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Agmon-Type Estimates for a Class of Difference Operators

Abstract.

We analyze a general class of self-adjoint difference operators \(H_\varepsilon = T_\varepsilon + V_\varepsilon\,\, {\rm on}\,\, \ell^2((\varepsilon {\mathbb{Z}})^d)\), where V ε is a one-well potential and ε is a small parameter. We construct a Finslerian distance d induced by H ε and show that short integral curves are geodesics. Then we show that Dirichlet eigenfunctions decay exponentially with a rate controlled by the Finsler distance to the well. This is analog to semiclassical Agmon estimates for Schrödinger operators.

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Correspondence to Markus Klein.

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Communicated by Christian Gérard.

Submitted: February 23, 2008. Accepted: May 23, 2008.

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Klein, M., Rosenberger, E. Agmon-Type Estimates for a Class of Difference Operators. Ann. Henri Poincaré 9, 1177–1215 (2008). https://doi.org/10.1007/s00023-008-0383-7

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  • DOI: https://doi.org/10.1007/s00023-008-0383-7

Keywords

  • Manifold
  • Integral Curve
  • Hamiltonian Vector
  • Semiclassical Limit
  • Finsler Manifold