On a continuum limit of discrete Schr\"odinger operators on square lattice

The norm resolvent convergence of discrete Schr\"odinger operators to a continuum Schr\"odinger 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.


Introduction
We consider a Schrödinger operator on H = L 2 (R d ), where d ≥ 1, and corresponding discrete Schrödinger operators: We set h > 0 be the mesh size, and we write H h = ℓ 2 (hZ d ), hZ d = (hz 1 , . . . , hz d ) z ∈ Z d , with the norm v 2 h = h d |v(hz)| 2 for v ∈ H h . We denote the standard basis of R d by e j = (δ ik ) d k=1 ∈ R d , j = 1, . . . , d. Our discrete Schrödinger operator is where We suppose Assumption A. V is a real-valued continuous function on R d , and bounded from below. (V (x) + M ) −1 is uniformly continuous with some M > 0, and there is c 1 > 0 such that , if |x − y| ≤ 1. The above assumption implies V is slowly varying in some sense, and uniformly continuous relative to the size of V (x). Under the assumption, H is essentially self-adjoint, and H h is self-adjoint. The assumption is satisfied if V is bounded and uniformly continuous. V (x) = a x µ with a, µ > 0, also satisfies the assumption.
For ϕ ∈ S(R d ), h > 0 and z ∈ hZ d , we set The adjoint operator is given by It is easy to observe that P * h is an isometry and hence P h is an orthogonal projection if and only if ϕ 1,z | z ∈ Z d is an orthonormal system. This condition is also equivalent to the condition: whereφ is the Fourier transform: 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. In this paper, we use P h to identify H h with a subspace of H. We suppose: Furthermore, if (V (x) + M ) −1 is uniformly Hölder continuous of order α ∈ (0, 1] (with some M > 0), then for any 0 < β < α, Here B(X) denotes the Banach space of the operators on a Banach space X. Combining this with the argument of Theorem VIII.23 (b) in [10], we obtain the following corollary. We denote the spectrum of a self-adjoint operator A by σ(A), and the spectral projection by E A (Ω) for Ω ⊂ R.
We denote the Hausdorff distance of sets X, Y ⊂ C by There are studies concerning continuum limits of NLS equations, in many cases, mainly with applications to numerical analysis. We refer Bambusi and Penati [2], Hong and Yang [4] and references therein. For linear discrete Schrödinger operators, Rabinovich [9] has studied the relation between the essential and discrete spectra of the discrete and continuum Schrödinger operators, provided V is bounded and uniformly continuous.
In Section 2, we give the proof of our main theorem, and proofs of several technical lemmas are given in Appendix.

Proof
We denote the discrete Fourier transform F h : F h is unitary, and its adjoint is given by

Convergence of the free Hamiltonian
The following formula is convenient in the following argument. It is wellknown in signal analysis (see, e.g., [7]), but we give a proof in Appendix for the completeness. , whereg is the periodic extension of g on R d .
Proof. We first note Let f ∈Ĥ and g = (|2πξ| 2 − µ) −1 f . Then we have, by using the above lemma, For the first term in the right hand side, we observe by Assumption B that |φ(hξ)| = 1 if |ξ| ≤ h −1 δ with some δ > 0. Then we learn For the second term, we note that the terms in the summation vanish except for n ∈ {0, ±1} d \ 0. Using the support condition ofφ again, we learn that ϕ(hξ)φ(hξ + n) = 0 if |ξ + h −1 n| ≤ h −1 δ with some δ > 0. Thus we can use the same argument to show that the second term is bounded by Ch 2 .
Proof. Since P * h is isometric, it suffices to estimate Then we compute, for f ∈ S(R d ), We note, as well as in the proof of Lemma 2.2,φ(hξ)φ(hξ − n) vanishes except for n ∈ {0, ±1} d .

Relative boundedness
In this section, we suppose V ≥ 1 without loss of generality. In particular, V (x) −1 is uniformly bounded, and Lemma 2.4. Suppose Assumption A. Then V is H-bounded, and hence H 0 is also H-bounded.

The first term in the right hand side is bounded since
and ∂ x is H 1/2 -bounded. We also note with some C > 0, and hence ∂ x W is H 1/2 -bounded. Thus we learn is bounded, and henceṼ is H-bounded.
, and the first term in the right hand side is uniformly bounded.
For the second term, we recall that Then we learn

Proof of Theorem 1.1
Lemma 2.6. If G is a uniformly continuous function, then If, in addition, G is uniformly Hölder continuous of order α ∈ (0, 1], then with any ε > 0.
Proof. We note By Schur's lemma, we have We set R(δ) := sup x,y∈R d ,|x−y|<δ |G(x) − G(y)| and we choose n > d. Then we have By the same computation, we also have Combining these and setting δ = h γ with γ ∈ (0, 1), we obtain By the assumption, R(δ) → 0 as δ → 0, and we conclude the first assertion.
If G is uniformly Hölder continuous of order α, then R(δ) ≤ Cδ α , and hence the right hand side of the above estimate is O(h αγ ) + O(h (1−γ)(n−d) ). We can choose γ very close to 1, and n very large so that αγ ≥ α − ε and (1 − γ)(n − d) ≥ α − ε, and we have the second assertion.
Proof of Theorem 1.1. We compute By Lemmas 2.2 and 2.4, we learn The other term is estimated as follows: where we have used Lemmas 2.4 and 2.5 for the second inequality. The two terms in the right hand side are estimated using Lemmas 2.3 and 2.6, respectively, to complete the proof.

A Appendix
Here we give the proofs of several technical lemmas.
Lemma A.1. Let ϕ ∈ S(R d ). Then, the following are equivalent.
We have used the Fourier inversion formula for the last equality. We also have and this implies (2.2).