Treffer: On a singular case in the Hyers–Ulam–Rassias stability of the Wigner equation: On a singular case in the Hyers-Ulam-Rassias stability of the Wigner equation.
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The author considers the Hyers-Ulam-Rassias stability of the Wigner equation in Hilbert spaces basing on a paper by \textit{J. Chmieliński} and \textit{S.-M. Jung} [ibid. 254, 309--320 (2001; Zbl 0971.39016)] and a preprint by \textit{S.-M. Jung}. Let \(E\), \(F\) be real or complex Hilbert spaces and \(f: E\to F\) satisfy that \[ ||\langle f(x)| f(y)\rangle|- |\langle x| y\rangle||\leq \varepsilon\| x\|^p\| y\|^p,\quad x,y\in E_p, \] where \(E_p= E\) for \(p\geq 0\) and \(E_p= E\setminus\{0\}\) for \(p< 0\). The stability of the Wigner equation about \(f\) for \(p\neq 1\) is established. For \(p= 1\), the stability is also established in finite Euclidean spaces for some special cases. About the Wigner equation and the stability, some interesting results are established by \textit{Th. M. Rassias} [Pitman Res. Notes Math. Ser. 376, 219--240 (1997; Zbl 0891.39024)].