A new estimate for the spectral function of the self-adjoint extension in \(L^{2}(\mathbb R)\) of the Sturm-Liouville operator with a uniformly locally integrable potential (Q2461861)

The author extends the results of \textit{L. V. Kritskov} [Differ. Equations 31, No. 12, 2008--2015 (1995); translation from Differ. Uravn. 31, No. 12, 2038--2045 (1995; Zbl 0873.35056)] on one-dimensional Schrödinger operators \(-d^2/dx^2+q(x)\) on the line with uniformly locally integrable potential. The author obtains a sharpened estimate of the function \(\theta(x,y,\lambda)\), as the spectral parameter \(\lambda\) tends to \(+\infty\), under a slightly stronger condition on the potential, namely \(\lim_{h\to 0+}\sup_{x\in{\mathbb R}}\int_x^{x+h}| q(t)| \,dt=0\).