Abstract
In this note, we prove the uniform resolvent estimate of the discrete Schr\"odinger operator with dimension three. To do this, we show a Fourier decay of the surface measure on the Fermi surface.
Highlights
We consider the three-dimensional discrete Laplacian H0u(x) = − (u(y) − u(x)). |x−y|=1We denote the Fourier expansion by Fd: u(ξ) = F u(ξ) = e−2πix·ξu(x), ξ ∈ T3 = R3/Z3. x∈Z3 it follows thatFdH0u(ξ) = h0(ξ)Fdu(ξ), h0(ξ) = 4 sin2(πξj). j=1 (1.1) {multh_0}We denote the set of the critical points of h0 by Cr(h0): Cr(h0) = {ξ ∈ T3 | ∇h0(ξ) = 0} = {ξ ∈ T3 | ξj ∈ {0, 1/2}, j = 1, 2, 3}
We show the uniform resolvent estimates for discrete Schrodinger operator with dimension three
In [13, Theorem 1.7 (iii)], it is shown that the resolvent R0(z) = (H0 − z)−1 for the discrete Schrodinger operator is not bounded from lp(Zd) to lp′(Zd) with p = 2d/(d + 2), p′ = p/(p − 1) and with d ≥ 5
Summary
We show the uniform resolvent estimates for discrete Schrodinger operator with dimension three. In [13, Theorem 1.7 (iii)] (see Lemma A.1), it is shown that the resolvent R0(z) = (H0 − z)−1 for the discrete Schrodinger operator is not bounded from lp(Zd) to lp′(Zd) with p = 2d/(d + 2), p′ = p/(p − 1) and with d ≥ 5. We improve this result and deal with the Fourier decay near the umbilic point. It follows from [14, Theorem 1.2 (i)] that the uniform resolvent estimates away form the diagonal line hold, that is, sup R0(z) B(lp(Z3),lq(Z3)) < ∞, z∈Dε\R for [14, Theorem 1.2 (ii)] implies that R0(z) is Holder continuous on B(lp(Z3), lp′(Z3)). Ikromov for pointing out a mistake of the earlier version of this manuiscript
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