Asymptotics of the heat kernels on 2D lattices

Abstract

We obtain asymptotic expansions of the spatially discrete 2D heat kernels, or Green’s functions on lattices, with respect to powers of time variable up to an arbitrary order and estimate the remainders uniformly on the whole lattice. Unlike in the 1D case, the asymptotics contains a time independent term. The derivation of its spatial asymptotics is the technical core of the paper. Besides numerical applications, the obtained results play a crucial role in the analysis of spatio-temporal patterns for reaction-diffusion equations on lattices, in particular rattling patterns for hysteretic diffusion systems.

Keywords

Discrete heat kernel Green’s function lattice dynamics asymptotics

1. Introduction

The paper deals with the spatially discrete heat kernels, or Green’s functions of the heat equation, on 2D lattices. Although one can easily obtain integral representations of the discrete Green’s function, using the discrete Fourier transform, it is not possible to express them via elementary functions. Therefore, their asymptotic expansions play an important role in applications. Asymptotics of lattice Green’s functions in the stationary (elliptic) case were studied beginning from 1950s, see [4], the subsequent papers [2,9,11,14–16], and the monograph [13, Chapter 8]. For parabolic operators, there is vast literature in the spatially continuous case. For example, large-time behavior of Green’s functions was treated in [17] (for small perturbations of the heat operator) and in [18,22] (for spatially periodic coefficients). A survey on the large time behavior of heat kernels for second-order parabolic operators on Riemannian manifolds can be found in [21]. In the spatially discrete case, the research directions include continuous-time random walks on general graphs (see, e.g., [12,20] and references therein) and on lattices in a random environment (see, e.g., [3]). In both cases, Gaussian bounds for the heat kernel are extensively studied. However, higher-order asymptotics of Green’s functions are not available in general. We mention [5,10], where an asymptotic expansion of Green’s function for particular parabolic equations on 1D lattices was obtained in terms of the Bessel functions. In the case of constant coefficients, we obtained asymptotic formulas as $t \to \infty$ for Green’s functions of general higher-order parabolic operators, with uniform estimates of the remainders on the whole lattices [6]. These formulas played a crucial role in explaining spatio-temporal patterns (rattling) for hysteretic reaction-diffusion equations on 1D lattices [7,8]. To generalize those results to 2D lattices, one needs explicit asymptotic expansions of 2D heat kernels, which is one of the main motivations for the present paper. Unlike in the 1D case, the general asymptotics in [6] contains a time independent term $Ω (x / ε)$ given in the integral form, where ε is the grid step. Derivation of its spatial asymptotics is a technical core of our paper. It turns out that, unlike the leading time-dependent term, $Ω (x / ε)$ is not rotationally symmetric but rather depends on the polar angle of x. Furthermore, it vanishes for $x \neq 0$ as $ε \to 0$ and, as such, is an artifact of spatial discretization (generally depending on the lattice structure).

The paper is organized as follows. In Section 2, we introduce general notation and define first and second Green’s functions on 2D lattices. In Section 3, we obtain a theorem on asymptotics of the first Green’s function, which directly follows from a general result in [6]. In Section 4, we formulate a theorem on asymptotics of the second Green’s function, which contains the above-mentioned term $Ω (x / ε)$ . Derivation of asymptotics of the latter is contained in Section 5. The Appendix contains auxiliary results on the Bessel functions of the first kind, which are used in the main part of the paper.

2. Notation

Green’s functions are special solutions of the heat equations on the 2D grid space, or lattice, $\begin{matrix} R_{ε}^{2} : = {x \in R^{2} : x = ε s, s \in Z^{2}}, ε > 0 . \end{matrix}$ The heat equations are defined via the Laplace operator on $R_{ε}^{2}$ given by $\begin{matrix} Δ_{ε} w (x) : = ε^{- 2} (w (x_{1} + ε, x_{2}) + w (x_{1} - ε, x_{2}) + w (x_{1}, x_{2} + ε) + w (x_{1}, x_{2} - ε) - 4 w (x)), x \in R_{ε}^{2} . \end{matrix}$ Let $δ^{ε} (x)$ denote the grid delta-function given by $\begin{matrix} (2.1) & δ^{ε} (0) = ε^{- 2}, δ^{ε} (x) = 0 \forall x \in R_{ε}^{2} ∖ {0} . \end{matrix}$

Definition 2.1.
We call the function $u_{ε} (x, t)$ , $x \in R_{ε}^{2}$ , $t ⩾ 0$ , the first discrete Green function if $u_{ε} (\cdot, t)$ is a rapidly decreasing grid function for all $t ⩾ 0$ , $u_{ε} (x, \cdot) \in C^{1} [0, \infty)$ for all $x \in R_{ε}^{2}$ , and $\begin{matrix} \{\begin{matrix} {\dot{u}}_{ε} (x, t) - Δ_{ε} u_{ε} (x, t) = 0, & x \in R_{ε}^{2}, t > 0, \\ u_{ε} (x, 0) = δ_{ε} (x), & x \in R_{ε}^{2} . \end{matrix} \end{matrix}$
Definition 2.2.
We call the function $v_{ε} (x, t)$ , $x \in R_{ε}^{2}$ , $t ⩾ 0$ , the second discrete Green function if $v_{ε} (\cdot, t)$ is a rapidly decreasing grid function for all $t ⩾ 0$ , $v_{ε} (x, \cdot) \in C^{1} [0, \infty)$ for all $x \in R_{ε}^{2}$ , and $\begin{matrix} \{\begin{matrix} \dot{v} (x, t) - Δ_{ε} v (x, t) = δ_{ε} (x), & x \in R_{ε}^{2}, t > 0, \\ v_{ε} (x, 0) = 0, & x \in R_{ε}^{2} . \end{matrix} \end{matrix}$

Using the discrete Fourier transform (see [6] for details), we obtain the explicit representations $\begin{array}{l} (2.2) & u_{ε} (x, t) = {\dot{v}}_{ε} (x, t) = \frac{1}{{(2 π)}^{2}} \int_{R_{π ε^{- 1}}} e^{- t ε^{- 2} A (η ε)} e^{i x η} d η, \\ (2.3) & v_{ε} (x, t) = \frac{1}{{(2 π)}^{2}} \int_{R_{π ε^{- 1}}} \frac{1 - e^{- t ε^{- 2} A (η ε)}}{ε^{- 2} A (η ε)} e^{i x η} d η, \end{array}$ where $\begin{matrix} (2.4) & A (ξ) : = 2 (2 - cos ξ_{1} - cos ξ_{2}) \end{matrix}$ is the symbol of the operator $- Δ_{1}$ and $\begin{matrix} R_{L} : = {η \in R^{2} : | η_{k} | ⩽ L, k = 1, 2} . \end{matrix}$ Remark 2.1.
Changing the variables in the integrals in (2.2) and (2.3), we obtain for $J = 0, 1, 2, \dots$ $\begin{matrix} (2.5) & \frac{\partial^{J} u^{ε} (x, t)}{\partial t^{J}} \equiv ε^{- 2 (J + 1)} \frac{\partial^{J} u^{1} (x^{'}, τ)}{\partial τ^{J}} |_{x^{'} = x / ε, τ = t / ε^{2}}, v^{ε} (x, t) \equiv v^{1} (\frac{x}{ε}, \frac{t}{ε^{2}}) . \end{matrix}$

3. Asymptotics of first Green’s function

First, we formulate a theorem on asymptotics of the first Green’s function $u_{ε} (x)$ .

Theorem 3.1.
For any $ε > 0$ , $t_{0} > 0$ , integer $N ⩾ 1$ , integer $J ⩾ 0$ , and all $x \in R_{ε}^{2}$ and $t ⩾ t_{0} ε^{2}$ , we have $\begin{matrix} (3.1) & \frac{\partial^{J} u_{ε} (x, t)}{\partial t^{J}} = \sum_{n = 0}^{N - 1} \frac{ε^{2 n}}{t^{n + 1 + J}} H_{J n} (\frac{x}{t^{1 / 2}}) + r_{u}^{ε} (J, N, t_{0}; x, t), \end{matrix}$ where $\begin{array}{l} (3.2) & H_{J n} (y) : = \{\begin{matrix} Δ^{J} H (y) & if n = 0, \\ finite linear combinations of derivatives of H (y) & if n = 1, \dots, N - 1, \end{matrix} \\ Δ = \frac{\partial^{2}}{\partial y_{1}^{2}} + \frac{\partial^{2}}{\partial y_{2}^{2}}, \\ (3.3) & H (y) = \frac{1}{4 π} e^{- | y |^{2} / 4}, \\ (3.4) & | r_{u}^{ε} (J, N, t_{0}; x, t) | ⩽ \frac{ε^{2 N} R_{u} (J, N, t_{0})}{t^{N + 1 + J}}, \end{array}$ and $R_{u} (J, N, t_{0}) ⩾ 0$ does not depend on $x \in R_{ε}^{2}$ , $t ⩾ t_{0} ε^{2}$ , and $ε > 0$ .
Proof.
The result follows from [6, Theorem 3.1]. Note that the corresponding asymptotic expansion in [6, Theorem 3.1] is obtained for parabolic problems with general higher-order elliptic parts and additionally contains fractional powers $t^{k / 2 + 1 + J}$ with odd k. The corresponding factors $H_{J, k / 2} (\cdot)$ correspond to homogeneous polynomials of odd degrees in the Taylor expansion of the symbol $A (ξ)$ of the elliptic part. In our case, these odd degree polynomials are absent, see (2.4). Hence, the corresponding factors $H_{J, k / 2} (\cdot)$ in the asymptotic expansion vanish too. □
Remark 3.1.

The first term in the asymptotics of $u_{ε} (x, t)$ is $\frac{1}{4 π t} e^{- | x |^{2} / (4 t)}$ , which coincides with Green’s function of the continuous heat operator.

All the functions $H_{J n} (y)$ entering the higher order terms can be found explicitly, see [6, Remark 3.1] for details. For example, $\begin{array}{l} H_{01} (y) = \frac{2}{4!} (\frac{\partial^{4}}{\partial y_{1}^{4}} + \frac{\partial^{4}}{\partial y_{2}^{4}}) H (y), \\ H_{02} (y) = \frac{2}{6!} (\frac{\partial^{6}}{\partial y_{1}^{6}} + \frac{\partial^{6}}{\partial y_{2}^{6}}) H (y) + \frac{4}{2! {(4!)}^{2}} {(\frac{\partial^{4}}{\partial y_{1}^{4}} + \frac{\partial^{4}}{\partial y_{2}^{4}})}^{2} H (y) . \end{array}$

4. Asymptotics of second Green’s function

In this section, we formulate a theorem on asymptotics of the second Green’s function $v_{ε} (x, t)$ . The proof of this theorem is given in Section 5.

For $y \in R^{2} ∖ {0}$ , we introduce the functions $\begin{matrix} (4.1) & \begin{array}{l} F_{0} (y) : = 2 \int_{r}^{\infty} \frac{H_{00} (ρ, φ)}{ρ} d ρ = \frac{1}{2 π} \int_{r}^{\infty} \frac{e^{- ρ^{2} / 4}}{ρ} d ρ, \\ F_{n} (y) : = - \frac{2}{r^{2 n}} \int_{0}^{r} ρ^{2 n - 1} H_{0 n} (ρ, φ) d ρ, n ⩾ 1 . \end{array} \end{matrix}$ Here and below we denote functions written in Cartesian and spherical coordinates by the same letter, e.g., $H_{0 n} (r, φ)$ stands for $H_{0 n} (y)$ .

In what follows, we set $B_{1} : = {θ \in R^{2} : | θ | < 1}$ . We also introduce the function $\begin{matrix} (4.2) & r_{π} (φ) : = distance from the origin to the boundary of the square R_{π} in the direction φ . \end{matrix}$

Theorem 4.1.
For any $ε > 0$ , $t_{0} > 0$ , integer $N ⩾ 1$ , and $t ⩾ t_{0} ε^{2}$ , the following holds.
If $x \in R_{ε}^{2} ∖ {0}$ , then $\begin{matrix} (4.3) & v_{ε} (x, t) = F_{0} (\frac{x}{t^{1 / 2}}) + Ω (\frac{x}{ε}) + \sum_{n = 1}^{N - 1} \frac{ε^{2 n}}{t^{n}} F_{n} (\frac{x}{t^{1 / 2}}) + r_{v}^{ε} (N, t_{0}; x, t), \end{matrix}$ where $F_{n} (y)$ are given by ( 4.1 ), $\begin{matrix} (4.4) & Ω (\frac{x}{ε}) = \frac{ε^{2}}{24 π} \frac{cos (4 ψ)}{r^{2}} + r_{Ω}^{ε} (x), \end{matrix}$ $(r, ψ)$ are the polar coordinates of x, $\begin{matrix} (4.5) & | r_{Ω}^{ε} (x) | ⩽ \frac{ε^{5 / 2} R_{Ω}}{r^{5 / 2}}, \end{matrix}$ and $R_{Ω} ⩾ 0$ does not depend on $x \in R_{ε}^{2} ∖ {0}$ and $ε > 0$ .

If $x = 0$ , then $\begin{matrix} v (0, t) = H (0) ln \frac{t}{ε^{2}} + S_{0} - \sum_{n = 1}^{N - 1} \frac{ε^{2 n}}{t^{n}} \frac{H_{0 n} (0)}{n} + r_{v}^{ε} (N, t_{0}; 0, t), \end{matrix}$ where $H (\cdot) = H_{00} (\cdot)$ and $H_{0 n} (\cdot)$ are the functions in ( 3.3 ) and ( 3.2 ), $\begin{matrix} (4.6) & S_{0} = \frac{1}{{(2 π)}^{2}} (π γ + \int_{B_{1}} (\frac{1}{A (θ)} - \frac{1}{| θ |^{2}}) d θ + \int_{0}^{2 π} ln r_{π} (φ) d φ), \end{matrix}$ $r_{π} (φ)$ is given by ( 4.2 ), and γ is the Euler–Mascheroni constant.
In both cases, $\begin{matrix} (4.7) & | r_{v}^{ε} (N, t_{0}; x, t) | ⩽ \frac{ε^{2 N} R_{v} (N, t_{0})}{t^{N}} \end{matrix}$ and $R_{v} (N, t_{0}) ⩾ 0$ does not depend on $x \in R_{ε}^{2}$ , $t ⩾ t_{0}$ , and $ε > 0$ .

Item 2 in Theorem 4.1 follows from [6, Theorem 5.2, part 2], in which the constant $S_{0}$ is given by $\begin{matrix} (4.8) & \begin{matrix} S_{0} & : = \frac{1}{{(2 π)}^{2}} (\int_{B_{1}} \frac{1 - e^{- | ξ |^{2}}}{| ξ |^{2}} d ξ - \int_{R^{2} ∖ B_{1}} \frac{e^{- | ξ |^{2}}}{| ξ |^{2}} d ξ \\ + \int_{R_{π}} (\frac{1}{A (θ)} - \frac{1}{| θ |^{2}}) d θ + \int_{0}^{2 π} ln r_{π} (φ) d φ) . \end{matrix} \end{matrix}$ Passing to the polar coordinates and using Lemma A.2, we see that $\begin{matrix} (4.9) & \begin{matrix} \int_{B_{1}} \frac{1 - e^{- | ξ |^{2}}}{| ξ |^{2}} d ξ - \int_{R^{2} ∖ B_{1}} \frac{e^{- | ξ |^{2}}}{| ξ |^{2}} d ξ & = 2 π (\int_{0}^{1} \frac{1 - e^{- ρ^{2}}}{ρ} d ρ - \int_{1}^{\infty} \frac{e^{- ρ^{2}}}{ρ} d ρ) \\ = π (\int_{0}^{1} \frac{1 - e^{- z}}{z} d z - \int_{1}^{\infty} \frac{e^{- z}}{z} d z) = π γ, \end{matrix} \end{matrix}$ where γ is the Euler–Mascheroni constant. Combining (4.8) and (4.9) yields (4.6).

The proof of item 1 is given in Section 5.
5. Proof of Theorem 4.1

5.1. Integral representation of $Ω (x)$

Taking into account Remark 2.1, it suffices to prove item 1 in Theorem 4.1 for $ε = 1$ . Therefore, from now on, we consider $x \in Z^{2}$ .

Due to [6, Theorem 5.2, part 1], we have $\begin{matrix} (5.1) & \begin{matrix} Ω (x) & = \frac{1}{{(2 π)}^{2}} (2 \int_{B_{1}} \frac{1 - e^{- | ξ |^{2}}}{| ξ |^{2}} d ξ - 2 \int_{R^{2} ∖ B_{1}} \frac{e^{- | ξ |^{2}}}{| ξ |^{2}} d ξ + \int_{R_{π}} (\frac{e^{i x θ}}{A (θ)} - \frac{1}{| θ |^{2}}) d θ \\ + \int_{0}^{2 π} ln (r_{π} (φ)) d φ) + \frac{1}{2 π} (ln r - ln 2) . \end{matrix} \end{matrix}$ Hence, our main goal is to prove (4.4), (4.5) for $Ω (x)$ given by (5.1).

We begin with simplifying (5.1). Writing $x \in R^{2} ∖ {0}$ in the polar coordinates $(r, ψ)$ and $θ \in R^{2}$ in the polar coordinates $(ρ, φ)$ , we have $\begin{matrix} (5.2) & \int_{R_{π}} (\frac{e^{i x θ}}{A (θ)} - \frac{1}{| θ |^{2}}) d θ = I_{1} (r, ψ) + I_{2} (r, ψ), \end{matrix}$ where $\begin{matrix} (5.3) & \begin{array}{l} I_{1} (r, ψ) : = \int_{- π}^{π} d φ \int_{0}^{r_{π} (φ)} cos (r ρ cos (φ - ψ)) (\frac{1}{A (ρ cos φ, ρ sin φ)} - \frac{1}{ρ^{2}}) ρ d ρ, \\ I_{2} (r, ψ) : = \int_{- π}^{π} d φ \int_{0}^{r_{π} (φ)} \frac{cos (r ρ cos (φ - ψ)) - 1}{ρ} d ρ . \end{array} \end{matrix}$ In the integral defining $I_{2}$ , we change the variable $z = r ρ$ and obtain $\begin{matrix} I_{2} (r, ψ) & = \int_{- π}^{π} d φ \int_{0}^{r_{π} (φ) r} \frac{cos (z cos (φ - ψ)) - 1}{z} d z \\ = \int_{- π}^{π} d φ \int_{0}^{1} \frac{cos (z cos (φ - ψ)) - 1}{z} d z + \int_{- π}^{π} d φ \int_{1}^{π r} \frac{cos (z cos (φ - ψ))}{z} d z \\ + \int_{- π}^{π} d φ \int_{π r}^{r_{π} (φ) r} \frac{cos (z cos (φ - ψ))}{z} d z - \int_{- π}^{π} d φ \int_{1}^{r_{π} (φ) r} \frac{1}{z} d z . \end{matrix}$ Using the integral representation of the Bessel function of the first kind $J_{0} (z)$ (see (A.3)) and the identity from Lemma A.2, we have $\begin{matrix} (5.4) & \begin{matrix} I_{2} (r, ψ) \\ = 2 π \int_{0}^{1} \frac{J_{0} (z) - 1}{z} d z + 2 π \int_{1}^{π r} \frac{J_{0} (z)}{z} d z + I_{3} (r, ψ) - \int_{- π}^{π} ln (r_{π} (φ)) d φ - 2 π ln r \\ = - 2 π γ + 2 π ln 2 + I_{4} (r) + I_{3} (r, ψ) - \int_{- π}^{π} ln (r_{π} (φ)) d φ - 2 π ln r, \end{matrix} \end{matrix}$ where $\begin{matrix} (5.5) & I_{3} (r, ψ) : = \int_{- π}^{π} d φ \int_{π r}^{r_{π} (φ) r} \frac{cos (z cos (φ - ψ))}{z} d z, I_{4} (r) : = - 2 π \int_{π r}^{\infty} \frac{J_{0} (z)}{z} d z . \end{matrix}$

Combining (5.1), (4.9), (5.2), (5.4), we obtain $\begin{matrix} (5.6) & Ω (x) = \frac{1}{{(2 π)}^{2}} (I_{1} (r, ψ) + I_{3} (r, ψ) + I_{4} (r)) . \end{matrix}$

5.2. Estimates of $I_{1}$ , $I_{3}$ , and $I_{4}$

We recall that $(ρ, φ)$ and $(r, ψ)$ stand for the polar coordinates of $θ \in R^{2}$ and $x \in Z^{2}$ , respectively. Set $\begin{matrix} (5.7) & Ψ_{1} : = [- π / 4, π / 4] \cup [3 π / 4, π] \cup [- π, - 3 π / 4], Ψ_{2} : = [0, 2 π) ∖ Ψ_{1} . \end{matrix}$ Thus, the set ${x \in Z^{2} : ψ \in Ψ_{j}}$ is the intersection of $Z^{2}$ with the bisector of opening $π / 2$ symmetric with respect to the axis $x_{j}$ .

Lemma 5.1.

If $ψ \in Ψ_{1}$ , then $\begin{matrix} (5.8) & I_{3} (r, ψ) = - \frac{2 \sqrt{2}}{π} sin (π r - \frac{π}{4}) r^{- 3 / 2} + {\hat{I}}_{3} (r, ψ) + O (r^{- 5 / 2}) as r \to \infty, \end{matrix}$ where $O (\cdot)$ is uniform with respect to $ψ \in [- π, π)$ and $\begin{matrix} (5.9) & {\hat{I}}_{3} (r, ψ) : = - \frac{1}{r cos ψ} \int_{R_{π} ∖ B_{π}} sin (x θ) {(\frac{1}{| θ |^{2}})}_{θ_{1}} d θ . \end{matrix}$ If $ψ \in Ψ_{2}$ , then $I_{3} (r, ψ) = I_{3} (r, ψ + π / 2)$ .

Proof.

Assume that $ψ \in Ψ_{1}$ and hence $| cos ψ | ⩾ 1 / \sqrt{2}$ . We rewrite $I_{3} (r, ψ)$ as follows: $\begin{matrix} (5.10) & I_{3} (r, ψ) = \int_{R_{π} ∖ B_{π}} \frac{cos (x θ)}{| θ |^{2}} d θ = \frac{1}{r cos ψ} \int_{R_{π} ∖ B_{π}} \frac{{(sin (x θ))}_{θ_{1}}}{| θ |^{2}} d θ . \end{matrix}$ Integrating by parts yields $\begin{matrix} (5.11) & I_{3} (r, ψ) = {\tilde{I}}_{3} (r, ψ) + {\hat{I}}_{3} (r, ψ), \end{matrix}$ where $\begin{matrix} {\tilde{I}}_{3} (r, ψ) : = - \frac{1}{π r cos ψ} \int_{- π}^{π} sin (π r cos (φ - ψ)) cos φ d φ, \end{matrix}$ and ${\hat{I}}_{3} (r, ψ)$ is defined in (5.9).

Using the integral representation (A.4) and Lemma A.5 with $n = 1$ , we obtain $\begin{matrix} (5.12) & \begin{matrix} {\tilde{I}}_{3} (r, ψ) & = - \frac{1}{π r cos ψ} \int_{- π}^{π} sin (π r cos φ) cos φ cos ψ d φ = - \frac{2}{r} J_{1} (π r) \\ = - \frac{2 \sqrt{2}}{π} sin (π r - \frac{π}{4}) r^{- 3 / 2} + O (r^{- 5 / 2}) . \end{matrix} \end{matrix}$

To estimate ${\hat{I}}_{3} (r, ψ)$ , we integrate by parts again and obtain ${\hat{I}}_{3} (r, ψ) = O (r^{- 2})$ . Together with (5.11) and (5.12), this proves the lemma whenever $ψ \in Ψ_{1}$ . If $ψ \in Ψ_{2}$ , it suffices to note that $I_{3} (r, ψ) = I_{3} (r, ψ + π / 2)$ . □

Lemma 5.2.

$I_{4} (r) = \frac{2 \sqrt{2}}{π} sin (π r - \frac{π}{4}) r^{- 3 / 2} + O (r^{- 5 / 2})$ as $r \to \infty$ .

Proof.

The proof follows from the asymptotic expansion in Lemma A.4. □

To write down an asymptotic formula for $I_{1} (r, ψ)$ (defined in (5.3)), we need the following notation for $θ \neq 0$ : $\begin{array}{l} (5.13) & D_{0} (θ) : = \frac{1}{A (θ)} - \frac{1}{| θ |^{2}}, \\ (5.14) & D (θ) : = \frac{\partial D_{0} (θ)}{\partial θ_{1}} = - \frac{2 sin θ_{1}}{{(4 - 2 cos θ_{1} - 2 cos θ_{2})}^{2}} + \frac{2 θ_{1}}{{(θ_{1}^{2} + θ_{2}^{2})}^{2}}, \end{array}$ and $\begin{matrix} (5.15) & E (ρ, φ) : = ρ D (ρ cos φ, ρ sin φ) . \end{matrix}$

Obviously, the functions $D_{0} (θ)$ and $D (θ)$ are infinitely differentiable for $θ \in R_{π} ∖ {0}$ , while $E (ρ, φ)$ is infinitely differentiable for $ρ > 0$ and $φ \in R$ and is $2 π$ periodic with respect to φ. It is also easy to see that $D_{0} (θ)$ is bounded near the origin. The behavior of $E (ρ, φ)$ near $ρ = 0$ is described in the following lemma.

Lemma 5.3.

For any integer $k_{1}, k_{2} ⩾ 0$ , the derivative $\frac{\partial^{k_{1} + k_{2}} E (ρ, φ)}{\partial ρ^{k_{1}} \partial φ^{k_{2}}}$ extends as a continuous function from ${ρ > 0, φ \in R}$ to ${ρ ⩾ 0, φ \in R}$ . Moreover, $\begin{matrix} (5.16) & lim_{ρ \to 0} E (ρ, φ) = \frac{{cos}^{3} φ - {cos}^{5} φ - {sin}^{4} φ cos φ}{3} = \frac{cos (3 φ) - cos (5 φ)}{24} . \end{matrix}$

Proof.

The assertion follows by expanding the sin and cos functions in the Taylor series about the origin and using (5.15). □

Now, for each fixed $ρ > 0$ , we represent $E (ρ, φ)$ by its Fourier series $\begin{matrix} (5.17) & E (ρ, φ) = a_{0} (ρ) + \sum_{n = 1}^{\infty} (a_{n} (ρ) cos (n φ) + b_{n} (ρ) sin (n φ)) . \end{matrix}$ This series converges to $E (ρ, φ)$ for every ρ and φ due to Lemma 5.3. Moreover, this lemma immediately implies the following result.

Lemma 5.4.

The Fourier coefficients $a_{0} (ρ)$ , $a_{n} (ρ)$ , and $b_{n} (ρ)$ and all their derivatives are continuous for $ρ > 0$ and extend as continuous function to $ρ ⩾ 0$ . Moreover, for any integer $M ⩾ 0$ , there exists a constant $c_{M} > 0$ such that $\begin{matrix} | a_{n} (ρ) |, | b_{n} (ρ) |, | a_{n}^{'} (ρ) |, | b_{n}^{'} (ρ) | ⩽ \frac{c_{M}}{n^{M}} for all n \in N, ρ \in (0, π] . \end{matrix}$

We have $a_{3} (0) = 1 / 24$ , $a_{5} (0) = - 1 / 24$ , and all the other Fourier coefficients of $E (0, φ)$ vanish.

In particular, Lemma 5.4 guarantees that the series in (5.17) converges absolutely and uniformly for $ρ \in [0, π]$ and $φ \in [- π, π)$ . In what follows, this will allow us to interchange the operations of summation and integration.

Now we are in a position to provide an asymptotics of $I_{1} (r, ψ)$ .

Lemma 5.5.

$I_{1} (r, ψ) = \frac{π cos (4 ψ)}{6 r^{2}} - {\hat{I}}_{3} (r, ψ) + O (r^{- 5 / 2})$ as $r \to \infty$ , where ${\hat{I}}_{3} (r, ψ)$ is given by ( 5.9 ) and $O (\cdot)$ is uniform with respect to $ψ \in [- π, π)$ .

Proof.

Assume that $ψ \in Ψ_{1}$ . Then, using integration by parts, we represent $I_{1} (r, ψ)$ as follows: $\begin{matrix} (5.18) & \begin{matrix} I_{1} (r, ψ) & = \frac{1}{r cos ψ} \int_{R_{π}} {(sin (x θ))}_{θ_{1}} D_{0} (θ) d θ \\ = - \frac{1}{r cos ψ} (Σ_{1} (r, ψ) + Σ_{2} (r, ψ)) - {\hat{I}}_{3} (r, ψ), \end{matrix} \end{matrix}$ where $\begin{matrix} Σ_{1} (r, ψ) : = \int_{B_{π}} sin (x θ) D (θ) d θ, Σ_{2} (r, ψ) : = \int_{R_{π} ∖ B_{π}} sin (x θ) {(\frac{1}{A (θ)})}_{θ_{1}} d θ, \end{matrix}$ $D_{0} (θ)$ and $D (θ)$ are given by (5.13) and (5.14), respectively, and ${\hat{I}}_{3} (r, ψ)$ is given by (5.9).

Using (5.17), we have $\begin{matrix} Σ_{1} (r, ψ) & = \int_{0}^{π} d ρ \int_{- π}^{π} sin (r ρ cos (φ - ψ)) E (ρ, φ) d φ \\ = \int_{0}^{π} (\sum_{n = 1}^{\infty} \int_{- π}^{π} sin (r ρ cos φ) [a_{n} (ρ) cos (n (φ + ψ)) + b_{n} (ρ) sin (n (φ + ψ))] d φ) d ρ . \end{matrix}$ Using the trigonometric addition formulas and the integral representation of the Bessel function $J_{n}$ (see (A.4)), we obtain $\begin{matrix} Σ_{1} (r, ψ) & = \int_{0}^{π} (\sum_{odd n \in N} \int_{- π}^{π} sin (r ρ cos φ) cos (n φ) d φ) [a_{n} (ρ) cos (n ψ) + b_{n} (ρ) sin (n ψ)] d ρ \\ = 2 π \sum_{odd n \in N} {(- 1)}^{(n - 1) / 2} (cos (n ψ) \int_{0}^{π} J_{n} (r ρ) a_{n} (ρ) d ρ + sin (n ψ) \int_{0}^{π} J_{n} (r ρ) b_{n} (ρ) d ρ) . \end{matrix}$ Therefore, making the change of variables $z = r ρ$ in the integrals, taking into account Lemma 5.4 and applying Lemma A.8, we conclude that $\begin{matrix} (5.19) & \begin{matrix} Σ_{1} (r, ψ) & = \frac{2 π}{r} (- cos (3 ψ) a_{3} (0) + cos (5 ψ) a_{5} (0)) + O (r^{- 3 / 2}) \\ = - \frac{π}{12 r} (cos (3 ψ) + cos (5 ψ)) + O (r^{- 3 / 2}) as r \to \infty, \end{matrix} \end{matrix}$ where $O (\cdot)$ is uniform with respect to ψ.

Now we estimate $Σ_{2} (r, ψ)$ . Using the $2 π$ -periodicity of $cos (x θ)$ and $\frac{1}{A (θ)}$ with respect to $θ_{1}$ , we obtain $\begin{matrix} (5.20) & Σ_{2} (r, ψ) = - \frac{1}{r cos ψ} \int_{R_{π} ∖ B_{π}} {(cos (x θ))}_{θ_{1}} {(\frac{1}{A (θ)})}_{θ_{1}} d θ = {\tilde{Σ}}_{2} (r, ψ) + {\hat{Σ}}_{2} (r, ψ), \end{matrix}$ where $\begin{array}{l} {\tilde{Σ}}_{2} (r, ψ) : = \frac{π}{r cos ψ} \int_{- π}^{π} cos (π r cos (φ - ψ)) f (φ) cos φ d φ, \\ {\hat{Σ}}_{2} (r, ψ) : = \frac{1}{r cos ψ} \int_{R_{π} ∖ B_{π}} cos (x θ) {(\frac{1}{A (θ)})}_{θ_{1} θ_{1}} d θ, \end{array}$ and $f (φ)$ is obtained from ${(\frac{1}{A (θ)})}_{θ_{1}}$ by substituting $θ_{1} = π cos φ$ , $θ_{2} = π sin φ$ . Expanding $f (φ) cos φ$ into the Fourier series $\begin{matrix} f (φ) cos φ = α_{0} + \sum_{n = 1}^{\infty} (α_{n} cos (n φ) + β_{n} sin (n φ)) \end{matrix}$ and using the integral representation (A.3), we have $\begin{matrix} {\tilde{Σ}}_{2} (r, ψ) & = \frac{π}{r cos ψ} \int_{- π}^{π} cos (π r cos φ) α_{0} d φ \\ + \frac{π}{r cos ψ} \sum_{even n = 2}^{\infty} \int_{- π}^{π} cos (π r cos φ) (α_{n} cos (n φ) cos (n ψ) + β_{n} cos (n φ) sin (n ψ)) d φ \\ = \frac{π^{2}}{r cos ψ} (α_{0} J_{0} (π r) + {(- 1)}^{n / 2} \sum_{even n = 2}^{\infty} (α_{n} cos (n ψ) J_{n} (π r) + β_{n} sin (n ψ) J_{n} (π r))) . \end{matrix}$ Thus, Lemma A.7 implies $\begin{matrix} (5.21) & {\tilde{Σ}}_{2} (r, ψ) = O (r^{- 3 / 2}), \end{matrix}$ where $O (\cdot)$ is uniform with respect to $ψ \in [- π, π)$ .

Finally, using integration by parts, we immediately obtain ${\hat{Σ}}_{2} (r, ψ) = O (r^{- 2})$ . Combining this with (5.18)–(5.21) yields $\begin{matrix} I_{1} (r, ψ) & = \frac{π}{12 r^{2} cos ψ} (cos (3 ψ) + cos (5 ψ)) - {\hat{I}}_{3} (r, ψ) + O (r^{- 5 / 2}) \\ = \frac{π cos 4 ψ}{6 r^{2}} - {\hat{I}}_{3} (r, ψ) + O (r^{- 5 / 2}) . \end{matrix}$ □

Now Theorem 4.1 follows from (5.6) and Lemmas 5.1, 5.2, and 5.5.

Footnotes

Acknowledgements

The author expresses his gratitude to Sergey Tikhomirov for numerous discussions and to an anonymous referee for meticulously reading the manuscript and providing useful suggestions. The research was supported by the DFG project SFB 910, the DFG Heisenberg Programme and by the “RUDN University Program 5-100”.

Auxiliary results

In this appendix, we collect some known facts about the Bessel functions of the first kind $J_{n} (z)$ as well as several corollaries that we need in the main part of the paper. We will consider the functions $J_{n} (z)$ only for $z > 0$ and integer $n ⩾ 0$ .

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Asymptotics of the heat kernels on 2D lattices

Abstract

Keywords

1. Introduction

2. Notation

5.1. Integral representation of Ω ( x )

5.2. Estimates of I 1 , I 3 , and I 4

Footnotes

Acknowledgements

Auxiliary results

References

5.1. Integral representation of $Ω (x)$

5.2. Estimates of $I_{1}$ , $I_{3}$ , and $I_{4}$