Weyl-Titchmarsh theory for Schrödinger operators with strongly singular potentials (Q2883871)

The authors develop the Weyl-Titchmarsh theory for the one-dimensional Schrödinger operator NEWLINE\[NEWLINEH = - \frac{d^{2}}{d x^{2}} + q (x), \;\;\;x \in (a, b),NEWLINE\]NEWLINE considered on the Hilbert space \(L^{2} (a, b),\) where the potential \(q\) is a real-valued function in general from the class \(L^{1}_{loc} (a, b).\) Under certain assumptions it is proved that the corresponding singular Weyl \(m\)-function \(M (z)\) admits an analytic extension to the upper half plane. Namely, it is proved the existence of entire functions NEWLINE\(\hat{g} (z) \) and \(E (z)\) such that NEWLINE\(\hat{g} (\lambda) >0\) for \(\lambda \in \mathbb{R}\), NEWLINE\((1 + \lambda^{2})^{-1}\hat{g} (\lambda)^{- 1} \in L^{1}(\mathbb{R}, d \rho)\) (\(\rho\) is the spectral measure associated with \(M (z)\)), and NEWLINEthe integral representation NEWLINENEWLINE\[NEWLINEM(z) = E(z) + \hat{g}(z) \int_{\mathbb{R}} \left( \frac{1}{\lambda - z} - \frac{\lambda}{1+\lambda^{2}} \right) NEWLINE\frac{d \rho (\lambda)}{\hat{g} (\lambda)}, \quad z \in \mathbb{C} \setminus \sigma (H)NEWLINE\]NEWLINE holds true. NEWLINENEWLINENEWLINE NEWLINEThe results are applied to perturbed spherical Schrödinger operators.