Spectral Properties of the Dirichlet Operator $\sum_{i=1}^d (-\partial_i^2)^s$ on Domains in d-Dimensional Euclidean Space

0.2 MB·English

by  Agapitos N. Hatzinikitas

Spectral Properties of the Dirichlet Operator
d ( ∂2)s on Domains in d-Dimensional
i=1 i
−
Euclidean Space
P
3
1 Agapitos N. Hatzinikitas,
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University of Aegean, School of Sciences,
2

n Department of Mathematics, Karlovasi 83200, Samos, Greece
a
J Email: [email protected]

1
2

] Abstract
h
p
In this article we investigate the distribution of eigenvalues of the Dirichlet pseudo-
-
h differential operator d ( ∂2)s, s (1,1] on an open and bounded subdomain Ω Rd
t i=1 − i ∈ 2 ⊂
a andpredictboundson thesumof thefirstN eigenvalues, thecounting function, theRiesz
m P
means and the trace of the heat kernel. Moreover, utilizing the connection of coherent
[
states to the semi-classical approach of Quantum Mechanics we determine the sum for

1 moments of eigenvalues of the associated Schro¨dinger operator.
v
6
0
8
4
Key words: Pseudo-differential Dirichlet operator, Spectral properties, Semi-classical approxi-
.
1
mation
0
3 PACS: 02.30 -f, 02.30.Rz, 03.65.Sq
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:
v
i
X
r
a
1
1 Introduction
In 1912 H. Weyl [16], in a brilliant solution to the asymptotic behaviour of the sequence of
eigenvalues for the Dirichlet Laplacian over the bounded domain Ω R2, proved that
⊂
k Ω
lim = | | (1)
k→∞ k 4π
E
where Ω is the surface area of Ω. Defining the counting function ( ) := ♯ , relation
n
| | N E {E ≤ E}
(1) is equivalent to
Ω
( ) = | | +o( ) as . (2)
N E 4πE E E → ∞
These reasults are now called Wel’s law. Shortly afterwards, he submitted two papers [17, 18]
which contain a generalization of (2) to the three dimensional scalar wave equation and the
extension to the vector Helmholtz wave equation describing the vibrations of the electric field
E~ in an empty cavity Ω with perfectly reflecting walls ∂Ω. Later he conjectured the existence
of a second asymptotic term of lower order in two and three dimensions
|Ω| |∂Ω|√ +o(√ ), d = 2,
4πE ∓ 4π E E E → ∞
( ) = (3)
N E 
 6|Ωπ2|E23 ∓ |1∂6Ωπ|E +o(E), d = 3, E → ∞
where ∂Ω denotes the length of the circumference of the domain in d = 2 and the surface
| |
area in d = 3 dimensions respectively. Also the minus sign refers to the Dirichlet boundary
condition u = 0 and the plus sign to the Neumann boundary condition ∂u/∂n = 0, x ∂Ω.
∂Ω
| ∈
These formulae were justified (under a global condition on the geometry of Ω) by V. Ivrii [7]
and R. Melrose [12] in 1980.
In 1959 R. Blumenthal and R. Getoor [4] obtained the following result
Ω
d d
( ) = | | 2s +o( 2s), s (0,1], (4)
N E (4π)d2Γ(1+ d)E E ∈ E → ∞
2
fortheasymptoticdistributionoftheeigenvalues forasymmetric stableprocessofindexα, with
infinitesimal generator the fractional Laplacian ( ∆)s , by applying Karamata’s Tauberian
Ω
− |
theorem.
G.Po´lya[14]in1961conjecturedforanarbitrarydomainandprovedonlyfortilingdomains,
i.e. domains whose congruent non-overlapping translations cover Rd without gaps, that
Ω S
d−1 d
( ) | | | | 2. (5)
N E ≤ (2π)d d E
(cid:18) (cid:19)
For general domains the conjecture is still open although extensions to product domains Ω
1
×
Ω Rd1+d2, where Ω Rd1, d 2 is a tiling domain and Ω Rd2, d 1 is an arbitrary
2 1 1 2 2
⊂ ⊂ ≥ ⊂ ≥
domain of finite Lebesgue measure, can be found in [9].
The closest result to Po´lya’s inequality for an arbitrary bounded domain in Rd is due to F.
Berezin [3] and, P. Li and S. Yau [10] who proved the sharp bound
k d 4π2 1
Xn=1En ≥ d+2(|Bd||Ω|)d2k1+d2, k ∈ N, |Bd| = d|Sd−1| (6)
2
where B is the volume of the d-ball.
d
| |
This paper is structured as follows:
In Sec.2 we provide the definition of the ~ dependent unitary Fourier transform operator
−
as well as that of the operator d ( ∂2)s through relation (8). The discreteness of the
i=1 − i
spectrum for theDirichlet problem onanopenandbounded d dimensional hypercube withthe
P −
assistanceofLemma(2.4)enableustoproveWeyl’slaw. NextweestimatethesumofthefirstN
eigenvalues andbyTheorem(2.6)weprovePo´lya’s inequality foratilingdomain. Asacorollary
we derive the lower bound of the aforementioned sum. Theorem (2.7) generalizes Po´lya’s
inequality for an open, bounded and simply connected subdomain of Rd and also predicts an
upper bound for the counting function.
In Sec.3 we estimate the Riesz’s mean of order ρ 0 and the partition function taking ad-
≥
vantageoftheirinterconnection throughaLaplacetransform. Upperboundsforbothquantities
are also found for ρ > 1.
In Sec.4 considering a particle moving freely in a subset of the phase space and using
the semi-classical approximation method we determine the sum of its eigenvalues. The result
coincides to the one derived fromWeyl’s law or Po´lya’s inequality after an appropriate rescaling
of the tiling domain. In the presence of a negatively valued potential V L1+2ds(Rd) by
∈
performing a similar calculation we exctract (64) for the sum of the eigenvalues.
In Sec.5 we begin with the definition and basic properties of coherent states. In the sequel,
by making a suitable choice for the normalized coherent states, we find the semi-classical limit
of the expectation value for the corresponding Schro¨dinger operator. Finally, Theorem (5.3)
establishes the semi-classical sum for moments of eigenvalues of the Schro¨dinger operator.
2 The counting function and the sum of eigenvalues for
the Dirichlet problem
Definition 2.1 Given ψ S(Rd), where S is the Schwartz space1, we denote by ~ the unitary
∈ F
~ dependent Fourier operator
−
~ : S(Rd) S(Rd) (7)
F →
defined by
1
( ~ψ)(p) = ψˆ(p) = e−~ihp,xiψ(x)dx. (8)
F (2π~)2d Rd
Z
The integral in (8) is understood as the limit ψˆ = lim ψˆ in the strong topology in L2(Rd),
n→∞ n
where
1 n
ψˆ (p) = e−~ihp,xiψ(x)dx, n Rd.
n (2π~)2d Z−n ∈
The inverse Fourier transformation is given by
1
( −1ψˆ)(x) = ψ(x) = e~ihp,xiψˆ(p)dp. (9)
F~ (2π~)d2 Rd
Z
1The linear space consisting of all ψ C∞(Rd) for which
∈
n m
supx∈Rd x (D )ψ)(x) < .
| | ∞
3
Definition 2.2 Let s (1,1], ψ : Rd R and
∈ 2 →
2s,~ : S(Rd) L2(Rd)
L →
where L2s,~ = − di=1(−~2∂i2)s 2 and ∂i denotes the partial derivative w.r.t. xi. We define the
operator 2s,~ by
L P
1
(L2s,~ψ)(x) := (2π~)2d Rd e~ihp,xikpk2sψˆ(p)dp
Z
= ~−1gˆ (x), gˆ(p) = p 2s( ~ψ)(p). (10)
F k k F
In (10) ~ is Planck’s constant and p(cid:0) 2s =(cid:1) d p 2s is the 2s norm3 corresponding to the
k k i=1| i| −
symbol of the pseudo-differential operator [15].
P
Note that ( ∆)s = ( d ∂2)s = d ( ∂2)s unless s = 1. Definition (2.2) is initiated by
− − i=1 i 6 i=1 − i
the anisotropic fractional diffusion equation
P P
d
∂ψ(x,t)
= D ( ∂2)sψ(x,t), (x,t) Rd [0, ] (11)
∂t − i − i ∈ × ∞
i=1
X
considered in [2].
Proposition 2.3 On the open and bounded hypercube Γ Rd, the eigenvalues for the homo-
d
⊂
geneous Dirichlet problem
d
E
( ∂2)sψ (x) = ψ (x), in Γ ; = n
− i n En n d En D
! 2s
i=1
X
ψ (x) = 0 on Γ (12)
n d
are given by
nπ 2s
= , n Zd (13)
En L ∈ +
(cid:13) (cid:13)
(cid:13) (cid:13)
where ψ ∞ forms an orthonormal ba(cid:13)sis i(cid:13)n L2(Γ ) with ψ (x) = c d ψ (x ), at least one
{ n}n=1 d n n j=1 nj j
n should not vanish, and D is a constant with dimensions [D ] = [M]1−2s([L]/[T])2(1−s).
j 2s 2s Q
Proof. The Fourier transformation of the boundary conditions requires p ’s to be discrete and
j
moreover applying Parseval’s identity to (12) it can be proved that the eigenvalues should
n
E
also be discrete and given by (13) provided one makes the substitution p = n π/L. 2
j j
Arranging the positive, real and discrete spectrum of in increasing order (including multi-
2s
L
plicities), we have
0 < (Γ ) < (Γ ) < (Γ ) < and lim (Γ ) = . (14)
1 d 2 d 3 d n d
E E E ··· n→∞E ∞
2The ~ dependence of the operator will be declared explicitly when needed.
3The Euclidean norm will be denoteLd by 2.
2
k·k
4
Remark. If Γ′ Γ Rd such that Γ′ = λd Γ where the scale factor λ (0,1) then the
d ⊂ d ⊂ | d| | d| ∈
nth eigenvalues satisfy
(Γ ) = λ2s (Γ′) (15)
En d En d
as can be checked by (13). The scaling property (15) can be generalized as follows: let Ω′
⊂
Ω Rd and Ω′ = λd Ω then
⊂ | | | |
(Ω) = λ2s (Ω′). (16)
n n
E E
Thisstatement canbeprovedusing(10)withgˆ(p) = p 2s( ~(χΩψ))(p)andmakingthechange
k k F
of variables x = λz.
The following Lemma [8] will be used repeatedly in our study.
Lemma 2.4 The integral formula
d
1
e−kxk2sdx = 2Γ 1+ (17)
2s
ZRd (cid:18) (cid:18) (cid:19)(cid:19)
holds and one may recover from it the volume
2Γ 1+ 1 d
B := Vol(B ) = 2s (18)
| d,2s| d,2s Γ 1+ d
(cid:0) (cid:0) 2s(cid:1)(cid:1)
of the convex unit ball defined as (cid:0) (cid:1)
1
n 2s
B = x Rd : x = x 2s 1 .
d,2s i
∈ k k | | ≤
!
(cid:26) i=1 (cid:27)
X
Proof. Starting from the left-hand side of (17) we have
d ∞ 1 ∞ d
e−kxk2sdx = 2d e−|xi|2sdxi = u21s−1e−udu
s
ZRd i=1(cid:18)Z0 (cid:19) (cid:18) Z0 (cid:19)
Y
d
1
= 2Γ 1+ (19)
2s
(cid:18) (cid:18) (cid:19)(cid:19)
where the factor 2d represents the number of orthants and the gamma function Γ is defined by
Euler’s integral of the second kind
∞
Γ(z) = e−ttz−1dt, Re z > 0. (20)
Z0
On the other hand, the same integral can be computed as
∞ ∞
e−kxk2sdx = e−udu dx = e−udu χ( u [0, ) : x u21s )dx
ZRd ZRd (cid:18)Zkxk2s (cid:19) Z0 (cid:18)ZRd { ∈ ∞ k k ≤ } (cid:19)
∞ ∞
= e−u u21sBd,2s du = Bd,2s u(1+2ds)−1e−udu
| | | |
Z0 Z0
d
= B Γ 1+ . (21)
d,2s
| | 2s
(cid:18) (cid:19)
5
2
Comparing the two expressions (19) and (21) we get the result (18).
Remark. If one uses the Euclidean norm 2 then (18) becomes
k·k2
d
1 2π2
B = S , S = . (22)
| d,2s|E d| d−1| | d−1| Γ(d)
2
Proposition 2.5 The number of eigenvalues in a d-dimensional, 2s-deformed hypersphere
n
of radius R = L 21s, s (1,1] asymptotically (ER ) is given by the counting function
πE ∈ 2 → ∞
N(E) = |(Γ2dπ|E)d2dds|Ad−1,2s|+o(E2ds), |Ad−1,2s| = 2s 2ΓΓ(1d+ 21s) d = d|Bd,2s| (23)
(cid:0) 2s (cid:1)
(cid:0) (cid:1)
where A represents the volume of the 2s-deformed unit sphere S and the little o( )
d−1,2s d−1
| | ·
symbol means a term that grows slower than ( ).
·
Proof. Using the definition of the counting function (i.e. the function that counts the number
of eigenvalues not exceeding a cut off value ) we have
E
( ) := 1 = ♯ n Zd :
N E { ∈ + En ≤ E}
EXn≤E
d
(1=3) ♯{n ∈ Zd+ : |ni|2s ≤ LπE21s = R}
i=1 (cid:18) (cid:19)
X
1
= 2dRd|Bd,2s|+o(E2ds), R → ∞
1 Γ A
d d−1,2s d d
= | || | 2s +o( 2s), (24)
(2π)d d E E E → ∞
2
where Lemma (18) has been applied.
Remarks.
1. Solving (23) w.r.t. := we obtain
N
E E
2s
N = (2π)2s Nd d +o( 2ds). (25)
E A Γ N
(cid:18)| d−1,2s|| d|(cid:19)
Relation (25) represents an extension of Blumenthal’s and Getoor’s result which in the
Euclidean norm case is given by (4). Summing the eigenvalues (25) using the finite series
formula [5]
n nq+1 nq
kq = + +o(nq)
q +1 2
k=1
X
we have
N 2s
S(N) := n = (2π)2s d d d 1+2ds +o( 1+2ds). (26)
E d+2s A Γ N N
n=1 (cid:18)| d−1,2s|| d|(cid:19)
X
6
(ii) Substituting the values s = 1 and d = 2 into relation (23) we recover Weyl’s asymptotic
formula (2) for a square while for the same values of s,d into (25) we confirm Po´lya’s
result (5).
Theorem 2.6 (Po´lya’s inequality for over tiling domains) If Ω Rd is a tiling do-
2s
L ⊂
main then
2s
nd d
(2π)2s . (27)
n
E ≥ A Ω
(cid:18)| d−1,2s|| |(cid:19)
Proof. Let Ω′ be another tiling subdomain of Ω such that Ω′ = λd Ω then (16) holds. Also
| | | |
suppose Γ1 is the unit hypercube in Rd and m be the number of congruent domains Ω′ filling
d
Γ1 without overlapping and leaving gaps. Then we obtain the following two relations
d
1
lim (m Ω′ ) = Γ1 = 1 lim (mλd) = and (28)
m→∞ | | | d| ⇒ m→∞ Ω
| |
1
′′ (Γ1) ′(Ω′) = (Ω) (29)
Enm d ≤ En λ2sEn
where by ( ) we denote the eigenvalue on the corresponding domain. By virtue of (25) and
E ·
combining (28), (29) we have
′′ (Γ1)
En(Ω) ≥ E(nnmm)2dds (nm)2dsλ2s. (30)
Taking the m limit we finally find
→ ∞
2s
nd d
(Ω) (2π)2s . (31)
n
E ≥ A Ω
(cid:18)| d−1,2s|| |(cid:19)
2
Corollary 2.1 If Ω Rd is a tiling domain then
⊂
2s
S(N) (2π)2s d d d 1+2ds. (32)
≥ d+2s A Γ N
(cid:18)| d−1,2s|| d|(cid:19)
2s
Proof. The function f(t) = td is increasing for t 0 and applying the inequality
≥
N−1 N N−1
f(n) f(t)dt f(n+1) (33)
≤ ≤
n=0 Z0 n=0
X X
with the help of (32), we show that
S(N) (2π)2s d 2ds N−1n2ds (2π)2s d 2ds N t2dsdt
≥ A Ω ≥ A Ω
(cid:18)| d−1,2s|| |(cid:19) n=1 (cid:18)| d−1,2s|| |(cid:19) Z0
X
2s
= (2π)2s d d d N1+2ds. (34)
d+2s A Ω
(cid:18)| d−1,2s|| |(cid:19)
2
7
Theorem 2.7 Let Ω be an open, bounded and simply connected set in Rd of finite volume Ω .
| |
Consider the homogeneous Dirichlet eigenvalue problem
d
E
( ∂2)sψ (x) = ψ (x), in Ω; = n
− i n En n En D
! 2s
i=1
X
ψ (x) = 0 on Ω
n
¯
ψ ,ψ = ψ (x)ψ (x)dx = δ , m,n. (35)
n m n m mn
h i ∀
ZΩ
Then
2s
S(N) (2π)2s d d d N1+2ds (36)
≥ d+2s A Ω
(cid:18)| d−1,2s|| |(cid:19)
and the following bound for the counting function valids
d
1 d+2s 2s Ad−1,2s Ω d
(z) | || |z2s. (37)
N ≤ (2π)d d d
(cid:18) (cid:19)
Proof. Consider the extension of ψ ’s by setting them identically zero outside their support,
n
namely
ψ (x), x Ω
φ (x) = n ∈ (38)
n 0, x Ω¯.
(cid:26) ∈
Define the function
N
F (p) := φˆ (p) 2 (39)
N n
| |
n=1
X
and by Plancherel’s theorem observe that
N N N
F (p)dp = φˆ (p) 2dp = φ (x) 2dx = 1 = (Ω). (40)
N n n
| | | | N
ZRd n=1ZRd n=1ZΩ n=1
X X X
Furthermore, using (10) and (35) we derive the expression
N N N
p 2sF (p)dp = p 2s φˆ (p) 2dp = φ , φ = = S(N). (41)
N n n 2s n n
k k k k | | h L i E
Rd Rd
Z n=1Z n=1 n=1
X X X
For every fixed p Rd, since exp(i p,x ) L2(Ω), it follows that
∈ h i ∈
∞
eihp,xi = c (p)φ (x), with c (p) = φ (x)eihp,xidx. (42)
m m m m
m=1 ZΩ
X
Thus from (39) we deduce
∞ ∞ 2
1 1 Ω
F (p) φˆ (p) 2 = c (p) = dx = | | (43)
N n m
≤ | | (2π)d (2π)d (2π)d
Xn=1 (cid:12)(cid:12)mX=1 (cid:12)(cid:12) ZΩ
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) 8 (cid:12)
which is the L2 norm of exp(i p,x ). The function F (p) that minimizes expression (41)
N,min
− h i
and satisfies (40) and (43) should have the form
1
F (p) = Ω χ( (0,r)) (44)
N,min (2π)d| | B
where (0,r) is the 2s-deformed ball with radius r obeying
B
dN
rd = . (45)
Ω A
d−1,2s
| || |
Plugging (44) into (41) with p = 2πk we arrive at the desired result.
To prove (37) we choose z [ , ] and using (36) we have
k k+1
∈ E E
2s
k k S(k) (2π)2s d d d k1+2ds. (46)
E ≥ ≥ d+2s A Ω
(cid:18)| d−1,2s|| |(cid:19)
But k = (z) so
N
2s
z k (2π)2s d d d 2ds (47)
≥ E ≥ d+2s A Ω N
(cid:18)| d−1,2s|| |(cid:19)
2
from which (37) follows.
Remark. Relation (36) for s = 1 is in agreement with Li’s and Yau’s result (6). In terms of
the counting function we obtain the upper bound
d
1 d+2 2 Ω
d
(z) | | z2. (48)
N ≤ (4π)d2 (cid:18) d (cid:19) Γ 1+ d2
(cid:0) (cid:1)
3 Riesz means and the partition function
It is generally believed that things get more manageable if one considers averaged or smoothed
versions of the counting function such as the Riesz mean or the trace of the heat kernel, the
so-called partition function.
Definition 3.1 The Riesz mean of order ρ 0 is defined for > 0 by
≥ E
ρ
d
R ( ) := Tr ( ∂2)s = ( )ρ (49)
ρ E − i Ω −E E −Ej +
!
Xi=1 − Xj
where x := ( x x)/2 denotes the positive and negative part of x R respectively.
±
| |± ∈
The Riesz mean reduces to the counting function when ρ 0+ while for ρ 1− is directly
→ →
realated to the sum of eigenvalues. This quantity describes the energy of non-interacting
fermionic particles trapped in Ω and plays an important role in physical applications. If is
j
E
considered to be a continuous variable then (49) is replaced by
∞ ∞
R ( ) = ( t)ρdN(t) = ρ ( t)ρ−1N(t)dt. (50)
ρ E E − + E − +
Z0 Z0
9
Relation (50) is a limiting case of the following iteration property [1]
1 ∞
R ( ) = ( t)δ−1R (t)dt, ρ 0,δ > 0 (51)
ρ+δ E B(1+ρ,δ) E − + ρ ≥
Z0
whereB(x,y) denotes the beta function defined by the functional relation
Γ(x)Γ(y)
B(x,y) = . (52)
Γ(x+y)
We point out that (51) is nothing but a Riemann-Liouville fractional integral transform. Sub-
stituting (24) into (50) for a tiling domain Ω Rd we learn that
⊂
Rρ(E) ∼ Lcρl,.d|Ω|Eρ+2ds as E → ∞ (53)
where the classical constant is given by
1 Γd(1+ 1 )Γ(1+ρ)
Lcl. = 2s . (54)
ρ,d πd Γ(1+ρ+ d )
2s
One can smooth the counting function even further and consider the partition function defined
by
∞
Z(t) := Tr ePdi=1(−∂i2)sΩt = e−Ej(Ω)t. (55)
(cid:16) (cid:17) Xj=1
If is a continuous variable then (55) is written as [6]
j
E
∞ ∞
Z(t) = e−EtdN( ) = t e−EtN( )d = tL[N( )](t) (56)
E E E ·
Z0 Z0
where L[f( )](t) = ∞e−ztf(z)dz istheLaplace transformofa suitable functionf : (0, )
· 0 ∞ →
R. Again using (24) into (56) we have
R
1 (2Γ(1+ 1 ))d
Z(t) Ω 2s (57)
∼ (2π)d| | t2ds
where L[zδ](t) = Γ(1+δ).
t1+δ
Remarks.
1. Utilizing Theorem (2.7) and applying the Laplace transform to inequality (37), it follows
immediately that
d
Z(t) 1 d+2s 2s |Ad−1,2s||Ω|Γ 1+ d t−2ds. (58)
≤ (2π)d d d 2s
(cid:18) (cid:19) (cid:18) (cid:19)
2. The Laplace transform of (49) for ρ > 1 and definition (20) of gamma function leads to
Γ(1+ρ) Γ(1+ρ)
L[R ( )](t) = e−Ejt = Z(t). (59)
ρ · t1+ρ t1+ρ
j
X
Combining (58) with (59) we obtain the inequality
d
Rρ(E) ≤ (2π1)d d+d2s 2s |Ad−1d,2s||Ω|ρB ρ,1+ 2ds Eρ+2ds. (60)
(cid:18) (cid:19) (cid:18) (cid:19)
10