Non-existence of positive eigenvalues of the Schrödinger operator in a domain with unbounded boundary (Q1181051)

Consider the Schrödinger equation (1) \((-\Delta+q-\lambda)\;u=0\) in \(D\subset\mathbb{R}^ n\) \((n\geq 2)\), where \(\lambda>0\), \(q\) complex valued. Let \(D_ \alpha=\{x\in\mathbb{R}^ n\mid x^ 1>| x|\cos(\alpha\pi/2)\}\), where \(1<\alpha<2\) and \(| x|=(x_ 1^ 2)+\cdots+x^ 2_ n)^{1/2}\). The author proves the following Theorem: Assume that \(D\) is larger that the half space \(x_ 1>0\) in the sense that there exists a constant \(c\) with \(1<c<2\) such that \(D\supset D_ c\), and assume that the following conditions are satisfied \[ q_ 1\text{ is real valued, of class } C^ 1(D),\text{ and } q_ 1(x)=o(1)\quad (| x|\to\infty\text{ in } D_ c).\leqno (i) \] \[ |\nabla q_ 1(x)|+| q_ 2(x)|=o(| x|^{-1})\quad (| x|\to\infty\text{ in } D_ c).\leqno (ii) \] (iii) There exist constants \(d\) and \(\delta>0\) such that \(1<d<c\), and \[ |\nabla q_ 1(x)| + | q_ 2(x)|=O(| x|{}^{-(2/c)-\delta})\quad (| x|\to\infty \text{ in } D_ c-D_ d). \] In addition assume that \(q\) is such that the unique continuation property holds for equation (1), i.e. if a solution \(u\) of (1) vanishes in an open set of \(D\), \(u\) vanishes in all of \(D\). Then if a solution \(u\) belongs to \(L^ 2(S)\), \(u\) vanishes identically: \(u\equiv 0\).