Discrete time-dependent wave equation for the Schrödinger operator with unbounded potential

2023/06/04 by Dasgupta, Aparajita, Mondal, Shyam Swarup, Ruzhansky, Michael +1
#22E30 #Analysis of PDEs (math.AP) #FOS: Mathematics #Primary 46F05 #Secondary 58J40

paper · doi:10.48550/arxiv.2306.02409

Abstract

In this article, we investigate the semiclassical version of the wave equation for the discrete Schrödinger operator, H_ℏ,V:=-ℏ^-2L_ℏ+V on the lattice ℏℤⁿ, where L_ℏ is the discrete Laplacian, and V is a non-negative multiplication operator. We prove that H_ℏ,V has a purely discrete spectrum when the potential V satisfies the condition |V(k)|→ ∞ as |k|→∞. We also show that the Cauchy problem with regular coefficients is well-posed in the associated Sobolev type spaces and very weakly well-posed for distributional coefficients. Finally, we recover the classical solution as well as the very weak solution in certain Sobolev type spaces as the limit of the semiclassical parameter ℏ→ 0.

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