The Schrödinger-Hill equation \(-y''(x)+q(x)y(x)=\mu \cdot y(x)\) on odd potentials q (Q1823366)

Denote by \(h^ s(s\geq 0)\) the space of all real sequences \((c_ n)_ n\) such that \(\sum n^{2s}c^ 2_ n<\infty\). The following facts are well-known: a) for any \(q\in L^ 2(0,1)\), the eigenvalues \(\mu_ n(q)\) of the Sturm-Liouville problem (*) \(-y''+q(x)y=\mu y\) (0\(\leq x\leq 1)\), \(y(0)=y(1)=0\) satisfy \(\mu_ n(q)-n^ 2\pi^ 2\in h^ 0=\ell^ 2;\) b) conversely, for any sequence \(\tilde u_ n\in h^ 0\) there exists a potential \(q\in L^ 2(0,1)\) such that \(n^ 2\pi^ 2+{\tilde \mu}_ n=\mu_ n(q)\); if q is odd, the eigenvalues \(\mu_ n(q)\) of (*) even satisfy \(\mu_ n(q)-n^ 2\pi^ 2\in h^ 1.\) The author proves that, if q is odd, then \(\mu_ n(q)-n^ 2\pi^ 2\not\in h^{3/2}\). Moreover, this result is sharp, since for \(q=\chi_{(0,1/2)}-\chi_{(1/2,1)}\) one has \(\mu_ n(q)-n^ 2\pi^ 2\in h^ s\) for any \(s\in [1,3/2)\). From a result of \textit{V. A. Marchenko} and \textit{I.V. Ostrovskij} [Math. Sb., Nov. Ser. 97, 540-606 (1975; Zbl 0327.34021)] it follows that, given an arbitrary sequence \({\tilde \mu}_ n\in h^ 1\), one may not find, in general, an odd potential q such that \(n^ 2\pi^ 2+{\tilde \mu}_ n=\mu_ n(q).\)