Divergence of spectral decompositions of Hill operators with two exponential term potentials

TL;DR

This paper investigates the spectral properties of Hill operators with specific exponential potentials, revealing divergence in spectral decompositions and the non-existence of a basis of root functions under certain boundary conditions.

Contribution

It demonstrates that for certain exponential potentials, the system of root functions does not form a basis in L^2, highlighting divergence phenomena in spectral decompositions.

Findings

Root functions do not form a basis under periodic boundary conditions.

Non-basis property occurs for specific parity conditions of r and s.

Divergence of spectral decompositions is established for these potentials.

Abstract

We consider the Hill operator $$ Ly = - y^{\prime \prime} + v(x)y, \quad 0 \leq x \leq \pi, $$ subject to periodic or antiperiodic boundary conditions ($bc$) with potentials of the form $$ v(x) = a e^{-2irx} + b e^{2isx}, \quad a, b \neq 0, r,s \in \mathbb{N}, r\neq s. $$ It is shown that the system of root functions does not contain a basis in $L^2 ([0,\pi], \mathbb{C})$ if $bc$ are periodic or if $bc$ are antiperiodic and $r, s$ are odd or $r=1$ and $s \geq 3. $