On the spectrum of a discrete Laplacian on ℤ with finitely supported potential

Abstract

We study spectral properties of the discrete Laplacian L = −Δ + V on ℤ with finitely supported potential V. We give sufficient and necessary conditions for L to satisfy that the number of negative (resp. positive) eigenvalues is equal to one of the points x on which V(x) is negative (resp. positive). In addition, we prove that L has at least one discrete eigenvalue. If ∑ _x∈ℤ V(x) = 0, then L has both negative and positive discrete eigenvalues.

Keywords:

• discrete Laplacian
• discrete eigenvalues
• infinite graph
• Green's function

AMS Subject Classifications:

• 39A12
• 47A75
• 34L40

Acknowledgements

Yu. Higuchi was supported in part by the Grant-in-Aid for Scientific Research (C) 20540133 and (B) 21340039 from Japan Society for the Promotion of Science. Osamu Ogurisu was supported by the Grant-in-Aid for Scientific Research (C) 20540204 from Japan Society for the Promotion of Science.