On spectral estimates for the Schrödinger operators in global dimension 2

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.

On continuous and discrete Hardy inequalities

“…The number of research papers in this area is enormous. We just mention [M], [T], [RS1], [RS2] and [FMT] as more closely related to our text.…”

“…In Sections 3 and 4 we consider Hardy type inequalities for discrete operators in dimensions d ≥ 3 and d = 2 respectively. We would like to mention the papers [RS1] and [RS2], where a very different method is used to obtain a discrete Hardy type inequality when d ≥ 3.…”

Spectral dimension and Bohr's formula for Schrödinger operators on unbounded fractal spaces

“…The paper [23] contains the justification of the physical conjecture by Madau and Wu on N − (V ) for 2D operators. The case d = 1 was studied in the relatively recent papers by K. Naimark, G. Rozenblum, M. Solomyak et al (see [37,52] and references therein). (2), which is Type (ii) of the fractafold considered in §6.2.…”

Bound states of discrete Schrödinger operators on one and two dimensional lattices

“…In d = 2 an estimate of type (1.3) (with v ≤ 0 and with ln(1 + |x|) in place of |x| γ ) for − ∆ + v has been obtained in [9] applying Markov processes. Analogous estimate in Z 2 (again with v ≤ 0 and with ln(1 + |x|) in place of |x| γ ) for − ∆ + v has been obtained in [10] using some careful estimates for the two dimensional continuous Schrödinger operators together with interpolation arguments. In this paper we establish (1.2)-(1.3) without using those techniques, rather adapting the methods of Klaus in [6].…”