Some notes to extend the study on random non-autonomous second order linear differential equations appearing in Mathematical Modeling

The objective of this paper is to complete certain issues from our recent contribution [J. Calatayud, J.-C. Cort\'es, M. Jornet, L. Villafuerte, Random non-autonomous second order linear differential equations: mean square analytic solutions and their statistical properties, Advances in Difference Equations, 2018:392, 1--29 (2018)]. We restate the main theorem therein that deals with the homogeneous case, so that the hypotheses are clearer and also easier to check in applications. Another novelty is that we tackle the non-homogeneous equation with a theorem of existence of mean square analytic solution and a numerical example. We also prove the uniqueness of mean square solution via an habitual Lipschitz condition that extends the classical Picard Theorem to mean square calculus. In this manner, the study on general random non-autonomous second order linear differential equations with analytic data processes is completely resolved. Finally, we relate our exposition based on random power series with polynomial chaos expansions and the random differential transform method, being the latter a reformulation of our random Fr\"obenius method.Comment: 15 pages, 0 figures, 2 table

Commutation Relations for Unitary Operators

Let $U$ be a unitary operator defined on some infinite-dimensional complex Hilbert space ${\cal H}$. Under some suitable regularity assumptions, it is known that a local positive commutation relation between $U$ and an auxiliary self-adjoint operator $A$ defined on ${\cal H}$ allows to prove that the spectrum of $U$ has no singular continuous spectrum and a finite point spectrum, at least locally. We show that these conclusions still hold under weak regularity hypotheses and without any gap condition. As an application, we study the spectral properties of the Floquet operator associated to some perturbations of the quantum harmonic oscillator under resonant AC-Stark potential

Relativistic kinematics beyond Special Relativity

In the context of departures from Special Relativity written as a momentum power expansion in the inverse of an ultraviolet energy scale M, we derive the constraints that the relativity principle imposes between coefficients of a deformed composition law, dispersion relation, and transformation laws, at first order in the power expansion. In particular, we find that, at that order, the consistency of a modification of the energy-momentum composition law fixes the modification in the dispersion relation. We therefore obtain the most generic modification of Special Relativity that preserves the relativity principle at leading order in 1/M.Comment: Version with minor corrections, to appear in Phys. Rev.

Mechanical systems subjected to generalized nonholonomic constraints

We study mechanical systems subject to constraint functions that can be dependent at some points and independent at the rest. Such systems are modelled by means of generalized codistributions. We discuss how the constraint force can transmit an impulse to the motion at the points of dependence and derive an explicit formula to obtain the ``post-impact'' momentum in terms of the ``pre-impact'' momentum.Comment: 24 pages, no figure