Let $s \in \mathbb{R}$ such that $0<s<1$. Consider the periodic fractional Laplacian $(-\Delta)^s$ in the real line defined via Fourier series as follows: if $f:[-\pi,\pi] \subset \mathbb{R} \longrightarrow \mathbb{C}$ is a $2\pi$-periodic function and is written as $$ f(x)=\sum_{n \in \mathbb{Z}} f_n e^{inx} $$ then $$ (-\Delta)^{s}f(x)=\sum_{n \in \mathbb{Z}} |n|^{2s} f_n e^{inx}. $$

For $w>0$, consider the linear operator $$ A(f)=(-\Delta)^{s} f+wf, $$ defined in $L^2_{per}([-\pi,\pi])$ and domain $H^{2s}_{per}([-\pi,\pi])$.

Question. Can we characterize the spectrum of $A$? Moreover, are we able to count the number of negative eigenvalues of $A$?

I think that, if we denote by $\sigma(A)$ the spectrum of $A$, then $\sigma(A)=\{-w+|n|^{2s}\; ; \; n \in \mathbb{Z} \}$. Is it correct?

About the number of negative eigenvalue, I think that we need take into account the parameter $w>0$. For instance, can we guarantee that the number of negative eigenvalues of $A$ is equal to $1$ when $w \in (0,w_0]$, for some $w_0>0$?

References are welcome.

Here, $H^r_{per}([-\pi,\pi])$ represent the fractional periodic Sobolev space of order $r \in \mathbb{R}$, with $H^0_{per}([-\pi,\pi])\equiv L^2_{per}([-\pi,\pi])$.


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