CUBO A Mathematical Journal

Vol.10, N
o
¯ 02, (47–59). July 2008

Green Function for a Two-Dimensional Discrete

Laplace-Beltrami Operator

Volodymyr Sushch

Koszalin University of Technology,

Sniadeckich 2, 75-453 Koszalin, Poland,

email: volodymyr.sushch@tu.koszalin.pl

ABSTRACT

We study a discrete model of the Laplacian in R
2

that preserves the geometric structure

of the original continual object. This means that, speaking of a discrete model, we do

not mean just the direct replacement of differential operators by difference ones but

also a discrete analog of the Riemannian structure. We consider this structure on the

appropriate combinatorial analog of differential forms. Self-adjointness and boundness

for a discrete Laplacian are proved. We define the Green function for this operator and

also derive an explicit formula of the one.

RESUMEN

Nosotros estudiamos un modelo discreto del Laplaciano en R
2

que preserva la estructura

geométrica del objeto continuo original. Esto significa hablando de un modelo discreto,

que nosotros no tenemos la intención de remplazar directamente el operador diferencial

por uno en diferencias como también un análogo discreto de la estructura Riemanniana.

Nosostros consideramos esta estructura sobre un apropriado análogo combinatório de

formas diferenciables. Provamos para un Laplaceano discreto que es auto adjunto y



48 Volodymyr Sushch CUBO
10, 2 (2008)

acotado. Definimos la función de Green de tal operador e conseguimos una formula

explicita para este.

Key words and phrases: Discrete Laplacian, difference equations, Green function.

Math. Subj. Class.: 39A12, 39A70

1 Introduction

We begin with a brief review of some well known definitions that are related to the contents of

this paper. Denote by Λ
r
(R

2
) the set of all differentiable complex-valued r-forms on R2, where

r = 0, 1, 2. Let ∗ : Λr(R2) → Λ2−r(R2) be the Hodge star operator. An inner product for r-forms

with compact support is defined by

(ϕ, ψ) =

∫

R2

ϕ ∧ ∗ψ, (1.1)

where the bar over ψ denotes complex conjugation. Let L2Λr(R2) denote the completion of Λr(R2)

with respect to the norm generated by the inner product (1.1). Let the exterior derivative d :

Λ
r
(R

2
) → Λr+1(R2) be defined as usual. We define the operator

d : L
2
Λ

r
(R

2
) → L

2
Λ

r+1
(R

2
) (1.2)

as the closure in the L2-norm the corresponding operation specified on smooth forms. The adjoint

of d, denoted by δ, is given by

(dϕ, ω) = (ϕ, δω), ϕ ∈ L2Λr(R2), ω ∈ L2Λr+1(R2).

Note that δ : L2Λr+1(R2) → L2Λr(R2). The following relations hold among ∗, d and δ. See for

instance [15].

∗
2

= (−1)
r(2−r)Id, δ = (−1)r ∗−1 d ∗ . (1.3)

The Laplacian is defined to be

−∆ = dδ + δd : L2Λr(R2) → L2Λr(R2). (1.4)

In this paper we develop some combinatorial structures that are analogs of objects in dif-

ferential geometry. We are interested in finding of a natural discrete analog of the Laplacian on

cochains. Speaking of a discrete model, we mean not only the direct replacement of differential

operators by difference ones but also discrete analogs of all essential ingredients of the Riemannian

structure over a properly introduced combinatorial object.

Our approach bases on the formalism proposed by Dezin [5]. We adapt the combinatorial

constructions from [5, 11, 12] and define discrete analogs of operators (1.2)–(1.4) in a similar



CUBO
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Green Function for a Two-Dimensional Discrete ... 49

way. In [5], Dezin study discrete Laplace operators in finite-dimensional Hilbert spaces, i.e. on

cochains given in domains with boundary. In this paper we extend these results on an infinite

complex of complex-valued cochains. We prove self-adjointness and boundness for the discrete

Laplace-Beltrami operator in infinite Hilbert spaces that are associated with L2Λr(R2). Spectral

properties are discussed. We define the Green function for the discrete Laplacian and derive the

one in an explicit form.

There are other geometric approaches to discretisation of the Hodge theory of harmonic forms

presented in [6, 7, 8]. In all these papers discrete models are given on the simplicial cochains of

triangulated closed Riemannian manifolds. See also [3, 4, 9] and references given there.

Classical references on second order difference equations are the books by Berezanski [2, ch.

7], Atkinson [1] and the most recent monograph by Teschl [14].

2 Combinatorial structures

Let us denote by C(2) the two-dimensional complex. This complex is defined by

C(2) = C
0
⊕ C

1
⊕ C

2,

where C
r

is a real linear space of r-dimensional chains. We follow the notation of [5, 12]. Let

{xk,s}, {e
1
k,s
, e2

k,s
}, {Ωk,s}, k,s ∈ Z, be the sets of basis elements of C

0, C1, C2 respectively. It

is convenient to introduce shift operators

τk = k + 1, σk = k − 1

in the set of indices. The boundary operator ∂ is defined by the rule

∂xk,s = 0, ∂Ωk,s = e
1
k,s

+ e
2
τ k,s

− e
1
k,τ s

− e
2
k,s
,

∂e
1
k,s

= xτ k,s − xk,s, ∂e
2
k,s

= xk,τ s − xk,s. (2.1)

The definition of ∂ is linearly extended to arbitrary chains. We call the complex C(2) a combina-

torial model of R
2
.

On the other hand, we can consider C(2) as the tensor product C(2) = C ⊗ C of the one-

dimensional complex C (combinatorial model of a real line). Then basis elements of C(2) can be

written as follows

xk ⊗ xs = xk,s, ek ⊗ xs = e
1
k,s
,

ek ⊗ es = Ωk,s, xk ⊗ es = e
2
k,s
,

where xk, ek are the basis elements of C.

Let us introduce an object dual to C(2). Namely, the complex of complex-valued functions

over C(2). The dual complex K(2) we can consider as the set of complex-valued cochains and it



50 Volodymyr Sushch CUBO
10, 2 (2008)

has the same structure as C(2), i.e. K(2) = K0 ⊕ K1 ⊕ K2. In other words, K(2) is a linear

complex space with basis elements

{xk,s, e
k,s

1 , e
k,s

2 , Ω
k,s

}.

The pairing (chain-cochain) operation is defined by the rule:

< xk,s, x
p,q

>=< Ωk,s, Ω
p,q

>=< e
1
k,s
, e

p,q

1 >=< e
2
k,s
, e

p,q

2 >= δ
p,q

k,s
, (2.2)

where δ
p,q

k,s
is Kronecker symbol. We call elements of the complex K(2) forms, emphasizing their

closeness to the corresponding continual objects, differential forms. Then the 0-, 1-, 2-forms

ϕ, ω = (u,v), η can be written as

ϕ =
∑

k,s

ϕk,sx
k,s, η =

∑

k,s

ηk,sΩ
k,s, ω =

∑

k,s

(uk,se
k,s

1 + vk,se
k,s

2 ), (2.3)

where ϕk,s, uk,s, vk,s, ηk,s ∈ C for any k,s ∈ Z. Operation (2.2) is extended to arbitrary forms

(2.3) by linearity. The boundary operator (2.1) in C(2) induces the dual operation dc in K(2):

< ∂a, α >=< a, dcα >, (2.4)

where a ∈ C(2), α ∈ K(2). We assume that the coboundary operator dc : Kr → Kr+1 is a discrete

analog of the exterior differentiation operator d (1.2).

If ϕ ∈ K0 and ω = (u,v) ∈ K1, then we have the following difference representations for dc:

< e
1
k,s
, d

c
ϕ >= ϕτ k,s − ϕk,s ≡ ∆kϕk,s,

< e
2
k,s
, d

c
ϕ >= ϕk,τ s − ϕk,s ≡ ∆sϕk,s

< Ωk,s, d
cω >= vτ k,s − vk,s − uk,τ s + uk,s ≡ ∆kvk,s − ∆suk,s. (2.5)

Note that if η ∈ K2, then dcη = 0.

Let us now introduce in K(2) a multiplication which is an analog of the exterior multiplication

∧ for differential forms. We denote this operation by ∪ and define it according to the rule:

xk,s ∪ xk,s = xk,s, e
k,s

2 ∪ e
k,τ s

1 = −Ω
k,s,

xk,s ∪ e
k,s

1 = e
k,s

1 ∪ x
τ k,s

= e
k,s

1 ,

xk,s ∪ e
k,s

2 = e
k,s

2 ∪ x
k,τ s

= e
k,s

2 ,

xk,s ∪ Ωk,s = Ωk,s ∪ xτ k,τ s = e
k,s

1 ∪ e
τ k,s

2 = Ω
k,s, (2.6)

supposing the product to be zero in all other cases. The ∪-multiplication is extended to discrete

forms by linearity. In terms of the theory of homologies, this is the so-called Whitney multiplication.

For arbitrary forms α,β ∈ K(2) we have the following relation

dc(α ∪ β) = dcα ∪ β + (−1)rα ∪ dcβ, (2.7)



CUBO
10, 2 (2008)

Green Function for a Two-Dimensional Discrete ... 51

where r is the degree of α. The proof of this can be found in Dezin [5, p. 147]. Relation (2.7) is

an analog of the corresponding continual relation for differential forms (see [15]).

Define a discrete analog of the Hodge star operator. Let εk,s denote an arbitrary basis element

of K(2). We introduce the operation ∗ : Kr → K2−r by setting

εk,s ∪ ∗εk,s = Ωk,s. (2.8)

Using (2.6) we get

∗xk,s = Ωk,s, ∗e
k,s

1 = e
τ k,s

2 , ∗e
k,s

2 = −e
k,τ s

1 , ∗Ω
k,s

= xτ k,τ s.

The operation ∗ is extended to arbitrary forms by linearity.

Let α ∈ Kr is an arbitrary r-form:

α =
∑

k,s

αk,sε
k,s. (2.9)

Denote by Kr0 the set of all discrete r-form with compact support on C(2). Let Ω be the following

”domain”

Ω =

∑

k,s

Ωk,s, k,s ∈ Z, (2.10)

where Ωk,s is a two-dimensional basis element of C(2). Note that if the sum (2.10) is finite and let

−N ≤ k,s ≤ N, N ∈ N, then we will write Ω = ΩN .

The relation

(α, β) =< Ω, α ∪ ∗β >, (2.11)

where α,β ∈ Kr0 , gives a correct definition of inner product in K(2). Using (2.2), (2.6) and (2.8),

this definition can be rewritten as follows

(α, β) =
∑

k,s

αk,sβk,s. (2.12)

For Ω = ΩN we will write

(α, β)N =< ΩN, α ∪ ∗β >=

N∑

k,s=−N

αk,sβk,s.

Let α ∈ Kr, β ∈ Kr+1. The relation

(dcα, β)N =< ∂ΩN, α ∪ ∗β > +(α, δ
cβ)N (2.13)

defines the operator δc : Kr+1 → Kr,

δcβ = (−1)r ∗−1 dc ∗ β,



52 Volodymyr Sushch CUBO
10, 2 (2008)

which is the formally adjoint operator to dc (see [5] for more details). It is obvious that the operator

δc can be regarded as a discrete analog of the codifferential δ (cf. (1.3)). Equation (2.13) is an

analog of the Green formula for the formally adjoint differential operators d and δ. It is easy to

check that for α ∈ Kr0, β ∈ K
r+1
0 we obtain

(d
c
α, β) = (α, δ

c
β). (2.14)

According to (2.5), we have δcϕ = 0 and

< xk,s, δ
cω >= −∆kuσk,s − ∆svk,σs, (2.15)

< e
1
k,s
, δ

c
η >= ∆sηk,σs, < e

2
k,s
, δ

c
η >= −∆kησk,s, (2.16)

where ϕ ∈ K0, ω ∈ K1 and η ∈ K2.

Therefore a discrete analog of the Laplace-Beltrami operator (1.4) can be defined as follows

−∆
c

= δcdc + dcδc : Kr → Kr. (2.17)

Obviously, if ϕ ∈ K0, then we have

−∆
c
ϕ = δ

c
d

c
ϕ. (2.18)

Combining (2.15) with (2.5) we can rewrite (2.18) as

< xk,s, −∆
cϕ >= 4ϕk,s − ϕτ k,s − ϕk,τ s − ϕσk,s − ϕk,σs. (2.19)

The same difference form of (2.17) can be drawn for the components ηk,s of η ∈ K
2

and for the

two components uk,s vk,s of ω ∈ K
1
.

3 Discrete Laplacian

Let us now introduce the linear space

H
r

= {α ∈ Kr :
∑

k,s

|αk,s|
2 < +∞, k,s ∈ Z}, (3.1)

where r = 0, 1, 2. Clearly, Hr is a Hilbert space with inner product (2.11) (or (2.12)) and with the

following norm

‖α‖ =
√

(α, α) =
( ∑

k,s

|αk,s|
2
) 1

2

. (3.2)

Note that if α ∈ Hr, then the set of complex-valued sequences (αk,s) is ℓ
2
(Z

2
). From now on we

regard dc, δc and −∆c as the following operators

d
c

: H
r
→ H

r+1
, δ

c
: H

r
→ H

r−1
, −∆

c
: H

r
→ H

r
,

where r = 0, 1, 2. It is convenient to suppose that H−1 = H3 = 0.



CUBO
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Green Function for a Two-Dimensional Discrete ... 53

Theorem 1. The operators

−∆
c

: H
r
→ H

r, r = 0, 1, 2, (3.3)

are bounded and self-adjoint. Moreover, ‖ − ∆c‖ = 8, where ‖ − ∆c‖ denotes the operator norm of

−∆c.

Proof. We begin by proving self-adjointness of −∆c for the case r = 0. Let ϕ,ψ ∈ H0 and

ω = (u,v) ∈ H1. Then the Green formula (2.13) can be rewritten as

(dcϕ, ω)N =

N∑

k=−N

(ϕk,τ Nvk,N − ϕk,−N vk,−τ N )+

+

N∑

s=−N

(ϕτ N,suN,s − ϕ−N,su−τ N,s) + (ϕ, δ
cω)N (3.4)

The substitution of dcψ for ω in (3.4) gives

(dcϕ, dcψ)N =

N∑

k=−N

(
ϕk,τ N (ψk,τ N − ψk,N ) − ϕk,−N (ψk,−N − ψk,−τ N )

)
+

+

N∑

s=−N

(
ϕτ N,s(ψτ N,s − ψN,s) − ϕ−N,s(ψ−N,s − ψ−τ N,s)

)
+ (ϕ, δcdcψ)N . (3.5)

Letting N → +∞ we get

(dcϕ, dcψ) = (ϕ, δcdcψ). (3.6)

It follows immediately that −∆c : H0 → H0 is self-adjoint.

The same proof remains valid for the case r = 2. Now we have −∆c = dcδc and the analog of

relation (3.5) results from the inner product (δcη, δcζ)N , where η,ζ ∈ H
2
. A trivial verification

shows that properties of (δcη, δcζ)N are completely similar to those of (d
cϕ, dcψ)N . Hence

(δ
c
η, δ

c
ζ) = (η, d

c
δ

c
ζ). (3.7)

Finally, let r = 1. In this case we have −∆c = dcδc + δcdc and we must study the sum

(dcω, dcϑ) + (δcω, δcϑ), where ω = (u,v) ∈ H1, ϑ = (f,g) ∈ H1. Taking in (2.13) α = ω and

β = dcϑ we obtain the analog of relation (3.5) for 1-forms

(dcω, dcϑ)N ≡

N∑

k,s=−N

(∆kvk,s − ∆suk,s)(∆kgk,s − ∆sfk,s) =

=

N∑

k=−N

[
uk,−N (∆kgk,−τ N − ∆Nfk,−τ N ) − uk,τ N (∆kgk,N − ∆N fk,N )

]
+

+

N∑

s=−N

[
vτ N,s(∆NgN,s − ∆sfN,s) − v−N,s(∆Ng−τ N,s − ∆sf−τ N,s)

]
+

+ (ω, δcdcϑ)N.



54 Volodymyr Sushch CUBO
10, 2 (2008)

Letting N → +∞ we obtain equation (3.6) for the 1-forms ω, ϑ ∈ H1. In the same manner we

can see that equation (3.7) holds for ω, ϑ ∈ H1. Adding we obtain

(dcω, dcϑ) + (δcω, δcϑ) = (ω, −∆cϑ). (3.8)

Thus it follows that

(−∆
cω, ϑ) = (ω, −∆cϑ).

For the rest of the proof let α ∈ Hr be an arbitrary r-form. Substituting (2.19) into (2.12) we

get

|(−∆
cα, α)| =

∣∣∣
∑

k,s

(4αk,s − ατ k,s − αk,τ s − ασk,s − αk,σs)αk,s

∣∣∣ ≤

≤ 4
∑

k,s

|αk,s|
2

+

∑

k,s

|ατ k,sαk,s| +
∑

k,s

|αk,τ sαk,s|+

+

∑

k,s

|ασk,sαk,s| +
∑

k,s

|αk,σsαk,s| ≤ 8‖α‖
2.

From this we conclude that ‖−∆c‖ ≤ 8. Since −∆c is self-adjoint, it follows easily that ‖−∆c‖ = 8

(see for instance [10, Ch. 3]).

Corollary 2. The operators (3.3) are positive, i.e. for any non-trivial r-form α ∈ Hr we have

(−∆
cα, α) > 0.

Proof. This follows from (3.6), (3.7) and (3.8).

Corollary 3. For any r, r = 0, 1, 2, we have

σ(−∆c) = [0, 8],

where σ(−∆c) denotes the spectrum of −∆c.

Proof. Straightforward.

4 Discrete analog of the Green function

Let ̺(−∆c) = C \ σ(−∆c) denotes the resolvent set of −∆c. In this section we try to describe the

resolvent operator (−∆c − λ)−1, λ ∈ ̺(−∆c), of the operator −∆c : H0 → H0. Let us introduce

a discrete form G(x,x̃) on C0 × C0 as follows

G(x,x̃) =
∑

k,s

Gk,s(x̃)x
k,s, where Gk,s(x̃) =

∑

m,n

Gk,s,m,nx
m,n



CUBO
10, 2 (2008)

Green Function for a Two-Dimensional Discrete ... 55

and Gk,s,m,n ∈ C for any k,s,m,n ∈ Z. Hence we have

G(x,x̃) =
∑

k,s

∑

m,n

Gk,s,m,nx
k,sxm,n. (4.1)

This is a so-called discrete double form (for details see [11]). Note that in the continual case (for

differential forms) this construction is due to De Rham [13]. It is obvious that the basis elements

xk,s and xm,n in (4.1) commute and G(x,x̃) = G(x̃,x). By analogy with (2.2), the double 0-form

G(x,x̃) can be written pointwise as

< xk,sxm,n, G(x,x̃) >= Gk,s,m,n. (4.2)

Let ϕ ∈ H0. Then we define a 0-form δm,n, m,n ∈ Z, by setting

(ϕ, δm,n) =
∑

k,s

ϕk,sδ
m,n

k,s
= ϕm,n, (4.3)

where δ
m,n

k,s
is the Kronecker delta. By analogy with the continual case, the 0-form δm,n defined by

(4.3) will be called a discrete analog of Dirac’s δ-function at the point xm,n. We can write δ
m,n

as

δ
m,n

=

∑

k,s

δ
m,n

k,s
x

k,s
= x

m,n
.

We need also the following double form

δ(x,x̃) =
∑

k,s

∑

m,n

δ
m,n

k,s
xk,sxm,n =

∑

m,n

δm,nxm,n =
∑

m,n

xm,nxm,n. (4.4)

Definition 4. The double form (4.1) is called the Green function for the operator −∆c : H0 → H0

if

Gk,s,m,n(λ) = (δ
k,s
, (−∆

c
− λ)

−1
δ

m,n
) (4.5)

for any k,s,m,n ∈ Z.

Of course,

(−∆
c
− λ)−1δm,n = Gm,n(x,λ).

It follows easily that

(−∆
c
− λ)xG(x,x̃,λ) = δ(x,x̃), (4.6)

where (−∆c − λ)x is the operator −∆
c − λ that acts with respect to x. Recall that in our

abbreviation x corresponds to k,s.

We have

(−∆
c
− λ)−1ϕ =

∑

k,s

∑

m,n

Gk,s,m,n(λ)ϕm,nx
k,sxm,n, ϕ ∈ H0, λ ∈ ̺(−∆c).



56 Volodymyr Sushch CUBO
10, 2 (2008)

Indeed, applying (−∆
c
− λ)x to the right-hand side gives

(−∆
c
− λ)x

∑

m,n

ϕm,nGm,n(x,λ)x
m,n

=

∑

m,n

ϕm,n(−∆
c
− λ)xGm,n(x,λ)x

m,n
=

=

∑

m,n

ϕm,nδ
m,nxm,n =

∑

m,n

ϕm,nx
m,nxm,n = ϕ.

We now try to write the Green function for −∆
c

in a somewhat more explicit way. For this

we construct a solution of the equation

−∆
cϕ = λϕ, λ ∈ C. (4.7)

The following construction is adapted from [14], where the Green function is studied for Jacobi

operators. By (2.19), equation (4.7) can be written pointwise (at the point xk,s) as

4ϕk,s − ϕτ k,s − ϕk,τ s − ϕσk,s − ϕk,σs = λϕk,s. (4.8)

Applying the transformation λ = −4µ + 4 we reduce (4.8) to the equation

1

4
(ϕτ k,s + ϕk,τ s + ϕσk,s + ϕk,σs) = µϕk,s. (4.9)

An easy computation shows that, substituting the ansatz ϕk,s = p
k+s

into (4.9), we obtain

ϕ
±

k,s
(µ) = (µ ± R(µ))

k+s
, (4.10)

where R(µ) = −
√
µ2 − 1 and

√
· denotes the standard brunch of the square root. It follows that

we can write the solutions ϕ±(λ) of (4.7) in the form

ϕ±(λ) =
∑

k,s

ϕ
±

k,s
(µ)xk,s, (4.11)

where ϕ
±

k,s
(µ) are given by (4.10) and µ = 1 − λ

4
. Obviously, λ ∈ [0, 8] if and only if µ ∈ [−1, 1].

By Corollary 3, λ ∈ ̺(−∆c) leads to µ ∈ C \ [−1, 1].

It is convenient to write ϕ
±

k,s
(µ) as

ϕ
±

k,s
(µ) = ϕ

±

k
(µ) · ϕ±

s
(µ), (4.12)

where ϕ
±

k
(µ) = (µ ± R(µ))k and k,s ∈ Z. Suppose µ ∈ C \ [−1, 1]. Then one has to examine

that the sequences (ϕ
±

k
(µ))k∈Z are square summable near ±∞ respectively, i.e these are ℓ

2
(Z) near

±∞.

Let H0
±

denotes the set of 0-forms whose restriction to K0
±

belongs to H0. Here K0+ (K
0
−

)

denotes the set of 0-cochains (2.9) with k + s > 0 (k + s < 0).

Proposition 5. Let λ ∈ ̺(−∆c). Then ϕ±(λ) ∈ H0
±
.

Proof. This follows immediately from (4.12).



CUBO
10, 2 (2008)

Green Function for a Two-Dimensional Discrete ... 57

Lemma 6. For any k,s ∈ Z the component ϕ+
k
(µ) · ϕ−

s
(µ) is a solution of (4.9)

Proof. It is easy to check that

ϕ
±

2 − 2µϕ
±

1 + 1 = 0.

It follows that ϕ
+
k
(µ) · ϕ−

s
(µ) satisfies equation (4.9). Indeed, putting this in (4.9), we have

ϕ
+
σk

(µ)ϕ−
σs

(µ)
[
(ϕ

+
2 − 2µϕ

+
1 + 1)ϕ

−

1 + (ϕ
−

2 − 2µϕ
−

1 + 1)ϕ
+
1

]
= 0.

Theorem 7. Let λ ∈ ̺(−∆c). Then the components (4.5) of the Green function are given by

Gk,s,m,n(λ) =
−1

4R(µ)





(
µ + R(µ)

)
|τ k−m|+|τ s−n|

for k = m, s > n
or k > m, s = n,(

µ + R(µ)
)
|σk−m|+|σs−n|

for k = m, s < n
or k < m, s = n,(

µ + R(µ)
)
|k−m|+|s−n|

for the all others,

(4.13)

where µ = 1 − λ
4
.

Proof. We must prove that

< xk,sxm,n, (−∆
c
− λ)xG(x,x̃) >= δ

m,n

k,s
, (4.14)

where G(x,x̃) is given by (4.1). Using (2.19) and (4.1), we can rewrite the left-hand side of (4.14)

as

(4 − λ)Gk,s,m,n − Gτ k,s,m,n − Gσk,s,m,n − Gk,τ s,m,n − Gk,σs,m,n =

= 4µGk,s,m,n − (Gτ k,s,m,n + Gσk,s,m,n + Gk,τ s,m,n + Gk,σs,m,n). (4.15)

The proof falls naturally into three parts.

Fix xm,n. First, let k = m and s = n. Substituting (4.13) into (4.15) and using (4.12), we

obtain

< xk,sxm,n, (−∆
c
− λ)xG(x,x̃) > =

1

R(µ)
(−µϕ

+
0 (µ)ϕ

+
0 (µ) + ϕ

+
1 (µ)ϕ

+
0 (µ)) =

=
−µ + ϕ

+
1

R(µ)
= 1.

Note that ϕ
±

0 (µ) = (µ ± R(µ))
0

= 1.

Now we show that (4.13) satisfies equation (4.8) for the cases k = m, s 6= n and k 6= m,

s = n. Check the case k = m, s > n. In this case the left-hand side of (4.14) is equal to

−1

4R(µ)

[
4µϕ

+
1 (µ)ϕ

+
s−n+1(µ) − ϕ

+
2 (µ)ϕ

+
s−n+1(µ) − ϕ

+
0 (µ)ϕ

+
s−n+1(µ)−

− ϕ
+
1 (µ)ϕ

+
s−n+2(µ) − ϕ

+
1 (µ)ϕ

+
s−n

(µ)
]

=
ϕ

+
s−n+1(µ)

2R(µ)
(ϕ

+
2 − 2µϕ

+
1 + 1) = 0.



58 Volodymyr Sushch CUBO
10, 2 (2008)

The proof for the case k = m, s < n (or k 6= m, s = n) is similar.

Finely, let k 6= m, s 6= n. We give the proof only for the case k > m, s < n. For the other

cases the proof runs similarly. In this case we have

Gk,s,m,n(λ) =
1

−4R(µ)
ϕ

+
k−m

(µ)ϕ
+
n−s

(µ) =
1

−4R(µ)
ϕ

+
k
(µ)ϕ−

m
(µ)ϕ+

n
(µ)ϕ−

s
(µ).

Here we use that ϕ
+
−k

= ϕ
−

k
for any k ∈ Z. By Lemma 6, ϕ

+
k
(µ)ϕ−

s
(µ) is a solution of (4.9) and so

is Gk,s,m,n(λ) at the point xk,s.

Thus, since G(x,x̃,λ) with components given by (4.13) satisfies (4.6) and, by Proposition 5,

G(x,x̃,λ) ∈ H0 with respect to x, it must be the Green function of −∆c : H0 → H0.

It should be noted that the consideration of −∆c : H2 → H2 does not differ from that carried

out for the 0-forms ϕ ∈ H0. In this case we consider (4.1) as a double form on C2 × C2 with

basis elements Ω
k,s

Ω
p,q

. Then we define the Green function as above and its components are given

by (4.13). The situation with −∆c : H1 → H1 is more difficult. In this case, having written

equation (4.7) pointwise (at the elements e1
k,s

and e2
k,s

), we obtain a pair of equations (4.8) which

corresponds to the components uk,s, vk,s of ω ∈ H
1
. Roughly speaking, here we must describe

the Green function for each component of the 1-form ω = (u,v). The reader can verify that the

components Gk,s,m,n(λ) of the Green function have again the form (4.13).

Received: December 2007. Revised: February 2008.

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Green Function for a Two-Dimensional Discrete ... 59

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