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Gaps and bands of one dimensional periodic Schrödinger operators. II. (English) Zbl 0649.34034
The authors consider Hill’s equation $$y''+(\lambda -q(x))y=0$$ where $$q(x+1)=q(x)$$ on R and $$q\in L^ 2(0,1]$$. In particular they restrict their attention to the class E of even potentials for which $$q(1-x)=q(x)$$ and the subclass $$E_ 0$$ of E for which $$S_ 0'q(x)dx=0$$. They then provide a new and shorter proof showing that the lengths in the spectrum uniquely determine q(x) in E. The earlier proof can be found in the part I of this article [ibid. 59, 258-312 (1984; Zbl 0554.34013)]. They then consider the sequence $$\sigma (q)=(\sigma_ 1(q),\sigma_ 2(q),...,\sigma_ n(q),...)$$ where $$\sigma_ n(q)=\mu_ n(q)\to \nu_ n(q)$$. Here the $$\{\mu_ n(q)\}$$ are the Dirichlet eigenvalues and the $$\{\nu_ n(q)\}$$ the Newmann eigenvalues. They show that the Jacobian dq$$\sigma$$ maps $$E_ 0\to \ell^ 2$$ isomorphically onto. As one of the applications of these techniques they show that the set of finite bound potentials is norm dense in $$L^ 2_ R[0,1]$$, a result due to V. A. Marĉenko [Sturm-Liouville Operators and their Applications (1977; Zbl 0399.34022)]. As another application they show that q(x) has period 1/k iff the n-th gap vanishes for $$k\nmid n$$. (For another proof of this result not restricted to even q(x) see also the reviewer’s [J. Math. Anal. Appl. 102, 599-605 (1984; Zbl 0557.34021)].) Finally, the authors derive some inequalities in band lengths.