Spectral uniqueness of radial semiclassical Schrödinger operators (Q2428811)

The authors study the problem of determining a potential \(V\) in the class \[ \{V\in C^\infty({\mathbb R}^n):-h^2\Delta+V \text{\;is\;selfadjoint\;on\;a\;domain\;containing}\;C_0^\infty({\mathbb R}^n) \},\;\;n\geq 2, \] from its eigenvalues. It is assumed also that \(V \in C^\infty({\mathbb R}^n)\) is radial near \(0\) in the sense that \(V(x) = R(|x|)\) for \(|x| \leq R_0\) for some \(R_0 > 0\), and \(R\) satisfies \(R(0) = 0\) and \(R'(r) > 0\) for \(r \in (0,R_0)\). Let \(\lambda_0 = R(R_0)\), and suppose \(V > \lambda_0\) for \(|x| > R_0\). In general it is impossible to determine a potential \(V\) from the spectrum of the nonsemiclassical Schrödinger operator \(-\Delta + V\). The paper under review gives a positive answer to the inverse problem for potentials \(V\) from aforementioned class, provided that the discrete spectrum of \(-h^2\Delta+V\) is known up to order \(o(h^2)\). The proof is based on the first two terms of a semiclassical trace formula and on the isoperimetric inequality.