On the Ground State Energies of Discrete and Semiclassical Schr\"odinger Operators

We study the infimum of the spectrum, or ground state energy, of a discrete Schr\"odinger operator on ᵈ parameterized by a potential V:Rᵈ≥ 0 and a frequency parameter θ∈ (0,1). We prove a general comparison result relating this ground state energy to that of a corresponding semiclassical Schr\"odinger operator on Rᵈ with parameter θ, arising from the same choice of potential. We show that for smooth periodic potentials these ground state energies are multiplicatively related by a factor approaching $1$ as θ→ 0, and that they are multiplicatively related by a dimension-dependent constant for a more general class of bounded potentials and all irrational θ.