Application of approximation theory by nonlinear manifolds in Sturm-Liouville inverse problems

We give here some negative results in Sturm-Liouville inverse theory, meaning that we cannot approach any of the potentials with $m+1$ integrable derivatives on $\mathbb{R}^+$ by an $\omega$-parametric analytic family better than order of $(\omega\ln\omega)^{-(m+1)}$. Next, we prove an estimation of the eigenvalues and characteristic values of a Sturm-Liouville operator and some properties of the solution of a certain integral equation. This allows us to deduce from [Henkin-Novikova] some positive results about the best reconstruction formula by giving an almost optimal formula of order of $\omega^{-m}$.Comment: 40 page

Conservation laws in the continuum 1/r21/r^2 systems

We study the conservation laws of both the classical and the quantum mechanical continuum $1/r^2$ type systems. For the classical case, we introduce new integrals of motion along the recent ideas of Shastry and Sutherland (SS), supplementing the usual integrals of motion constructed much earlier by Moser. We show by explicit construction that one set of integrals can be related algebraically to the other. The difference of these two sets of integrals then gives rise to yet another complete set of integrals of motion. For the quantum case, we first need to resum the integrals proposed by Calogero, Marchioro and Ragnisco. We give a diagrammatic construction scheme for these new integrals, which are the quantum analogues of the classical traces. Again we show that there is a relationship between these new integrals and the quantum integrals of SS by explicit construction.Comment: 19 RevTeX 3.0 pages with 2 PS-figures include

On asymptotic stability of the Skyrmion

We study the asymptotic behavior of spherically symmetric solutions in the Skyrme model. We show that the relaxation to the degree-one soliton (called the Skyrmion) has a universal form of a superposition of two effects: exponentially damped oscillations (the quasinormal ringing) and a power law decay (the tail). The quasinormal ringing, which dominates the dynamics for intermediate times, is a linear resonance effect. In contrast, the polynomial tail, which becomes uncovered at late times, is shown to be a \emph{nonlinear} phenomenon.Comment: 4 pages, 4 figures, minor changes to match the PRD versio

Induced Time-Reversal Symmetry Breaking Observed in Microwave Billiards

Using reciprocity, we investigate the breaking of time-reversal (T) symmetry due to a ferrite embedded in a flat microwave billiard. Transmission spectra of isolated single resonances are not sensitive to T-violation whereas those of pairs of nearly degenerate resonances do depend on the direction of time. For their theoretical description a scattering matrix model from nuclear physics is used. The T-violating matrix elements of the effective Hamiltonian for the microwave billiard with the embedded ferrite are determined experimentally as functions of the magnetization of the ferrite.Comment: 4 pages, 4 figure

Solution of Some Integrable One-Dimensional Quantum Systems

In this paper, we investigate a family of one-dimensional multi-component quantum many-body systems. The interaction is an exchange interaction based on the familiar family of integrable systems which includes the inverse square potential. We show these systems to be integrable, and exploit this integrability to completely determine the spectrum including degeneracy, and thus the thermodynamics. The periodic inverse square case is worked out explicitly. Next, we show that in the limit of strong interaction the "spin" degrees of freedom decouple. Taking this limit for our example, we obtain a complete solution to a lattice system introduced recently by Shastry, and Haldane; our solution reproduces the numerical results. Finally, we emphasize the simple explanation for the high multiplicities found in this model

Efficient Algorithm for Asymptotics-Based Configuration-Interaction Methods and Electronic Structure of Transition Metal Atoms

Asymptotics-based configuration-interaction (CI) methods [G. Friesecke and B. D. Goddard, Multiscale Model. Simul. 7, 1876 (2009)] are a class of CI methods for atoms which reproduce, at fixed finite subspace dimension, the exact Schr\"odinger eigenstates in the limit of fixed electron number and large nuclear charge. Here we develop, implement, and apply to 3d transition metal atoms an efficient and accurate algorithm for asymptotics-based CI. Efficiency gains come from exact (symbolic) decomposition of the CI space into irreducible symmetry subspaces at essentially linear computational cost in the number of radial subshells with fixed angular momentum, use of reduced density matrices in order to avoid having to store wavefunctions, and use of Slater-type orbitals (STO's). The required Coulomb integrals for STO's are evaluated in closed form, with the help of Hankel matrices, Fourier analysis, and residue calculus. Applications to 3d transition metal atoms are in good agreement with experimental data. In particular we reproduce the anomalous magnetic moment and orbital filling of Chromium in the otherwise regular series Ca, Sc, Ti, V, Cr.Comment: 14 pages, 1 figur

On linearization of super sine-Gordon equation

Two sets of super Riccati equations are presented which result in two linear problems of super sine-Gordon equation. The linear problems are then shown to be related to each other by a super gauge transformation and to the super B\"{a}cklund transformation of the equation.Comment: 9 Page

Integrable equations in nonlinear geometrical optics

Geometrical optics limit of the Maxwell equations for nonlinear media with the Cole-Cole dependence of dielectric function and magnetic permeability on the frequency is considered. It is shown that for media with slow variation along one axis such a limit gives rise to the dispersionless Veselov-Novikov equation for the refractive index. It is demonstrated that the Veselov-Novikov hierarchy is amenable to the quasiclassical DBAR-dressing method. Under more specific requirements for the media, one gets the dispersionless Kadomtsev-Petviashvili equation. Geometrical optics interpretation of some solutions of the above equations is discussed.Comment: 33 pages, 7 figure

Fibre bundle formulation of nonrelativistic quantum mechanics. III. Pictures and integrals of motion

We propose a new systematic fibre bundle formulation of nonrelativistic quantum mechanics. The new form of the theory is equivalent to the usual one but it is in harmony with the modern trends in theoretical physics and potentially admits new generalizations in different directions. In it a pure state of some quantum system is described by a state section (along paths) of a (Hilbert) fibre bundle. It's evolution is determined through the bundle (analogue of the) Schr\"odinger equation. Now the dynamical variables and the density operator are described via bundle morphisms (along paths). The mentioned quantities are connected by a number of relations derived in this work. In this third part of our series we investigate the bundle analogues of the conventional pictures of motion. In particular, there are found the state sections and bundle morphisms corresponding to state vectors and observables respectively. The equations of motion for these quantities are derived too. Using the results obtained, we consider from the bundle view-point problems concerning the integrals of motion. An invariant (bundle) necessary and sufficient conditions for a dynamical variable to be an integral of motion are found.Comment: 19 standard (11pt, A4) LaTeX 2e pages. The packages AMS-LaTeX and amsfonts are required. New references and comments are added. Minor style chages. Continuation of quant-ph/9803083, quant-ph/9803084 and quant-ph/9804062. For continuation of the series view http://www.inrne.bas.bg/mathmod/bozhome

Conditional symmetries and Riemann invariants for inhomogeneous hydrodynamic-type systems

A new approach to the solution of quasilinear nonelliptic first-order systems of inhomogeneous PDEs in many dimensions is presented. It is based on a version of the conditional symmetry and Riemann invariant methods. We discuss in detail the necessary and sufficient conditions for the existence of rank-2 and rank-3 solutions expressible in terms of Riemann invariants. We perform the analysis using the Cayley-Hamilton theorem for a certain algebraic system associated with the initial system. The problem of finding such solutions has been reduced to expanding a set of trace conditions on wave vectors and their profiles which are expressible in terms of Riemann invariants. A couple of theorems useful for the construction of such solutions are given. These theoretical considerations are illustrated by the example of inhomogeneous equations of fluid dynamics which describe motion of an ideal fluid subjected to gravitational and Coriolis forces. Several new rank-2 solutions are obtained. Some physical interpretation of these results is given.Comment: 19 page