Helmholtz Solutions for the Fractional Laplacian and Other Related Operators

Abstract

We show that the bounded solutions to the fractional Helmholtz equation, $(-\Delta {)}^{s}u=u$ for $0<s<1$ in ${\mathbb{R}}^n$, are given by the bounded solutions to the classical Helmholtz equation $(-\Delta )u=u$ in ${\mathbb{R}}^n$ for $n\ge 2$ when $u$ is additionally assumed to be vanishing at $\infty$. When $n=1$, we show that the bounded fractional Helmholtz solutions are again given by the classical solutions $A\cos x+B\sin x$. We show that this classification of fractional Helmholtz solutions extends for $1<s\le 2$ and $s\in \mathbb{N}$ when $u\in C^{\infty }({\mathbb{R}}^n)$. Finally, we prove that the classical solutions are the unique bounded solutions to the more general equation $\psi (-\Delta )u=\psi (1)u$ in ${\mathbb{R}}^n$, when $\psi$ is complete Bernstein and certain regularity conditions are imposed on the associated weight $a(t)$.

Errors in the statements of Lemmas 2.2, 5,2 have been corrected, some corrected typos