On the geometry of the domain of the solution of nonlinear Cauchy problem

We consider the Cauchy problem for a second order quasi-linear partial differential equation with an admissible parabolic degeneration such that the given functions described the initial conditions are defined on a closed interval. We study also a variant of the inverse problem of the Cauchy problem and prove that the considered inverse problem has a solution under certain regularity condition. We illustrate the Cauchy and the inverse problems in some interesting examples such that the families of the characteristic curves have either common envelopes or singular points. In these cases the definition domain of the solution of the differential equation contains a gap.Comment: accepted for publication in the book Lie groups, differential equations and geometry in Springer Unip

Distributions of Conductance and Shot Noise and Associated Phase Transitions

For a chaotic cavity with two indentical leads each supporting N channels, we compute analytically, for large N, the full distribution of the conductance and the shot noise power and show that in both cases there is a central Gaussian region flanked on both sides by non-Gaussian tails. The distribution is weakly singular at the junction of Gaussian and non-Gaussian regimes, a direct consequence of two phase transitions in an associated Coulomb gas problem.Comment: 5 pages, 3 figures include

Fluctuations and control in the Vlasov-Poisson equation

In this paper we study the fluctuation spectrum of a linearized Vlasov-Poisson equation in the presence of a small external electric field. Conditions for the control of the linear fluctuations by an external electric field are established.Comment: 8 pages late

Nonlocal Gravity: Modified Poisson's Equation

The recent nonlocal generalization of Einstein's theory of gravitation reduces in the Newtonian regime to a nonlocal and nonlinear modification of Poisson's equation of Newtonian gravity. The nonlocally modified Poisson equation implies that nonlocality can simulate dark matter. Observational data regarding dark matter provide limited information about the functional form of the reciprocal kernel, from which the original nonlocal kernel of the theory must be determined. We study this inverse problem of nonlocal gravity in the linear domain, where the applicability of the Fourier transform method is critically examined and the conditions for the existence of the nonlocal kernel are discussed. This approach is illustrated via simple explicit examples for which the kernels are numerically evaluated. We then turn to a general discussion of the modified Poisson equation and present a formal solution of this equation via a successive approximation scheme. The treatment is specialized to the gravitational potential of a point mass, where in the linear regime we recover the Tohline-Kuhn approach to modified gravity.Comment: 27 pages, 4 figures; v2: minor improvements, accepted for publication in J. Math. Phy

Inversion of Gamow's Formula and Inverse Scattering

We present a pedagogical description of the inversion of Gamow's tunnelling formula and we compare it with the corresponding classical problem. We also discuss the issue of uniqueness in the solution and the result is compared with that obtained by the method of Gel'fand and Levitan. We hope that the article will be a valuable source to students who have studied classical mechanics and have some familiarity with quantum mechanics.Comment: LaTeX, 6 figurs in eps format. New abstract; notation in last equation has been correcte

Fredholm's Minors of Arbitrary Order: Their Representations as a Determinant of Resolvents and in Terms of Free Fermions and an Explicit Formula for Their Functional Derivative

We study the Fredholm minors associated with a Fredholm equation of the second type. We present a couple of new linear recursion relations involving the $n$th and $n-1$th minors, whose solution is a representation of the $n$th minor as an $n\times n$ determinant of resolvents. The latter is given a simple interpretation in terms of a path integral over non-interacting fermions. We also provide an explicit formula for the functional derivative of a Fredholm minor of order $n$ with respect to the kernel. Our formula is a linear combination of the $n$th and the $n\pm 1$th minors.Comment: 17 pages, Latex, no figures connection to supplementary compound matrices mentioned, references added, typos correcte

On hybrid states of two and three level atoms

We calculate atom-photon resonances in the Wigner-Weisskopf model, admitting two photons and choosing a particular coupling function. We also present a rough description of the set of resonances in a model for a three-level atom coupled to the photon field. We give a general picture of matter-field resonances these results fit into.Comment: 33 pages, 12 figure

Critical strength of attractive central potentials

We obtain several sequences of necessary and sufficient conditions for the existence of bound states applicable to attractive (purely negative) central potentials. These conditions yields several sequences of upper and lower limits on the critical value, $g_{\rm{c}}^{(\ell)}$, of the coupling constant (strength), $g$, of the potential, $V(r)=-g v(r)$, for which a first $\ell$-wave bound state appears, which converges to the exact critical value.Comment: 18 page

The target problem with evanescent subdiffusive traps

We calculate the survival probability of a stationary target in one dimension surrounded by diffusive or subdiffusive traps of time-dependent density. The survival probability of a target in the presence of traps of constant density is known to go to zero as a stretched exponential whose specific power is determined by the exponent that characterizes the motion of the traps. A density of traps that grows in time always leads to an asymptotically vanishing survival probability. Trap evanescence leads to a survival probability of the target that may be go to zero or to a finite value indicating a probability of eternal survival, depending on the way in which the traps disappear with time