Spectral analysis of a class of second-order non-self-adjoint differential operators
In this article, we study the spectrum and the eigenfunction expansion of a differential operator L, generated by the differential expression \(\ell (y)\equiv -y''+q(x)y\) in the space \(L_ 2(-\infty,\infty)\), under the assumption that the coefficient is \(q(x)=\sum^{\infty}_{n=1}q_ ne^{inx}\) and the series \(\sum^{\infty}_{n=1}| q_ n| =q\) converges. Clearly q(x) is a periodic function on (-\(\infty,\infty)\) (the period is equal to \(2\pi)\), and it admits holomorphic continuation to the upper half plane. Therefore, the operator L is not self-adjoint, and becomes self-adjoint only in the trivial case, for q(x)\(\equiv 0\). It turns out that the spectrum of L is purely continuous, fills the semiaxis [0,\(\infty)\), and there may be second-order spectral singularities on the continuous spectrum, which must coincide with numbers of the form \((n/2)^ 2\). For the generalized eigenfunctions corresponding to the spectral singularities, we may introduce the idea of generalized norming numbers \(\{s_ n\}\). We prove that from these we may effectively reconstruct \(\{q_ n\}\).
scientific article; zbMATH DE number 3916759