Polynomial growth or decay of eigenfunctions of second-order elliptic operators
We study the asymptotic behaviors as \(| x| \to \infty\) of the not identically vanishing solution \(u(x)\in H^ 2_{loc}(\Omega)\) of second-order elliptic equation in an unbounded domain \(\Omega =\{x|\) \(| x| >R_ 0\}\subset R^ n\) \[ - \sum^{n}_{i,j=1}(\frac{\partial}{\partial x_ i}+\sqrt{-1}b_ i(x))a_{ij}(x)(\frac{\partial}{\partial x_ j}+\sqrt{-1}b_ j(x))u(x)+(q_ 1(x)+q_ 2(x))u(x)=0, \] where the matrix \((a_{ij}(x))\) is uniformly positive definite, \(b_ i(x)\) (1\(\leq i\leq n)\) and \(q_ 1(x)\) are real-valued functions, and \(q_ 2(x)\) is a complex-valued function.
scientific article; zbMATH DE number 4071514