# Spectrum of a linear operator depending of the fractional laplacian operator

Let $$s \in \mathbb{R}$$ such that $$0. Consider the periodic fractional Laplacian $$(-\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 $$H^0_{per}([-\pi,\pi])\equiv L^2_{per}([-\pi,\pi])$$.