On a continuum limit of discrete Schrödinger operators on square lattice

Shu Nakamura,Yukihide Tadano

doi:10.4171/jst/343

Shu Nakamura, Yukihide Tadano

Open Access

https://doi.org/10.4171/jst/343

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Abstract

The norm resolvent convergence of discrete Schrödinger operators to a continuum Schrödinger operator in the continuum limit is proved under relatively weak assumptions. This result implies, in particular, the convergence of the spectrum with respect to the Hausdorff distance.

Highlights

We consider a Schrodinger operatorH = H0 + V (x), H0 = −△, x ∈ Rd, on H = L2(Rd), where d ≥ 1, and corresponding discrete Schrodinger operators: We set h > 0 be the mesh size, and we writeHh = l2(hZd), hZd = z ∈ Zd, with the norm v 2 h = hd|v(hz)|2 for v ∈ Hh
We denote the standard basis of Rd by ej =dk=1 ∈ Rd, j = 1, . . . , d
V is a real-valued continuous function on Rd, and bounded from below. (V (x) + M )−1 is uniformly continuous with some M > 0, and there is c1 > 0 such that c−1 1(V (x) + M ) ≤ V (y) + M ≤ c1(V (x) + M ), if |x − y| ≤ 1

Summary

Introduction

The assumption is satisfied if V is bounded and uniformly continuous. The adjoint operator is given by h > 0, z ∈ hZd. φh,z(x)v(z), h > 0, v ∈ Hh. It is easy to observe that Ph∗ is an isometry and Ph is an orthogonal projection if and only if φ1,z | z ∈ Zd is an orthonormal system. This claim is well-known, but we give its proof in Appendix for the completeness (Lemma A.1) By this observation, we learn that there is a large class of φ’s satisfying the above condition. B(X) denotes the Banach space of the operators on a Banach space X Combining this with the argument of Theorem VIII. (b) in [10], we obtain the following corollary.

Convergence of the free Hamiltonian

Relative boundedness

A Appendix