Next order energy asymptotics for Riesz potentials on flat tori

# Next order energy asymptotics for Riesz potentials on flat tori

## Abstract.

Let be a lattice in with positive co-volume. Among -periodic -point configurations, we consider the minimal renormalized Riesz -energy . While the dominant term in the asymptotic expansion of as goes to infinity in the long range case that (or ) can be obtained from classical potential theory, the next order term(s) require a different approach. Here we derive the form of the next order term or terms, namely for they are of the form and where we show that the constant is independent of the lattice .

###### Key words and phrases:
Periodic energy, Convergence factor, Ewald summation, Completely monotonic functions, Lattice sums, Epstein Hurwitz Zeta function
###### 2000 Mathematics Subject Classification:
Primary: 52C35, 74G65; Secondary: 40D15.
This research was supported, in part, by the U. S. National Science Foundation under the grant DMS-1412428 and DMS-1516400.

## 1. Preliminaries

Let be a nonsingular matrix with -th column and let denote the lattice generated by . The set

 Ω=ΩΛ:={w:w=d∑j=1αjvj,αj∈[0,1),j=1,2,…,d}.

is a fundamental domain of the quotient space ; i.e., the collection of sets tiles . The volume of , denoted by , equals and is called the co-volume of (in fact, any measurable fundamental domain of has the same volume). We will let denote the the dual lattice of which is the lattice generated by .

For an interaction potential , we consider the -energy of an -tuple

 (1) EF(ωN):=N∑k=1N∑j=1j≠kF(xk−xj),

and for a subset , we consider the -point minimal -energy

 (2) EF(A,N):=infωN∈ANEF(ωN).

In this paper we are mostly concerned with -periodic potentials , that is, for all . For such an , the energy is -periodic in each component and so, without loss of generality, we may assume that ; i.e., . Specifically, in this paper we consider periodized Riesz potentials and periodized logarithmic potentials and (or, equivalently ) as we next describe.

For , we consider the periodic potential generated by the Riesz -potential as follows

 (3) ζΛ(s;x):=∑v∈Λ1|x+v|s,s>d,x∈Rd,

which is finite for and equals when . Then can be considered to be the energy required to place a unit charge at location in the presence of unit charges placed at with charges interacting through the Riesz -potential. For , the sum on the right side of (3) is infinite for all . In , -periodic energy problems for a class of long range potentials are considered and it is shown that for the case of the Riesz potential with , the appropriate energy problem can be obtained through analytic continuation. Specifically, it follows from Theorems 1.1 and 3.1 of  that the potential defined by

 (4) Fs,Λ(x):=∑v∈Λ∫∞1e−|x+v|2tts2−1Γ(s2)dt+1|Λ|∑w∈Λ∗∖{0}e2πiw⋅x∫10πd2td2e−π2|w|2tts2−1Γ(s2)dt,

is, for fixed , an entire function of , and

 (5) Fs,Λ(x)=ζΛ(s;x)+2πd2|Λ|−1Γ(s2)(d−s),s>d,

showing that (4) provides an analytic continuation of to (note that has a simple pole at for ). We refer to (the analytically extended) as the Epstein Hurwitz zeta function for the lattice . We shall also need the Epstein zeta function defined for by

 (6) ζΛ(s):=∑v∈Λ∖{0}1|v|s,

and continued analytically as above for . We remark that (4) is derived from the formula

 (7) r−s=∫∞0e−tr2ts2−1Γ(s2)dt,(r>0),

together with the Poisson Summation Formula.

In , analytic continuation and periodized Riesz potentials are connected through the use of convergence factors; i.e., a parametrized family of functions such that

• for , decays sufficiently rapidly as so that

 Fs,a,Λ(x):=∑v∈Λfs(x+v)ga(x+v)

converges to a finite value for all , and

• for all .

For example, the family of Gaussians is a convergence factor for Riesz potentials. In , it is shown that for a large class of convergence factors (including the Gaussian convergence family) one may choose (depending on the convergence factor ) such that

 (8) Fs,Λ(x)=lima→0+(Fs,a,Λ(x)−Ca).

Then, for , represents the energy required to place a unit charge at location in the presence of unit charges placed at with charges interacting through the potential . This leads us to consider, for , the periodic Riesz -energy of associated with the lattice defined by

 (9) Es,Λ(ωN):=∑1≤k,j≤Nk≠jFs,Λ(xk−xj),

as well as the minimal -point periodic Riesz -energy

 (10) Es,Λ(N):=EFs,Λ(Rd;N)=infωN∈(Rd)NEs,Λ(ωN).

We shall also consider the periodic logarithmic potential associated with generated from the logarithmic potential using convergence factors as above and resulting in the definition

 (11) Flog,Λ(x):=∑v∈Λ∫∞1e−|x+v|2tdtt+1|Λ|∑w∈Λ∗∖{0}e2πiw⋅x∫10πd2td2e−π2|w|2tdtt.

Comparing (11) and (4), it is not difficult to obtain (cf. ) the relations

 (12) Flog,Λ(x)=lims→0Γ(s2)Fs,Λ(x)=2(ddsFs,Λ(x))∣∣∣s=0=2ζ′Λ(0;x)+2πd/2d|Λ|−1,

where the prime denotes differentiation with respect to the variable . We then define the periodic logarithmic energy of ,

 (13) Elog,Λ(ωN):=∑1≤k,j≤Nk≠jFlog,Λ(xj−xk),

and also the -point minimal periodic logarithmic energy for ,

 (14) Elog,Λ(N):=infωN∈(Rd)NElog,Λ(ωN).

For , the kernel is positive definite and integrable on and so there is a unique probability measure (called the Riesz -equilibrium measure) that minimizes the continuous Riesz -energy

 Is,Λ(μ):=∬ΩΛ×ΩΛKs,Λ(x,y)dμ(x)dμ(y)

over all Borel probability measures on . From the periodicity of and the uniqueness of the equilibrium measure, it follows that where denotes Lebesgue measure restricted to and normalized so that ; i.e., is the normalized Haar measure for . The periodic logarithmic kernel is conditionally positive definite and integrable and it similarly follows that is the unique equilibrium measure minimizing the periodic logarithmic energy

 Ilog,Λ(μ):=∬ΩΛ×ΩΛKlog,Λ(x,y)dμ(x)dμ(y)

over all Borel probability measures on .

lt is not difficult to verify (cf. ) that

 (15) ∫ΩΛζΛ(s;x) dλd(x)=0,0

and

 (16) ∫ΩΛζ′Λ(0;x) dλd(x)=0,

from which we obtain

 (17) Is,Λ(λd)=2πd2|Λ|−1Γ(s2)(d−s),0

and

 (18) Ilog,Λ(λd)=2πd/2d|Λ|−1.

It then follows (cf. ) that

 (19) limN→∞Es,Λ(N)N2=2πd2|Λ|−1Γ(s2)(d−s),0

and

 (20) limN→∞Elog,Λ(N)N2=2πd/2d|Λ|−1.

## 2. Main Results

Our main result is the following asymptotic expansion of the periodic Riesz and logarithmic minimal energy as .

###### Theorem 1.

Let be a lattice in with co-volume . Then, as ,

 (21) Es,Λ(N) =2πd2|Λ|−1Γ(s2)(d−s)N2+Cs,d|Λ|−s/dN1+sd+o(N1+sd),0

where and are constants independent of .

Petrache and Serfaty establish in  a result closely related to (21) for point configurations interacting through a Riesz potential and confined by an external field for values of the Riesz parameter and Sandier and Serfaty prove in  a result closely related to (22) for the case that and .

For comparison, when it is known that the leading order term of is the same as that of .

###### Theorem 2 (, ).

Let be a lattice in with co-volume . For , there is a positive and finite constant such that

 (23) limN→∞Es,Λ(N)N1+s/d =limN→∞Es(ΩΛ;N)N1+s/d=Cs,d|Λ|−s/d,s>d, (24) limN→∞Ed,Λ(N)N2logN =limN→∞Ed(ΩΛ,N)N2logN=2πd/2dΓ(d2)|Λ|−1,.

By considering scaled lattice configurations (see Lemma 7) of the form for a lattice of co-volume 1, we obtain the following upper bound for that holds both for and where is as in Theorem 1 as well as for where is as in Theorem 1.

###### Corollary 3.

Let be a -dimensional lattice with co-volume 1. Then,

 (25) Cs,d≤{ζΛ(s),s>0,s≠d,2ζ′Λ(0),s=log.

The constant for appearing in (23) is known only in the case where and denotes the classical Riemann zeta function. For dimensions , and , it has been conjectured (cf. [2, 1] and references therein) that for is also given by an Epstein zeta function, specifically, that for denoting the equilateral triangular (or hexagonal) lattice, the lattice, the lattice, and the Leech lattice (all scaled to have co-volume 1) in the dimensions and 24, respectively. In , it is shown that periodized lattice configurations for these special lattices are local minima of the energy for a large class of energy potentials that includes periodic Riesz -energy potentials for .

Finally, we would like to say a little more about periodizing Riesz potentials. It is elementary to verify that

 |x+v|−s−|v|−s=O(|v|−(s+1)),(|v|→∞).

Let be a bounded set containing an open neighborhood of the origin and for let (the set scaled by ). Then, we have

 (26) ζ(s;x)−ζ(s)=|x|−s+(1/2)limL→∞∑v∈CL∖{0}|x+v|−s−|v|−s,(x∉Λ),

for . Further assuming that is centrally symmetric (i.e., ) and using the fact (see (53) Section 5) that

 |x+v|−s+|x−v|−s−2|v|−s=O(|v|−(s+2)),(|v|→∞),

shows that (26) holds for and so we obtain (up to a constant depending only on ) the same periodic Riesz potential given in (5); i.e., the convergence factors procedure and the limit in (26) give the same result. However, (26) breaks down for . In this case the energy is dominated by long range contributions from translates near the boundary of and (26) no longer holds. In fact, as a consequence of Theorem 4 below, the right hand side of (26) is for all . For the case that , the unit ball in centered at the origin, we find that (26) can be ‘renormalized’ by dividing by

 (27) DL:=∑v∈CL∖{0}|v|−s,

but this leads to a non-periodic potential.

###### Theorem 4.

Let , , and for , let be given by (27). Then

 (28) limL→∞D−1L⎡⎣|x|−s+(1/2)∑v∈CL∖{0}|x+v|−s−|v|−s⎤⎦=−(s/d)(d−s−2)|x|2,

uniformly in on compact subsets of .

Remarks:

• Theorem 4 implies that for an -point configuration the energy sum

 (29)

as .

• Since the first term on the right side of (29) vanishes if , the dominant term of the left hand side of (29) is not determined by (28) when ; i.e., all we know is that this term is .

## 3. The Epstein Hurwitz Zeta Function

In this section, we will review some relevant terminology and notation involving special functions that will be crucial for our analysis in Section 4.

An argument utilizing the integral representation of the Riesz kernel given in (7) together with the Poisson Summation Formula can be used to establish the following lemma (cf., [9, Section 1.4]).

###### Lemma 5.

The Epstein zeta function can be analytically continued to through the following formula :

 ζΛ(s)=2Γ(s2)⎛⎜⎝2πd2|Λ|−1s−d−1s⎞⎟⎠+∑v∈Λ∖{0}∫∞1e−|v|2tts2−1Γ(s2)dt+1|Λ|∑w∈Λ∗∖{0}∫10πd2td2e−π2|w|2tts2−1Γ(s2)dt.

Remark. From and , it follows that and for any lattice .

Next we establish the following relation between the Epstein and Epstein Hurwitz zeta functions.

###### Lemma 6.

Let be a sublattice of . Then for any , it holds that

 (30) ∑x∈Λ′∩ΩΛ∖{0}ζΛ(s;x)=ζΛ′(s)−ζΛ(s).
###### Proof.

It is sufficient to prove that (30) holds for , since the general result follows from the fact that both sides of this relation are analytic on . For , we have by definition

 ζΛ′(s) =∑x∈Λ′∖{0}1|x|s=∑x∈Λ′∩ΩΛ∖{0}∑v∈Λ1|x+v|s+∑v∈Λ∖{0}1|v|s =∑x∈Λ′∩ΩΛ∖{0}ζΛ(s;x)+ζΛ(s),

thus proving the lemma. ∎

Using the above lemma and scaling properties of Epstein zeta functions we obtain the following:

###### Lemma 7.

For every and , it holds that

 (31) ∑x∈1mΛ∩ΩΛ∖{0}ζΛ(s;x)=(ms−1)ζΛ(s).

Therefore,

 (32) ∑x,y∈1mΛ∩ΩΛx≠yζΛ(s;x−y)=md(ms−1)ζΛ(s).

We will also require the following two lemmas, which establish continuity properties of the Epstein Hurwitz Zeta function with respect to the lattice.

###### Lemma 8.

Let be a sequence of matrices such that in norm as . Fix any distinct and in and suppose and are sequences in converging to and , respectively. Then for any compact set , converges to and converges to uniformly for in as .

###### Proof.

Let and . Notice that is finite since is entire. Let be large enough so that . Using (4), we have

 (33) |ζPmΛ(s;Pm(xm−ym))−ζPΛ(s;P(x−y))|=|Fs,PmΛ(Pmxm−Pmym)−Fs,PΛ(Px−Py)|≤∫∞1∑v∈Λ∣∣e−|Pm(xm−ym+v)|2t−e−|P(x−y+v)|2t∣∣|ts2−1||Γ(s2)|dt+∫10∑w∈Λ∗∖{0}∣∣∣e2πiw⋅(xm−ym)e−π2|(P−1m)Tw|2t−e2πiw⋅(x−y)e−π2|(P−1)Tw|2t∣∣∣πd2|ts−d2−1||Γ(s2)|dt≤∫∞1∑v∈Λ∣∣e−|Pm(xm−ym+v)|2t−e−|P(x−y+v)|2t∣∣tR2−1infs∈K|Γ(s2)|dt+∫10∑w∈Λ∗∖{0}∣∣∣e2πiw⋅(xm−ym)e−π2|(P−1m)Tw|2t−e2πiw⋅(x−y)e−π2|(P−1)Tw|2t∣∣∣πd2tr−d2−1infs∈K|Γ(s2)|dt.

As in , it is elementary to establish that integrals of the form

 ∫∞1∑v∈Λe−|P(x−y+v)|2ttR2−1dt and ∫10∑w∈Λ∗∖{0}e−π2|(P−1)Tw|2tπd2tr−d2−1dt

are finite and thus, by dominated convergence, it follows that the expressions in (33) tend to zero as .

Cauchy’s integral formula for derivatives then implies that uniformly for as . ∎

We remark that the proof of Lemma 8 shows that converges to as uniformly for in any compact set of .

###### Corollary 9.

Let be a sequence of matrices such that in norm as and suppose or . Then, for all , we have as .

###### Proof.

Let be such that is an optimal -point configuration. Then,

 limsupm→∞Es,PmΛ(N)≤limsupm→∞Es,PmΛ(Pmω∗N)=Es,PΛ(Pω∗N)=Es,PΛ(N),

where the next to last equality follows from Lemma 8.

Next let be such that is an optimal -point configuration for . Let be a subsequence such that

 limk→∞Es,PmkΛ(PmkωmkN)=liminfm→∞Es,PmΛ(N).

Using the compactness of in the ‘flat torus’ topology, we may assume without loss of generality that converges to some -point configuration ; i.e., as for each . Then we have

 liminfm→∞Es,PmΛ(N)=limk→∞Es,PmkΛ(PmkωmkN)=Es,PΛ(P~ωN)≥Es,PΛ(N),

where the next to last equality follows from Lemma 8. ∎

Finally, the following result expresses continuity properties of the Epstein zeta function with respect to the lattice similar to the results in Lemma 8 for the Epstein Hurwitz zeta function.

###### Lemma 10.

Let be a sequence of matrices such that in norm as . Then for any compact set , converges to uniformly in and hence for all as .

###### Proof.

Using Lemma 5, a similar argument as in the proof of Lemma 8 implies that converges uniformly to on compact sets . The convergence of the derivatives then follows from Cauchy’s integral formula for derivatives. ∎

## 4. Proof of Theorem 1

Throughout this section and the next we shall assume that denotes a -dimensional lattice in with fundamental domain , co-volume 1, and generating matrix . Then Theorem 1 follows from a simple rescaling. We shall find it convenient to use what we call the classical periodic Riesz -potential which, for , differs from only by the constant . Similarly, we call the classical periodic logarithmic potential. The energies associated with these potentials are given by

 (34) Ecps,Λ(ωN):=∑j≠kζΛ(s;xj−xk),(s>0),

and, similarly,

 (35) Ecplog,Λ(ωN):=2∑j≠kζ′Λ(0;xj−xk),

and we denote the respective minimal -point energies by and .

From (5), we obtain

 (36) Es,Λ(N)=Ecps,Λ(N)+2πd2Γ(s2)(d−s)N(N−1),

and

 (37) Elog,Λ(N)=Ecplog,Λ(N)+2πd2dN(N−1),

Define

 g–s,d(Λ) :=liminfN→∞Ecps,Λ(N)N1+s/d, ¯¯¯gs,d(Λ) :=limsupN→∞Ecps,Λ(N)N1+s/d, g–log,d(Λ) :=liminfN→∞Ecplog,Λ(N)+2dNlogNN, ¯¯¯glog,d(Λ) :=limsupN→∞Ecplog,Λ(N)+2dNlogNN.

Our use of these quantities is motivated by the proof of the main results in , and indeed the general strategy of our proofs is similar to that of . More precisely, we shall prove and