Estimates for the Hill operator. I
The author studies one-dimensional Schrödinger operators \(-d^2/dx^2 + q(x)\) with periodic potentials \(q(x)=q(x+1)\), normalized by \(\int_0^1 q(x) dx = 0\). He proves various estimates involving the \(L_2(0,1)\) norm of the potential and some spectral data. Typical of his results is the following: Let \(\gamma = (\gamma_n)\) be a sequence of the gap lengths, and let \(\|\gamma\|\) be the \(\ell_2\) norm of this sequence. Then the inequalities \(\|\gamma\|\leq 6 \|q\|(1+\|q\|^{1/3})\) and \(\|q\|\leq 4 \|\gamma\|(1+ \|\gamma \|^{1/3})\) hold.
scientific article; zbMATH DE number 1437337
scientific article