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Helmholtz Solutions for the Fractional Laplacian and Other Related Operators

We show that the bounded solutions to the fractional Helmholtz equation, $(-Δ)^s u= u$ for $0<s<1$ in $\mathbb{R}^n$, are given by the bounded solutions to the classical Helmholtz equation $(-Δ)u= u$ 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 $ψ(-Δ) u= ψ(1)u$ in $\mathbb{R}^n$, when $ψ$ is complete Bernstein and certain regularity conditions are imposed on the associated weight $a(t)$.