Brelot spaces of Schrödinger equations (Q1817368)

Let \(\mu\) be a signed measure on a region \(W\) in \(\mathbb{R}^d\), \(d \geq 2\), and let \({\mathcal H}^\mu\) denote the sheaf of continuous real solutions of the Schrödinger equation \((\Delta -\mu)u =0\) (in the distributional sense). It is shown that \((W,{\mathcal H}^\mu)\) may be a Brelot space, hence Harnack inequalities may hold for the functions \(u \in {\mathcal H}^\mu_+(W)\), even if \(\mu\) is not a (local) Kato measure. To understand the basic idea let us assume for simplicity that \(W = \mathbb{R}^d\), \(d\geq3\), and that the global potential \(G^{|\mu|}\) is bounded. Then \((\mathbb{R}^d,{\mathcal H}^\mu)\) is a harmonic space if and only if \(G^\mu=G^{\mu^+}-G^{\mu^-}\) is continuous. This is known since 1990 [\textit{J.-M. Keuntje}, Dissertation Bielefeld (1990; Zbl 0744.31006)]. A familiar argument leading to Harnack inequalities is the following: There exists a constant \(C>0\) such that, for every ball \(B\) in \(\mathbb{R}^d\) and for every superharmonic function \(s\geq 0\) on \(B\), the inequality \(G^{s|\mu|}_B\leq C|G^{1_B|\mu |}|_{\infty}s\) holds. If \(\mu\) is a Kato measure, then we may take \(B\) so small that \(C|G^{1_B|\mu |}|_{\infty}<\frac13\) and then a combination of the classical Harnack inequality and a simple geometric series argument yields the desired Harnack inequality for solutions \(u\geq 0\) of \(\Delta u-u\mu =0\). Suppose now that \(G^{|\mu|}\) is not continuous (i.e., that \(\mu\) is not a Kato measure), but that \(G^\mu\) is continuous (such measures exist [\textit{I. Netuka}, Czech. Math. J. 25, 309-316 (1975; Zbl 0309.31019)]). Choosing \(\alpha > 0\) such that \(\alpha C|G^{|\mu|}|_\infty<\frac13\) it is obvious from the above considerations that \(\alpha\mu\) is not a Kato measure and, nevertheless, \((\mathbb{R}^d,{\mathcal H}^\mu)\) is a Brelot space.