Dirichlet problem for the Schrödinger operator in a half-space with boundary data of arbitrary growth at infinity
The paper deals with the Dirichlet problem \[ L_cu(X) = 0 \;\text{for} \;X = (x,x_{n+1}) \in \mathbb{R}^{n+1}_+, \;u(x) = f(x) \;\text{for} \;x \in \mathbb{R}^n = \partial \mathbb{R}^{n+1}_+, \tag{\(*\)} \] where \(L_c u= - \Delta u+cu\). The potential \(c\) is supposed to be nonnegative and in \(L^p_{\text{loc}}(\mathbb{R}^{n+1}_+)\) with some \(p \geq (n+1)/2\) for \(n\geq 3\), \(p=2\) for \(n=1,2\). Denoting the essentially self-adjoint extension of \(L_c\) from \(C^{\infty}_0 (\mathbb{R}^{n+1}_+)\) to \(L^2(\mathbb{R}^{n+1}_+)\) again \(L_c\) and \(G(X,Y)\) its Green's function, the author introduces the Poisson \(c\)-kernel \(P_c(X,y)\) as the inner normal derivative \(\partial G(X,y)/\partial n(y)\) and proves that the problem \((*)\) has the solution \(u_0 (X) = \int_{\mathbb{R}^n} f(y) P_c (X,y)dy\) provided \(f\) is locally summable in \(\mathbb{R}^n\). He also deals with the uniqueness of the solution and with the case of radial potentials, \(c(X)=c(|X|)\).
scientific article; zbMATH DE number 1095826