Growth order of eigenfunctions of Schrödinger operators with potentials admitting some integral conditions. I: General theory
The authors consider the lower bounds of the growth order of a solution \(u(x)\in H^2_{\text{loc}}(\Omega)\) of a second-order elliptic equation: \[ -\sum^n_{j=1}(\partial_j+ \sqrt{-1}b_j(x))^2u(x)+ (q_1(x)+ q_2(x))u(x)= 0\tag{\(*\)} \] for \(x\in\Omega:= \{x|x\in\mathbb{R}^n, |x|>R_0\}\), where \(b_j(x)\) and \(q_1(x)\) are real functions, \(q_2(x)\) is a complex-valued function. To obtain the lower bounds of the growth order, is one of the main tools to show the nonexistence of eigenvalues embedded in the continuous spectrum. Among many other assumptions, \(q_j(x)\), \(j=1,2\), are supposed to satisfy: for any \(w(x)\in H^1_{\text{loc}}(\Omega)\), \(q_j(x)|w(x)|\), \(\partial_rq_1(x)\in L^1_{\text{loc}}(\Omega)\). Under some additional assumptions on \(q_j(x)\) the author proves lower bounds for \(u(x)\).
scientific article; zbMATH DE number 1002130
scientific article