An estimate, accurate to within a constant factor and uniform in \(\mathbb{R}^N\) for \(N=2\) and \(N=3\), of the increment of the spectral function with respect to the spectral parameter for a Schrödinger operator with a potential satisfying the Kato condition when the spectral function is taken on the diagonal (Q1385924)

In [\textit{V. A. Il'in}, Spectral theory of differential operators, New York (1995), transl. from the Russian (Moskva, Nauka, 1991; Zbl 0759.47026)], a method is presented that makes it possible to reduce the investigation of the convergence of a spectral expansion that is associated with a selfadjoint extension of the Schrödinger operator (or a general second-order elliptic operator) to the establishment of an estimate of the increment of the spectral function with respect to the spectral parameter, when the spectral function corresponds to this extension and is taken on the diagonal. Thus, of special interest is the establishment of an estimate, accurate to within a constant factor and uniform in the whole space \(\mathbb{R}^N\), of the increment of the spectral function with respect to the spectral parameter, when the spectral function corresponds to a selfadjoint extension of the studied elliptic operator in \(\mathbb{R}^N\) and is taken on the diagonal. In the present paper, this problem is solved for the spaces \(\mathbb{R}^N\) of dimension \(N= 2\) and \(N=3\) and in the case of an extension, selfadjoint in \(\mathbb{R}^N\), of the Schrödinger operator \(Mu= -\Delta u+q(x)u\), whose potential \(q(x)\) satisfies the so-called Kato condition.