Inverse theory of Schrödinger matrices (Q1961744)

The author discusses 3-dimensional Schrödinger matrices \[ H(b_1,b_2,b_3)=\left(\begin{matrix} b_1 & 1 & 0\\ 1 & b_2 & 1\\ 0 & 1 & b_3\end{matrix}\right), \] \((b_1,b_2,b_3)\in\mathbb{R}^3\) under the context of conjectures of \textit{F. Gesztesy} and \textit{B. Simon} [J. Anal. Math. 73, 267-297 (1997; Zbl 0924.15005)] concerning the properties of the mapping \(F:\mathbb{R}^3 \to\mathbb{R}^3\), \((b_1,b_2,b_3) \mapsto(t_1,t_2,t_3)\) where \((t_1,t_2,t_3)\) are eigenvalues of the matrix \(H(b_1,b_2,b_3)\) (it is known that \(t_1,t_2,t_3\) are always real and distinct). One of the Gesztesy-Simon conjectures says that the image \(W\) of \(F\) is closed and for \((t_1,t_2,t_3)\) from interior of \(W\), \(F^{-1}(t_1,t_2,t_3)\) contains \(3!\) points (the conjecture de facto concerns arbitrary dimension and for 2-dimensional matrices is true what is a trivial exercise noticed by Gesztesy and Simon). The author proves that the set \[ C=\biggl\{ \bigl(-\sqrt{2+t^2},0, \sqrt{2+t^2} \bigr);\;t\in(-1,1)\setminus \{0\}\biggr\} \] is contained in the interior of the image of \(F\) and \(F^{-1}(T)= \{(-t,0,t),\;(t,0,-t)\}\) for \(T\in C\) which gives a counterexample to the above conjecture.