periodic solutions for a fractional ODE

Question

Can any one give a reference on what are the periodic solutions of the linear fractional ODE $(-\frac{d^2}{dx^2} )^s u= u$ on $x\in (0, T)$ with $u(0)= u(T)$ and $s\in (0, 1)$.

An example of a particular case of $s$ being a fraction will help.

Answer 1

I am afraid I cannot give a reference, but at least I can indicate that the answer is not as simple as one may expect by looking at the classical case $s = 1$.

If $c(x) = 0$, then, under reasonable assumptions on $u$, and with proper understanding of the equation, there is either one (if $s < 1/2$) or two (if $s \geqslant 1/2$) solutions, namely $u(x) = 0$ and $u(x) = x$.

If $c(x) = 1$, then there are two solutions: $u(x) = \sin x$ and $u(x) = \cos x$.

If $c(x) = -1$, then there is no solution.

By reasonable assumptions and proper understanding you can substitute the following: $u$ is function that defines a tempered distribution on $\mathbb{R}$, and the equation holds in the sense of tempered distributions: for any Schwartz class function $\phi$ one has $$\int_{\mathbb{R}} u(x) ((-\Delta)^s \phi(x) - c(x) \phi(x)) dx = 0.$$

With the above definitions, you can get the conclusions listed above by a relatively simple argument in the Fourier variable.

In order to consider more general functions $u$, a more general definition of $(-\Delta)^s$ is needed. This is sometimes possible, but I am aware of no such definition that would help solve equations similar to $(-\Delta)^s u(x) = -u(x)$.