Arbitrary Positive Energy Research Articles

General Theory of Constructing Potential with Bound States in the Continuum

Abstract We present a general theory of potentials that support bound states at positive energies (bound states in the continuum). On the theoretical side, we prove that, for systems described by nonlocal potentials of the form $V(r,r^{\prime })$, bound states at positive energies are as common as those at negative energies. At the same time, we show that a local potential of the form $V(r)$ rarely supports a positive-energy bound state. On the practical side, we show how to construct a (naturally nonlocal) potential that supports an arbitrary normalizable state at an arbitrary positive energy. We demonstrate our theory with numerical examples both in momentum and coordinate spaces with emphasis on the important role played by nonlocal potentials. Finally, we discuss how to observe bound states at positive energies, and where to search for nonlocal potentials that may support them.

Theoretical and numerical study of the blow up in a nonlinear viscoelastic problem with variable-exponent and arbitrary positive energy

Theoretical and numerical study of the blow up in a nonlinear viscoelastic problem with variable-exponent and arbitrary positive energy

Blow-up analysis of a nonlinear pseudo-parabolic equation with memory term

This paper deals with the blow-up phenomena for a nonlinear pseudo-parabolic equation with a memory term $u_{t}-\triangle{u}-\triangle{u}_{t}+\int_{0}^{t}g(t-\tau)\triangle{u}(\tau)d\tau = |{u}|^{p}{u}$ in a bounded domain, with the initial and Dirichlet boundary conditions. We first obtain the finite time blow-up results for the solutions with initial data at non-positive energy level as well as arbitrary positive energy level, and give some upper bounds for the blow-up time $T^{*}$ depending on the sign and size of initial energy $E(0)$. In addition, a lower bound for the life span $T^{*}$ is derived by means of a differential inequality technique if blow-up does occur.

Black hole mining in the RST model

We consider the possibility of mining black holes in the 1 + 1-dimensional dilaton gravity model of Russo, Susskind and Thorlacius. The model correctly incorporates Hawking radiation and back-reaction in a semiclassical expansion in 1/N, where N is the number of matter species. It is shown that the lifetime of a perturbed black hole is independent of the addition of any extra apparatus when realized by an arbitrary positive energy matter source. We conclude that mining does not occur in the RST model and comment on the implications of this for the black hole information paradox.

Blowup of the solution to the Cauchy problem with arbitrary positive energy for a system of Klein–Gordon equations in the de Sitter metric

The ϕ4 model of a scalar (complex) field in the metric of an expanding universe, namely, in the de Sitter metric is considered. The initial energy of the system can have an arbitrarily high positive value. Sufficient conditions for solution blowup in a finite time are obtained. The existence of blowup is proved by applying H.A. Levine’s modified method is used.

Finite time blow up of the solutions to boussinesq equation with linear restoring force and arbitrary positive energy

Finite time blow up of the solutions to Boussinesq equation with linear restoring force and combined power nonlinearities is studied. Sufficient conditions on the initial data for nonexistence of global solutions are derived. The results are valid for initial data with arbitrary high positive energy. The proofs are based on the concave method and new sign preserving functionals.

A blowing up wave equation with exponential type nonlinearity and arbitrary positive energy

We consider in two space dimensions, the nonexistence of global solutions to the class of Klein–Gordon equations∂ttu−Δu+u=fa(u),(t,x)∈[0,T⁎)×Ω. We give sufficient conditions on the data such that the solution of the above Klein–Gordon equation blows up in finite time.

New Model for the Study of Liquid–Vapor Phase Transitions

A new model is proposed for the study of the liquid–vapor phase transition. The potential energy of a given configuration of N molecules is defined by U(r1, ···, rN) = [W(r1, ···, rN) − Nυ0]ε/υ0≤0, where W is the volume covered by N interpenetrating spheres each of volume υ0 and each centered on one molecule, and where ε is an arbitrary positive energy. This continuum model proves to have a line of symmetry comparable with those found hitherto only in lattice models. The line is that of the diameter, or mean orthobaric density ρ = 12ρ1 + 12ρg, below the critical point, and continues through the one-phase region to infinite temperature. The existence of this line allows some of the properties to be obtained explicitly, the most unusual of which is that the diameter has a singularity comparable with that in Cυ; hence the law of the rectilinear diameter is not obeyed. An exact solution of the model is obtained in one dimension, in which there is no phase transition, and a mean-field solution in three dimensions. The latter preserves the symmetry. The model is shown to be thermodynamically equivalent to a two-component system in which the pair potential between molecules of like species is zero, while that between molecules of unlike species implies a mutually excluded volume of υ0. In this transcription the symmetry of the model becomes obvious.