The structure of naked singularity in self-similar gravitational collapse: II

Generalizing the results of Joshi and Dwivedi inComm. Math. Phys. 146, 333 (1992), it is pointed out that strong curvature naked singularities could occur in the self-similar gravitational collapse of any form of matter satisfying the weak energy condition for the positivity of mass-energy density.     

Symmetries of the heat equation on the lattice

Discrete versions of the heat equation on two-dimensional uniform lattices are shown to possess the same symmetry algebra as their continuum limits. Solutions with definite symmetry properties are presented.     

Action variables for Toda Lattices

This Letter contains constructions of complex action variables for both the full Kostant-Toda Lattice in sl(n, ) and the generalized nonperiodic tridiagonal Toda lattice associated to an arbitrary complex semisimple Lie algebra g. The main tool is the explicit factorization solution for certain Hamiltonian flows. The Letter also contains a generalization of the standard factorization solution theorem necessary for the analysis of the full Kostant-Toda lattice.     

KMS states for the Weyl algebra

We consider the possible automorphism groups for the Weyl algebra overR, resp.T, and classify those for which KMS states, unique or not unique, exist.     

Anisotropic Bianchi type V perfect fluid cosmological models in Lyra’s geometry

The law of variation for mean Hubble’s parameter with average scale factor, in an anisotropic Bianchi type V cosmological space–time, is discussed within the frame work of Lyra’s manifold. The variation of Hubble’s parameter, which gives a constant value of deceleration parameter, generates two types of solutions for the average scale factor; one is the power-law and the other one is of exponential form. Using these two forms, new classes of exact solutions of the field equations have been found for a Bianchi type V space–time filled with perfect fluid in Lyra’s geometry by considering a time-dependent displacement field. The physical and kinematical behaviors of the singular and non-singular models of the universe are examined. Exact expressions for look-back time, luminosity distance and event horizon versus redshift are also derived and their significance are discussed in detail. It has been observed that the solutions are compatible with the results of recent observations.     

The One-Dimensional Schrödinger–Newton Equations

We prove an existence and uniqueness result for ground states of one-dimensional Schrödinger–Newton equations.     

A general class of (essentially) iso-spectral perturbations

We consider (essentially) iso-spectral perturbations of operators of the formH _A=A ^* A withA being a densely defined closed linear operator from a Hilbert space to another Hilbert space . We perturbH _A by perturbingA asA+B withB being a linear operator from to . Two classes ofB are defined so as to obtain (essentially) iso-spectral perturbations ofH _A. The abstract results are applied to Schrödinger operators. Our approach gives also a mathematical unification for the so-called factorization method in quantum mechanics.     

Separation of acoustic waves in isentropic flow perturbations

The present contribution investigates the mechanisms of sound generation and propagation in the case of highly-unsteady flows. Based on the linearisation of the isentropic Navier-Stokes equation around a new pathline-averaged base flow, it is demonstrated for the first time that flow perturbations of a non-uniform flow can be split into acoustic and vorticity modes, with the acoustic modes being independent of the vorticity modes. Therefore, we can propose this acoustic perturbation as a general definition of sound.     

Covariant differential operators for densities and the KdV equation as a flow on diffeomorphism groups

Covariant differential operators of arbitrary order, which are analogues of Hill's operator and act on densities of weights , are constructed and the KdV equation is formulated as a flow on a diffeomorphism group.     

The conjugate operator method for strongly singular operators

In this Letter, we adapt the version of the conjugate operator method for Hamiltonians defined as quadratic forms developed by Boutet de Monvel-Berthier and Georgescu, to study a class of self-adjoint operators of the form , whereH is conjugate to a self-adjoint operatorA but itself is not. The spectral theory for such operators is considered and applications to strongly singular second-order operators as the wave propagators in inhomogeneous and stratified media are given.     

Perturbed periodic Hamiltonians: Essential Spectrum and exponential decay of eigenfunctions

Spectral properties of – +V(x), whereV(x) lies in a neighbourhood of the periodic case and describes various models of disorder, are studied. We prove the exponential decay of generalized eigenfunctions corresponding to energies in the resolvent set of the unperturbed periodic Hamiltonian, as well as the stability of the essential spectrum for the dislocation disorder in two dimensions.     

Adiabatic perturbations for compactons under dissipation and numerically-induced dissipation

Compacton propagation under dissipation shows amplitude damping and the generation of tails. The numerical simulation of compactons by means of dissipative schemes also show the same behaviors. The truncation error terms of a numerical method can be considered as a perturbation of the original partial differential equation and perturbation methods can be applied to its analysis. For dissipative schemes, or when artificial dissipation is added, the adiabatic perturbation method yields evolution equations for the amplitude loss in the numerical solution and the amplitude of the numerically-induced tails. In this paper, such methods are applied to the K(2,2)$K(2\text{,}2)$ Rosenau–Hyman equation, showing a very good agreement between perturbative and numerical results.     

Non-commutative functional calculus: Unbounded operators

In a recent work, Colombo (in press) [1], we developed a functional calculus for bounded operators defined on quaternionic Banach spaces. In this paper we show how the results from the above-mentioned work can be extended to the unbounded case, and we highlight the crucial differences between the two cases. In particular, we deduce a new eigenvalue equation, suitable for the construction of a functional calculus for operators whose spectrum is not necessarily real.     

On a variational-perturbational method for large perturbations

Using a perturbation theoretical argument an effective variational ansatz for the ground state is given. The formula works for arbitrary coupling strengths. The anharmonic oscillator is presented as an example and a comparison with other methods is made.     

On log-Sobolev inequalities for infinite lattice systems

For a system on an infinite lattice, we show that a Gibbs measure for a smooth local specification ={E } satisfying the Dobrushin uniqueness theorem also satisfies log-Sobolev inequality, provided it is satisfied for one-dimensional measures E _l .     

The operator method for perturbations in the theory of infrared molecular spectra

The application of contact transformations of Hamiltonians with a degenerate zeroth approximation to an approximate separation of variables is studied. Unitary transformations are formed that interrelate various effective Hamiltonians by the method of contact transformations of Van Vleck, Jorgensen, Pedersen, Primas, ENRST and Soliverez with an accuracy up to 0(⁵). The relation between the contact transformations and the perturbation theory of Rayleigh-Schrödinger is studied, and it is shown that the intermediate denominators in effective Hamiltonians vanish in a nondegenerate case. A generalized variant is constructed of contact transformations and of transformation yielding an even effective Hamiltonian from which the results of the above methods in particular cases follow.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, No. 7, pp. 82–90, July, 1977.The authors would like to thank V. N. Bryukhanov and P. I. Gaev for useful discussions.     

Computation of the resonant frequency of a dielectric resonator by a differential method

A differential method to compute, with high accuracy, the resonant frequency of a dielectric body with an axis of revolution and placed between two conducting plates is described. The propagation equations, projected on a Fourier basis, are integrated by the Adams-Moulton predictor-corrector algorithm and matched to the modified Bessel expansion of the field. Numerical results obtained for the cylinder shape are compared with those obtained by a modal theory, exhibiting an accuracy better than 10^–4. Results for a dielectric sphere are given as another application.     

A novel method for nonlinear fractional differential equations using symbolic computation

In this paper, a new approach, namely an ansatz method is applied to find exact solutions for nonlinear fractional differential equations in the sense of modified Riemann–Liouville derivative. Based on a nonlinear fractional complex transformation, a certain fractional partial differential equation can be turned into another ordinary differential equation of integer order. For illustrating the validity of this method, we apply it to solve the fractional-order biological population model and the space–time fractional modified equal width equation, and as a result, some dark soliton solutions for them are established.     

Localization for random potentials supported on a subspace

We consider the lattice Schrödinger operator acting onl ² ( ^d ) with random potential (independent, identically distributed random variables), supported on a subspace of dimension 1 v <d. We use the multiscale analyses scheme to prove that this operator exhibits exponential localization at the edges of the spectrum for any disorder or outside the interval [-2d, 2d] for sufficiently high disorder.