Discrete Green’s functions

Abstract

Abstract. Let G(P; Q) be the discrete Green's function over a discrete A-convex region a of the plane; i.e., a(P)GXx(P; Q) + c{P)Gvi(P; Q) = -btP; Q)/h' for P G O», G(P; Q) = 0 for P G dSlk. Assume that a(P) and c{P) are Holder continuous over Q and positive. We show that \D^GiP; Q)\ g Am/Pp>Q and |5<™>G(P; Q)\ g BmdiQ)/p%\ where D™ is an mth order difference quotient with respect to the components of P or Q, and Í5<m) denotes an mth order difference quotient only with respect to the components of P.

Entanglement entropy in excited states of the quantum Lifshitz model

“…In the continuum, the answer is then given by (C.18). On the lattice, we need the half-plane lattice Green's function, which is [63] G(x, y; x 0 , y 0 ) = L(x − x 0 , y + y 0 ) − L(x − x 0 , y − y 0 ), (C. 24) where L is a discrete version of the logarithm…”

On the decay of the inverse of matrices that are sum of Kronecker products

“…We report in Figure 1 a typical such pattern, obtained for M = −tridiag(1, −2, 1) (here and later in the paper, the underlined number lies on the matrix diagonal), corresponding to the finite difference discretization of the two-dimensional negative Laplace operator −(u xx + u yy ) in the domain [0, 1] × [0, 1]. This non-monotonic behavior has been observed in the literature ( [19]), and explained in detail for the case of the discrete Laplacian, for which precise estimates are available [9], [18], [22]; bounds stemming from an algebraic analysis were also determined in [19]. The situation is far less understood when M is any tridiagonal SPD matrix, or more generally any banded SPD matrix.…”