2017/09/26 by Volker Bach, Bach, Volker, W. de Siqueira Pedra +3
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #FOS: Physical sciences #Mathematical Physics (math-ph) #Numerical methods in inverse problems #Spectral Theory in Mathematical Physics
paper · doi:10.48550/arxiv.1709.09200
openalex publication_date 2017/09/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In dimension d≥ 3, a variational principle for the size of the pure point spectrum of (discrete) Schrödinger operators H(\mathfrake,V) on the hypercubic lattice ℤd, with dispersion relation \mathfrake and potential V, is established. The dispersion relation \mathfrake is assumed to be a Morse function and the potential V(x) to decay faster than |x|-2(d+3), but not necessarily to be of definite sign. Our estimate on the size of the pure-point spectrum yields the absence of embedded and threshold eigenvalues of H(\mathfrake,V) for a class ot potentials of this kind. The proof of the variational principle is based on a limiting absorption principle combined with a positive commutator (Mourre) estimate, and a Virial theorem. A further observation of crucial importance for our argument is that, for any selfadjoint operator B and positive number λ>0, the number of negative eigenvalues of λB is independent of λ.