Exponential estimates of solutions of pseudodifferential equations on the lattice h ℤn: Applications to the lattice Schrödinger and Dirac operators

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Abstract

The main aim of the paper is the application of the calculus of pseudodifferential operators on the lattice ℤn ={x ∈ ℝn:x=hy,y ∈ ℤndepending on a small parameter h > 0 to exponential estimates of solutions of difference equations on (hℤ)n. As an application of general results we obtain the local exponential decreasing of eigenfunctions of Schrödinger and Dirac operators on the lattice h→ 0 for h → 0. These results can be considered as an analog of the well known tunnel effect for Schrödinger operators on ℝn. © 2010 Birkhäuser / Springer Basel AG.
Original languageAmerican English
Pages (from-to)233-253
Number of pages207
JournalJournal of Pseudo-Differential Operators and Applications
DOIs
StatePublished - 11 Mar 2010

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Dirac Operator
Electron tunneling
Difference equations
Pseudodifferential Operators
Tunnel
Eigenvalues and eigenfunctions
Small Parameter
Estimate
Difference equation
Eigenfunctions
Calculus
Analogue
Operator

Cite this

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abstract = "The main aim of the paper is the application of the calculus of pseudodifferential operators on the lattice ℤn ={x ∈ ℝn:x=hy,y ∈ ℤndepending on a small parameter h > 0 to exponential estimates of solutions of difference equations on (hℤ)n. As an application of general results we obtain the local exponential decreasing of eigenfunctions of Schr{\"o}dinger and Dirac operators on the lattice h→ 0 for h → 0. These results can be considered as an analog of the well known tunnel effect for Schr{\"o}dinger operators on ℝn. {\circledC} 2010 Birkh{\"a}user / Springer Basel AG.",
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AB - The main aim of the paper is the application of the calculus of pseudodifferential operators on the lattice ℤn ={x ∈ ℝn:x=hy,y ∈ ℤndepending on a small parameter h > 0 to exponential estimates of solutions of difference equations on (hℤ)n. As an application of general results we obtain the local exponential decreasing of eigenfunctions of Schrödinger and Dirac operators on the lattice h→ 0 for h → 0. These results can be considered as an analog of the well known tunnel effect for Schrödinger operators on ℝn. © 2010 Birkhäuser / Springer Basel AG.

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