be applicable to other classes of linear differential operators.

Levinson's theorem for graphs

We prove an analog of Levinson's theorem for scattering on a weighted (m + 1)-vertex graph with a semi-infinite path attached to one of its vertices. In particular, we show that the number of bound states in such a scattering problem is equal to m minus half the winding number of the phase of the reflection coefficient (where each so-called half-bound state is counted as half a bound state).

Asymptotic integration of nonoscillatory differential equations: a unified approach

We consider the equation [r(t)x?]??+?f(t)x?=?0 as a perturbation of the equation [r(t)y?]??+?g(t)y?=?0, where the latter is assumed to be nonoscillatory at infinity. The functions r and g are real-valued, r is positive, and f is complex-valued. The problem of the asymptotic integration of the perturbed equation in comparison with solutions of the unperturbed equation has been studied by many mathematicians, including Hartman and Wintner, Trench, ?Sim?sa, Chen, and Chernyavskaya and Shuster. Here we apply a unified approach. Working in a matrix setting, we use preliminary and so-called conditioning transformations to bring the system in the form $$ \vec{z} = \left[ {\Lambda (t) + R(t)} \right]\vec{z} $$ , where ? is a certain diagonal matrix and R is an absolutely integrable perturbation. This allows us to use Levinson's fundamental theorem to find the asymptotic behavior of solutions and, in addition, to estimate the error involved. This method allows us to derive these known results in a more unified setting and to weaken the hypotheses in some instances.

Levinson's theorem and higher degree traces for Aharonov-Bohm operators

We study Levinson-type theorems for the family of Aharonov-Bohm models from different perspectives. The first one is purely analytical involving the explicit calculation of the wave-operators and allowing to determine precisely the various contributions to the left hand side of Levinson's theorem, namely, those due to the scattering operator, the terms at 0-energy and at energy +∞. The second one is based on non-commutative topology revealing the topological nature of Levinson's theorem. We then include the parameters of the family into the topological description obtaining a new type of Levinson's theorem, a higher degree Levinson's theorem. In this context, the Chern number of a bundle defined by a family of projections on bound states is explicitly computed and related to the result of a 3-trace applied on the scattering part of the model.

A Borg–Levinson Theorem for Higher Order Elliptic Operators

We establish the Borg–Levinson theorem for elliptic operators of higher order with constant coefficients. The case of incomplete spectral data is also considered.

On the solution of the relativistic quasipotential equation for the superposition of a local potential and of the sum of nonlocal separable quasipotentials

A solution has been found for the finite-difference quasipotential equation with the total quasi-potential that describes the interaction of two relativistic spinless particles of unequal mass. The total interaction, which is the superposition of a local potential and of the sum of nonlocal separable quasi-potentials, is centrally symmetric, does not depend on energy, and admits existence of true bound states. The treatment has been performed in the context of the relativistic quasipotential approach to quantum field theory. Exact expressions for the phase shift increments have been found and their properties have been investigated, the conditions for existence of bound states have been determined, and a generalization of the Levinson theorem is given.

Asymptotics Near ±m of the Spectral Shift Function for Dirac Operators with Non-Constant Magnetic Fields

We consider a 3-dimensional Dirac operator H 0 with non-constant magnetic field of constant direction, perturbed by a sign-definite matrix-valued potential V decaying fast enough at infinity. Then we determine asymptotics, as the energy goes to +m and −m, of the spectral shift function for the pair (H 0 + V, H 0). We obtain, as a by-product, a generalized version of Levinson's Theorem relating the eigenvalues asymptotics of H 0 + V near +m and −m to the scattering phase shift for the pair (H 0 + V, H 0).

Asymptotic Behavior of Generalized Eigenvectors of Jacobi Matrices in the Critical (“Double Root”) Case

This paper is concerned with asymptotic behavior of generalized eigenvectors of a class of Hermitian Jacobi matrices J in the critical case. The last means that the fraction \frac {q_n} {λ_n} generated by the diagonal entries q_n of J and its subdiagonal elements λ_n has the limit ±2 . In other words, the limit transfer matrix as n → ∞ contains a Jordan box (= double root in terms of Birkhoff–Adams theory). This is the situation where the asymptotic Levinson theorem does not work and one has to elaborate more special methods for asymptotic analysis. It should be mentioned that the critical case exactly corresponds to spectral phase transition phenomena, where the spectral structure changes dramatically (from discreet spectrum to pure absolutely continuous one) whenever the parameters in matrix entries cross singular surfaces, see J. Janas and S. Naboko [Spectral properties of selfadjoint Jacobi matrices coming from birth and death processes, Oper. Theory Adv. Appl. 127 (2001), pp. 387–397]. A Jordan box is the limit transfer matrix for all values of the spectral parameter λ simultaneously, it describes the “moment” of spectral phase transition. An application to the case of λ_n = n^α_(1 + r_n) , q_n = −2n^α (1 + p_n) with small perturbations r_n, p_n and α\in (0, 1] is studied.

A new look at an energetic (e,2e) reaction: Binary versus recoil

We analyze the recoil-to-binary peak ratio in an energetic (e,2e) reaction on the valence ns sub-shell of noble gas atoms. Dramatic qualitative change in this ratio dependence on the ejected electron energy can be explained by variation of reflectivity of the short-range Hartree-Fock potential. The reflectivity increases profoundly from lighter (He) to heavier (Ne and Ar) noble gas atoms because of modification of the scattering phases due to occupation of the target p orbitals (Levinson theorem). This effect is further modified due to strong inter-shell correlations in Ar. These theoretical predictions are confirmed experimentally.