On a paper by Gesztesy, Simon, and Teschl concerning isospectral deformations of ordinary Schrödinger operators.
Let \(q:\mathbb{R}\to\mathbb{R}\) be a function with period 1. If the spectrum of \(H(q):=-d^2/dx^2+q\) has exactly \(n\) gaps, then the set of one-periodic functions \(p:\mathbb{R}\to\mathbb{R}\) with \(\sigma(H(p))=\sigma(H(q))\) forms an \(n\)-dimensional torus. Results of this type can be traced back to a paper by \textit{O. Steinmann} [Äquivalente periodische Potentiale. Helv. Phys. Acta 30, 515--520 (1957; Zbl 0079.23001)]. \textit{A. Finkel}, \textit{E. Isaacson} and \textit{E. Trubowitz} showed [SIAM J. Math. Anal. 18, 46--53 (1987; Zbl 0622.34021)] that \(p-q\) can be written as the second logarithmic derivative of a Wronskian of Dirichlet eigenfunctions and Floquet solutions associated with \(H(q)\). (There is also an extension to the situation that \(\sigma(H(q))\) has infinitely many gaps, which is the generic case.) In [J. Anal. Math. 70, 267--324 (1996; Zbl 0951.34061)] F. Gesztesy, B. Simon, and G. Teschl remarkably succeeded in extending this description to the case where \(p,q\) are no longer periodic. In the paper under review, their isospectrality result is rendered more transparent by proving the following factorisation. If \(\mu,\nu\) are numbers, not necessarily distinct, in a gap of \(\sigma(H(q)\), then there exists an operator \(B\) with a bounded inverse such that \[ \bigl(H(q)-\mu\bigr) \bigl(H(q)-\nu\bigr)=B^*B,\quad H(p)B=BH(q). \] As a consequence, \(H(q)\) is a function of \(|B|\) and so commutes with \(|B|\) and \(H(p)\) and \(H(q)\) are equivalent via the unitary operator \(B|B|^{-1}\). The operator \(B\) is the closure of a second-order differential operator the coefficients of which are derivatives up to the second-order of the Wronskian of certain eigensolutions associated with \(H(q)\).
scientific article; zbMATH DE number 1895732
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