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Authors
Alberto Torchinsky
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We solve the Cauchy problem for the $n$-dimensional wave equation using elementary properties of the Bessel functions.
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We solve the Cauchy problem for the $n$-dimensional wave equation using elementary properties of the Fourier transform.
We revisit two classical formulas for the Bessel function of the first kind, due to von Lommel and Weber-Schafheitlin, in a probabilistic setting. The von Lommel formula exhibits a family of solutions to the van Dantzig problem involving the generalized semi-circular distributions and the first hitting times of a Bessel process with positive parameter, whereas the Weber-Schafheitlin formula allows one to construct non-trivial moments of Gamma type having a signed spectral measure. Along the way, we observe that the Weber-Schafheitlin formula is a simple consequence of the von Lommel formula, the Fresnel integral and the Selberg integral.
The use of the umbral formalism allows a significant simplification of the derivation of sum rules involving products of special functions and polynomials. We rederive in this way known sum rules and addition theorems for Bessel functions. Furthermore, we obtain a set of new closed form sum rules involving various special polynomials and Bessel functions. The examples we consider are relevant for applications ranging from plasma physics to quantum optics.
We study the Cauchy problem for the 3D Gross-Pitaevskii equation. The global well-posedness in the natural energy space was proved by Gerard cite{Gerard}. In this paper we prove scattering for small data in the same space with some additional angular regularity, and in particular in the radial case we obtain small energy scattering.
We consider the $L_t^2L_x^r$ estimates for the solutions to the wave and Schrodinger equations in high dimensions. For the homogeneous estimates, we show $L_t^2L_x^infty$ estimates fail at the critical regularity in high dimensions by using stable Levy process in $R^d$. Moreover, we show that some spherically averaged $L_t^2L_x^infty$ estimate holds at the critical regularity. As a by-product we obtain Strichartz estimates with angular smoothing effect. For the inhomogeneous estimates, we prove double $L_t^2$-type estimates.
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