## Dirichlet problem for the Schrödinger operator in a half-space with boundary data of arbitrary growth at infinity.(English)Zbl 0879.35039

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, $$c(X)=c(|X|)$$.
Reviewer: A.Kufner (Praha)

### MSC:

 35J10 Schrödinger operator, Schrödinger equation 35J25 Boundary value problems for second-order elliptic equations

### Keywords:

Dirichlet problem; Schrödinger operator; Poisson kernel