New estimate for the spectral function of self-adjoint extension in \(L^2(\mathbb R)\) of the Sturm-Liouville operator with a uniformly locally summable potential (Q2455307)

Consider a self-adjoint extension \(\mathcal L\) in \(L_2(\mathbb R)\) of the Sturm-Liouville operator \(ly=-y''+q(x)y\), whose potential \(q\) is uniformly locally summable, i.e., satisfies the condition \[ \sup_{x\in R}\int_x^{x+h}| q(t)| \,dt<+\infty,\quad h>0. \] Several estimates are obtained for the absolute value \(| \theta(x,y,\lambda)-\theta_0(x,y,\lambda)| \) of the difference between the spectral function of \(\mathcal L\) and the spectral function \(\theta_0(x,y,\lambda)=\sin(\sqrt{\lambda}(x-y))/\pi(x-y)\) of the operator \(\mathcal L_0\) with the potential \(q(x)\equiv0\) for large values of \(\lambda\). These estimates are based on the inequality \[ \sup_{x,y\in \mathbb R}| \theta(x,y,\lambda)-\theta_0(x,y,\lambda)| \leq\sup_{x,y\in \mathbb R}\frac3{\pi}\biggl| \int_0^1w_q(x,y,t)\frac{\sin\sqrt{\lambda}\,t}t\,dt\biggr| +\frac{95\sqrt{2M}}{\sqrt{\ln\lambda}} \] for \(\lambda\geq e^{8M}\). Here, \[ w_q(x,y,t)=\begin{cases} \sum_{n=1}^{\infty}(-1)^nw_{q,n}(x,y,t),&| x-y| \leq t,\\ 0,&| x-y| >t,\end{cases} \] where \[ \begin{multlined} w_{q,1}(x,y,t)=\frac12\int_{\frac12(x+y-t)}^{\frac12(x+y+t)}q(\tau)\,d\tau,\quad w_{q,n}(x,y,t)=\\ \tfrac12\iint_{D(x,y,t)}q(z)w_{q,n-1}(z,y,\tau)\,d\tau\,dz,\quad (n>1)\end{multlined} \] and \(D(x,y,t)\) is the rectangle in the plane \((\tau,z)\) with vertices at \(K(0,y)\), \(L(\frac{x+t-y}2,\frac{x+t+y}2)\), \(M(t,x)\) and \(N(\frac{y-x+t}2,\frac{y+x-t}2)\).