Application of fractional powers of a singular Schrödinger operator to the study of a differential equation in a Banach space (Q2304400)

The author considers the second-order differential equation \(\frac{d^{2}u(t)}{dt^{2}}+Lu(t)=f(t)\), \(t\in[0, T]\), with the initial conditions \(u(0)=u_{0}\) and \(u_{t}(0)=v_{0}\) in a Banach space, where \(L(x, D):=-\Delta+q(x)\) is an elliptic Schrödinger operator with singular coefficient and domain \(W_{p}^{2}(\mathbb{R}^{n})\). The potential \(q(x)\) admits a singularity of the form \(\left|D^{\beta}q(x)\right|\leq \frac{C}{|x|^{1+|\beta|+\tau}}\), where \(C\) is a constant, \(\beta=(\beta_{1},\beta_{2},\dots,\beta_{n})\) is a multiindex, \(0\leq |\beta|:=\sum_{i=1}^{n}\beta_{i}\leq n\), and \(0\leq\tau<1\). The main results of the paper are included in Theorems 1, 2 and 3. Under appropriate conditions, the author in Theorem~3 shows that the sequence of Fourier approximations \(u_{N}(t)\) converges to the solution of the initial value problem in the norm of the space \(L_{p}(\mathbb{R}^{n})\) uniformly with respect to \(t\in[0,T]\).