To Mikhail Shl emovich Birman on the occasion of his 70-th birthday, as a sign of friendship and admiration

No abstract

The problem of finding eigenvalue estimates for the Schrödinger operator turns out to be most complicated for the dimension 2. Some important results for this case have been obtained recently. In the paper, these results are discussed, and their counterparts are established for the operator on the combinatorial and metric graphs corresponding to the lattice Z 2 .License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 496 G. ROZENBLUM AND M. SOLOMYAK interested in estimates of the typewhere the functional Φ(V ) is homogeneous of order 1 with respect to V , so that (1.2) automatically impliesNote that the term 1 in (1.2) and in (1.3) reflects the well-known fact that for all the cases under study the operator H αV has at least one negative eigenvalue for any α > 0. The authors express their gratitude to the referee for useful remarks and suggestions.

No abstract

Abstract. We define and study Toeplitz operators in the space of Herglotz solutions of the Helmholtz equation in R d . As the most traditional definition of Toeplitz operators via Bergman-type projection is not available here, we use an approach based upon the reproducing kernel nature of the Herglotz space and sesquilinear forms, which results in a meaningful theory. For two important patterns of sesquilinear forms we discuss a number of properties, including the uniqueness of determining the symbols from the operator, the finite rank property, the conditions for boundedness and compactness, spectral properties, certain algebraic relations.