The author provides new classes of potentials \(V\) for which the Schrödinger operator \(H =\Delta+V\) (\(\Delta\) being the positive Laplacian) on \(\mathbb{R}^\nu\) has no positive eigenvalues. It is well known that this is not always true but there is a long series of positive results and the present paper is an extension of those obtained by R. Froese and I. Herbst in [Commun. Math. Phys. 87, 429–447 (1982; Zbl 0509.35061)]. In the latter paper, an important rôle playes the Mourre estimate for \(H\) w.r.t. the generator of dilations, which is also a crucial tool in the Mourre commutator theory for Schrödinger operators. In a recent paper [“On the limiting absorption principle for a new class of Schroedinger Hamiltonians”, Preprint, arXiv:1510.03543] the author introduced a new class of “conjugate operators” (i.e., playing the rôle of the generator of dilations) that allows to extend the limiting absorption principle for the Schrödinger operator \(H\) to a larger set of potentials \(V\) than the one previously known. Therefore it was quite natural to try to use this new class to extend the results of Froese and Herbst [loc. cit.]. This is precisely what the author carries out in the present paper.
Recall that Froese and Herbst first proved exponential bounds on possible eigenvectors with positive energy and then used them to exclude their existence. In the present paper, this structure is conserved and the results are generalized in two directions: a larger class of \(\Delta\)-compact potentials \(V\) is considered on one hand, and, on the other hand, the author treats a class of \(\Delta\)-form-compact potentials \(V\). Not only the results by Froese and Herbst are extended but also the strategy of proof. This leads to slightly weaker exponential bounds on possible eigenvectors with positive energy but for a much larger class of potentials \(V\). Then the author formulates results on the absence of positive eigenvalues under somehow technical assumptions, but he provides immediatly concrete examples of potentials \(V\) for which those assumptions are satisfied. We refer to Section 2 of the paper for a clear and detailed presentation of theses results. In Section 2 again, each result is a real extension of a previous one since the author provides an example of a potential for which the actual result applies and the previous one does not.
We mention that the author also treats a certain kind of oscillating potentials that were studied in [the reviewer and A. Mbarek, Doc. Math. 22, 727–776 (2017; Zbl 1370.35092)]. He improves the result on the absence of embedded eigenvalue in [loc. cit., Zbl 1370.35092] but does not cover it completely. He also points out some statement that is uncorrectly justified in [loc. cit., Zbl 1370.35092] and corrects it. Finally the results on the absence of positive eigenvalues allow to slightly improve the results on the limiting absorption principle obtained by the author in [loc. cit., arXiv:1510.03543].