Point interactions in acoustics: one dimensional models

A one dimensional system made up of a compressible fluid and several mechanical oscillators, coupled to the acoustic field in the fluid, is analyzed for different settings of the oscillators array. The dynamical models are formulated in terms of singular perturbations of the decoupled dynamics of the acoustic field and the mechanical oscillators. Detailed spectral properties of the generators of the dynamics are given for each model we consider. In the case of a periodic array of mechanical oscillators it is shown that the energy spectrum presents a band structure.Comment: revised version, 30 pages, 2 figure

Spectral Properties and Linear Stability of Self-Similar Wave Maps

We study co--rotational wave maps from $(3+1)$--Minkowski space to the three--sphere $S^3$. It is known that there exists a countable family $\{f_n\}$ of self--similar solutions. We investigate their stability under linear perturbations by operator theoretic methods. To this end we study the spectra of the perturbation operators, prove well--posedness of the corresponding linear Cauchy problem and deduce a growth estimate for solutions. Finally, we study perturbations of the limiting solution which is obtained from $f_n$ by letting $n \to \infty$.Comment: Some extensions added to match the published versio

Jahn-Teller effect versus Hund's rule coupling in C60N-

We propose variational states for the ground state and the low-energy collective rotator excitations in negatively charged C60N- ions (N=1...5). The approach includes the linear electron-phonon coupling and the Coulomb interaction on the same level. The electron-phonon coupling is treated within the effective mode approximation (EMA) which yields the linear t_{1u} x H_g Jahn-Teller problem whereas the Coulomb interaction gives rise to Hund's rule coupling for N=2,3,4. The Hamiltonian has accidental SO(3) symmetry which allows an elegant formulation in terms of angular momenta. Trial states are constructed from coherent states and using projection operators onto angular momentum subspaces which results in good variational states for the complete parameter range. The evaluation of the corresponding energies is to a large extent analytical. We use the approach for a detailed analysis of the competition between Jahn-Teller effect and Hund's rule coupling, which determines the spin state for N=2,3,4. We calculate the low-spin/high-spin gap for N=2,3,4 as a function of the Hund's rule coupling constant J. We find that the experimentally measured gaps suggest a coupling constant in the range J=60-80meV. Using a finite value for J, we recalculate the ground state energies of the C60N- ions and find that the Jahn-Teller energy gain is partly counterbalanced by the Hund's rule coupling. In particular, the ground state energies for N=2,3,4 are almost equal

Static and dynamic Jahn-Teller effect in the alkali metal fulleride salts A4C60 (A = K, Rb, Cs)

We report the temperature dependent mid- and near-infrared spectra of K4C60, Rb4C60 and Cs4C60. The splitting of the vibrational and electronic transitions indicates a molecular symmetry change of C604- which brings the fulleride anion from D2h to either a D3d or a D5d distortion. In contrast to Cs4C60, low temperature neutron diffraction measurements did not reveal a structural phase transition in either K4C60 and Rb4C60. This proves that the molecular transition is driven by the molecular Jahn-Teller effect, which overrides the distorting potential field of the surrounding cations at high temperature. In K4C60 and Rb4C60 we suggest a transition from a static to a dynamic Jahn-Teller state without changing the average structure. We studied the librations of these two fullerides by temperature dependent inelastic neutron scattering and conclude that both pseudorotation and jump reorientation are present in the dynamic Jahn-Teller state.Comment: 13 pages, 10 figures, to be published in Phys. Rev.

Multiple classical limits in relativistic and nonrelativistic quantum mechanics

The existence of a classical limit describing interacting particles in a second-quantized theory of identical particles with bosonic symmetry is proved. This limit exists in addition to a previously established classical limit with a classical field behavior, showing that the limit $\hbar \to 0$ of the theory is not unique. An analogous result is valid for a free massive scalar field: two distinct classical limits are proved to exist, describing a system of particles or a classical field. The introduction of local operators in order to represent kinematical properties of interest is shown to break the permutation symmetry under some localizability conditions, allowing the study of individual particle properties.Comment: 13 page

Local energy decay of massive Dirac fields in the 5D Myers-Perry metric

We consider massive Dirac fields evolving in the exterior region of a 5-dimensional Myers-Perry black hole and study their propagation properties. Our main result states that the local energy of such fields decays in a weak sense at late times. We obtain this result in two steps: first, using the separability of the Dirac equation, we prove the absence of a pure point spectrum for the corresponding Dirac operator; second, using a new form of the equation adapted to the local rotations of the black hole, we show by a Mourre theory argument that the spectrum is absolutely continuous. This leads directly to our main result.Comment: 40 page

Cosmic recall and the scattering picture of Loop Quantum Cosmology

The global dynamics of a homogeneous universe in Loop Quantum Cosmology is viewed as a scattering process of its geometrodynamical equivalent. This picture is applied to build a flexible (easy to generalize) and not restricted just to exactly solvable models method of verifying the preservation of the semiclassicality through the bounce. The devised method is next applied to two simple examples: (i) the isotropic Friedman Robertson Walker universe, and (ii) the isotropic sector of the Bianchi I model. For both of them we show, that the dispersions in the logarithm of the volume ln(v) and scalar field momentum ln(p_phi) in the distant future and past are related via strong triangle inequalities. This implies in particular a strict preservation of the semiclassicality (in considered degrees of freedom) in both the cases (i) and (ii). Derived inequalities are general: valid for all the physical states within the considered models.Comment: RevTex4, 19 pages, 3 figure

Singular factorizations, self-adjoint extensions, and applications to quantum many-body physics

We study self-adjoint operators defined by factorizing second order differential operators in first order ones. We discuss examples where such factorizations introduce singular interactions into simple quantum mechanical models like the harmonic oscillator or the free particle on the circle. The generalization of these examples to the many-body case yields quantum models of distinguishable and interacting particles in one dimensions which can be solved explicitly and by simple means. Our considerations lead us to a simple method to construct exactly solvable quantum many-body systems of Calogero-Sutherland type.Comment: 17 pages, LaTe

Quantum healing of classical singularities in power-law spacetimes

We study a broad class of spacetimes whose metric coefficients reduce to powers of a radius r in the limit of small r. Among these four-parameter "power-law" metrics we identify those parameters for which the spacetimes have classical singularities as r approaches 0. We show that a large set of such classically singular spacetimes is nevertheless nonsingular quantum mechanically, in that the Hamiltonian operator is essentially self-adjoint, so that the evolution of quantum wave packets lacks the ambiguity associated with scattering off singularities. Using these metrics, the broadest class yet studied to compare classical with quantum singularities, we explore the physical reasons why some that are singular classically are "healed" quantum mechanically, while others are not. We show that most (but not all) of the remaining quantum-mechanically singular spacetimes can be excluded if either the weak energy condition or the dominant energy condition is invoked, and we briefly discuss the effect of this work on the strong cosmic censorship hypothesis.Comment: 14 pages, 1 figure; extensive revision