Functional Analysis and Its Applications

, Volume 44, Issue 4, pp 259–269 | Cite as

On spectral estimates for Schrödinger-type operators: The case of small local dimension

  • G. V. Rozenblum
  • M. Z. Solomyak


The behavior of the discrete spectrum of the Schrödinger operator - Δ -V is determined to a large extent by the behavior of the corresponding heat kernel P(t; x,y) as t → 0 and t→ ∞. If this behavior is power-like, i.e.,
$$ \left\| {P(t; \cdot , \cdot )} \right\|_{L^\infty } = O(t^{ - \delta /2} ),t \to 0,\left\| {P(t; \cdot , \cdot )} \right\|_{L^\infty } = O(t^{ - D/2} ),t \to \infty , $$
then it is natural to call the exponents δ and D the local dimension and the dimension at infinity, respectively. The character of spectral estimates depends on a relation between these dimensions. The case where δ < D, which has been insufficiently studied, is analyzed. Applications to operators on combinatorial and metric graphs are considered.


eigenvalue estimates Schrödinger operator metric graph local dimension dimension at infinity 


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© Springer Science+Business Media, Inc. 2010

Authors and Affiliations

  1. 1.Department of MathematicsChalmers University of Technology and The University Of GothenburgGothenburgSweden
  2. 2.Department of MathematicsWeizmann InstituteRehovotIsrael

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