A Liouville property for Schrödinger operators
The authors consider the problem whether a Schrödinger equation of type \[ \Delta u - u\mu = 0 \] admits a bounded nonvanishing solution or not depending on how the measure \(\mu\) is chosen. Here \(\Delta\) is the Laplace-Beltrami operator on a Riemannian manifold and \(\mu\) is a Kato measure or, more generally, a smooth measure, i.e. a countable sum of Radon measures charging no polar sets. The measure \(\mu\) is called big if no bounded non-trivial solutions (of appropriate smoothness depending on the kind of the measure) exist. The main result of the article is a complete characterization of ``bigness'' of \(\mu\), namely \(\mu\) is not big if and only if it is the sum of two measures, one having support in a so-called non-thick set, the other having a Green potential which is not identically \(\infty\). Moreover, the authors give a sufficient criterion of Wiener type, which is easily applicable to check whether a measure \(\mu\) is big. Since their approach is mainly based on fundamental properties of harmonic and superharmonic functions, they are able to extend their result in a second part of the paper to the more abstract setting of harmonic spaces. Then in this way it is also possible to cover much more general operators than the Laplace operator, including Hörmander type operators and general elliptic and parabolic operators.
scientific article; zbMATH DE number 1336963