I am studying the Lax pair formulation and I come across finding the spectrum of the operator $L: L^2(\mathbb{R}) \mapsto L^2(\mathbb{R})$,
$$L(t) \psi(t) = - \partial_x^2 \psi(x,t) + u(x,t) \psi(x,t),$$
where $u=u(x,t)$ is a solution of the KdV equation. I am confused about finding some information about the eigenvalues of this operator. I calculated the eigenvalues by Fourier transform and I got a result namely a $\lambda (k) = k^2,\quad k \in \mathbb{R}$. However, in the book, it says that there must be $N$ discrete eigenvalues $0 > \lambda_1>\lambda_2>\dots > \lambda_N$ and a continuum $[0,\infty)$ of spectral values $\lambda (k) = k^2,\quad k \in \mathbb{R}$ and corresponding to the eigenvalues there are N eigenfunctions $\psi_k\in L^2(\mathbb R),\,\,\, k=1,\dots,N$ and $\psi(k) \in L^\infty \setminus L^2$ for $k \in \mathbb R$.
I do not understand how he gets all this information and It is not mentioned in the book I am studying. Could you please show me how to find such information about this operator and any similar one since I face it a lot in my PDE study.