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On the periodic spectrum of the 1-dimensional Schrödinger operator. (English) Zbl 0703.34085
Consider Hill’s equation \(-y''(x)+q(x)y(x)=\lambda y(x)\) (x\(\in {\mathbb{R}})\) with q a potential in \(L^ 2[0,1]\). Denote by \(\lambda_ k=\lambda_ k(q)\) (k\(\geq 0)\) the union of the periodic and antiperiodic spectrum of \((-d^ 2/dx^ 2)+q\) on the interval [0,1], written in nondecreasing order and with multiplicities. Introduce the set f potentials ISO(q), having the same periodic spectrum as q and the set G(q) of potential p having the same gaps as q, i.e. \(\lambda_ 0(q)=\lambda_ 0(p)\) and \(\lambda_{2k}(q)-\lambda_{2k-1}(q)=\lambda_{2k}(p)-\lambda_{2k- 1}(p)\) (k\(\geq 1)\). This paper presents an elementary proof of the following result, due to J. Garnett and E. Trubowitz: Theorem. For all q in \(L^ 2[0,1]\), \(ISO(q)=G(q)\).
Reviewer: Th.Kappeler
34L05 General spectral theory of ordinary differential operators
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