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The spectral expansion for a nonself-adjoint Hill operator with a locally integrable potential. (English) Zbl 1003.34075

It is wellknown that the spectral theory of operators $L=\frac{{d}^{2}}{d{x}^{2}}+q\left(x\right)$ with a complex 1-periodic potential can be reduced to the study of operators ${L}_{t}=L|\left[0,1\right]$, which are defined by the $t$-periodic boundary conditions $y\left(1\right)={e}^{it}y\left(0\right)$ and ${y}^{\text{'}}\left(1\right)={e}^{it}{y}^{\text{'}}\left(0\right)$. For the spectral expansion, the authors determine first the asymptotics of the eigenvalues and eigenfunctions of the operators ${L}_{t}$, $t\ne 0$, $\pi$. Their key result is the estimate

${\psi }_{n,t}\left(x\right)=expi\left(2\pi n+t\right)x+O\left(\frac{1}{n}\right)$

on the eigenfunctions. This is used to show that $\psi ={\psi }_{n,t}^{0}$ and the associated functions ${\psi }_{n,t}^{k}=\left({L}_{t}-{\lambda }_{n}\right){\psi }_{n,t}^{k-1}$ form a Riesz basis of ${L}^{2}\left[0,1\right]$ by constructing a biorthonormal system with eigenfunctions of ${L}_{\overline{t}}$. Using this result, a spectral representation of functions with compact support can be derived.

##### MSC:
 34L10 Eigenfunctions, eigenfunction expansions, completeness of eigenfunctions (ODE) 34L30 Nonlinear ordinary differential operators 34B30 Special ODE (Mathieu, Hill, Bessel, etc.)
##### Keywords:
periodic operator; eigenfunction expansion