Pointwise Lower Bounds on the Heat Kernels of Higher Order Schrödinger Operators

0.09 MB·English

by  Narinder Claire

Pointwise Lower Bounds on the Heat Kernels of
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Higher Order Schro¨dinger Operators
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n
a Narinder S Claire
J

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] Abstract
P
A WeapplyDavies’methodforobtainingpointwiselowerboundsontheheatkernelsofhigher-order
. differential operators to obtain pointwise lower bounds in the presence of a polynomialy bounded
h
t potential.
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m
AMS Subject Classification : 35K41
[
Keywords : Heat Kernel, Parabolic, Elliptic, Schr¨odinger , Lower Bounds
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v
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6 1 Introduction
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1 There has been extensive work on the theory for heat kernels of second order elliptic operators but
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the situation for higher order ellipitic operators is considerably different. Higher order heat kernels are
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/ not necessarily positive on the off-diagonal. Despite the limitations of the approaches often used in the
h
t case of second order operators,Davies[1] provided a powerfull tool obtaining the necessary bounds and
a
m establishedalowerboundfortheheatkernelofhigherorderellipticoperatorswithboundedmeasurable
: coefficients on L2 RN where 2m>N. Robinson and Ter Elst[2] obtained bounds for operators on a
v
i Lie Group . Their(cid:16)meth(cid:17)od however required the conservation of probability.
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In this exposition we show that Davies’ theory can also be exploited to obtain pointwise lower bounds
r
a
for heat kernels of the Schr¨odinger operator with a polynomialy bounded potential.
We define
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2m m
hxi:= (|x|2+ρ) where ρ=max{1, −1 }
N
(cid:18) (cid:19)
p
We will assume that H is an elliptic differential operator of order 2m on L2 RN with constant of
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ellipticity c and that for the potential V there are positive constants c and γ such that
1 2 (cid:0) (cid:1)
V (x)≤c hxiγ
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If Q is the quadratic form of H defined with domain equal to the Sobolev space Wm,2(R) where
0 0 0
c−11k(−∆)m2 fk22 ≤Q0(f)≤c1k(−∆)m2 fk22 (1.1)
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Then we define Q, the quadratic form associated with H :=H +V
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Q(f):=Q(f)+kV 21fk2
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Our main result is the following theorem:
Theorem 1.1. If k(t,x,y), the heat kernel generated by H is continuous in all three variables and is
in L2 RN for all x and t>0 and there are constants σ,µ>0 and λ<1 such that
(cid:0) (cid:1) hxi−µ
k(t,x,x)<σ (1.2)
tλ
then we have the following estimates:
Whenever hxiγt>1 we have
hxi−µ (1−2Nm)γ−µ
σ exp −chxi 1−λ t <k(t,x,x) (1.3)
tλ  
 
and when hxiγt<1 we have
hxi−µ −µ 2Nm−λ
σ exp −chxi1−λ t 1−λ <k(t,x,x) (1.4)
tλ
!
where c is some positive constant..
We define the following functions for convenience.
hxi−µ
u(t,x):=σ
tλ
k(t,x,x)
δ(t,x):=
u(t,x)
We summarise some of the key aspects of Davies’ theory [1].
Lemma 1.2 (Davies [1]). Given 0<α,s<1 and p defined by
p+(1−p)αs=s (1.5)
k(ts,x,x)<(u(αts,x))1−p(u(t,x))p k(t,x,x) p
u(t,x)
(cid:18) (cid:19)
Proof. The three line lemma is applied to the analytic function z →k(z,x,y)
k(ts,x,x) < (k(αts,x,x))1−pk(t,x,x)p
< (u(αts,x))1−pk(t,x,x)p
= (u(αts,x))1−p(u(t,x))p k(t,x,x) p
u(t,x)
(cid:18) (cid:19)
and the proof is complete.
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Definition 1.3. If ω is the distribution such that hf,ω i=f(x) then we define G (x,x)
x x t
∞
G (x,x):=h(tH +1)−1ω ,ω i=h e−(tH+1)sds ω ,ω i
t x x x x
Z
0
Lemma 1.4 (Davies [1]).
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G (x,x) u(αts,x)
t < δ(t,x)p ds+e−1δ(t,x) (1.6)
u(t,x) u(t,x)
Z
0
Proof. By lemma 1.2 and using the fact that u(t,x) is a decreasing function we have
k(t,x,x) p
k(ts,x,x)<u(αts,x) (1.7)
u(t,x)
(cid:18) (cid:19)
Hence
1 1
k(t,x,x) p
h e−(tH+1)sds ω ,ω i≤ u(αts,x) ds
x x
u(t,x)
Z Z (cid:18) (cid:19)
0 0
Then the estimate
∞
h e−(tH+1)sds ω ,ω i≤e−1k(t,x,x)
x x
Z
1
and dividing by u(t,x) completes the proof.
Lemma 1.5.
|g(x)|2
G (x,x)= sup { :g 6=0}
t g∈DomQ tQ(g)+kgk22
Proof.
G (x,x) = h(tH +1)−1ω ,ω i
t x x
−1 −1
= h(tH +1) 2 ωx,(tH +1) 2 ωxi
= sup {|h(tH +1)−12 ωx,fi|2 :kfk2 =1}
f∈L2(Ω)
= sup {|(tH +1)−21 f(x)|2 :kfk2 =1}
f∈L2(Ω)
|g(x)|2
= sup { :g 6=0}
1
g∈DomQ k(tH +1)2 gk2
|g(x)|2
= sup { :g 6=0}
g∈DomQ tQ(g)+kgk22
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2 Some Inequalities
Lemma 2.1. If δ and λ are both positive and less than 1 then
(1−λ)1−λeλ−1 lnδ−1 λ−1 >δ
Proof. Follows from maximising the function δ lnδ(cid:0)−1 1−λ(cid:1)over lambda.
Lemma 2.2. For positive α< 1 and p defined(cid:0)as in l(cid:1)emma 1.2
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1
u(αts,x)δ(t,x)pds+e−1δ < Γ(1−λ) +(1−λ)1−λeλ−2 lnδ−1 λ−1
u(t,x) α
Z (cid:18) (cid:19)
0 (cid:0) (cid:1)
Proof.
1 1
u(αts,x)(δ(t,x))pds+e−1δ = (αs)−λ(δ(t,x))pds+e−1δ
u(t,x)
Z Z
0 0
substitution of τ =αs and with p>α we have
1 α
(αs)−λ(δ(t,x))pds< 1 τ−λexp(−|lnδ|τ)dτ
α
Z Z
0 0
a further change of variable y =τlnδ−1 the RHS becomes
lnδ−1 λ−1 α
y−λe−ydy
α
(cid:0) (cid:1) Z
0
we use
α
Γ(1−λ)> y−λe−ydy
Z
0
along with lemma 2.1 to complete the proof.
3 Estimation of the Greens function
If g (y) is an appropriate test function then we can estimate the Greens function with the evaluation
x
|g(x)|2
tQ(g)+kgk2
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We are primarily concerned with behaviour of the heat kernel away from the origin, and thus will con-
sider test functions to reflect this. The choice of test functions is the only significantdifference between
the analysisto find lowerbounds forelliptic operatorsin[1]andour analysishere for Schr¨odingeroper-
ators. We requiretest functions that aremeasurablysensitiveto the potentialaswellas the differential
operator.
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Definition 3.1. We let ψ ∈C∞(R) such that ψ(s)=1 when |s|≤1 and ψ(s)=0 when |s|≥2
c
Definition 3.2. We define g (y)∈C∞ RN
x c
(cid:0) (cid:1) x−y
g (y)=ψ (3.8)
x hxiβ
(cid:18) (cid:19)
for some β(x) such that β(x)<1 for all x∈RN
It is straight forward to show that
k((−∆)mgx)k∞ ≤chxi−β2m
for some poistive c.
Lemma 3.3. There is some positive constant c>0 such that
tQ(g )+kg k2 ≤chxiβN thxi−2mβ +thxiγ +1 (3.9)
x x 2
(cid:0) (cid:1)
Proof. There is some positive constant c such that
Q (g )≤chxiβ(N−2m)
0 x
and moreover there is a positive constant c such that
kV 12gxk22 ≤chxiγ+βN
4 Heat kernel for thxiγ > 1
Lemma 4.1. There is a positive constant c such that
tQ(g )+kg k2 ≤chxi(1−2Nm)γt
x x 2
becomes
Proof. For thxiγ >1 inequality (3.9) reduces to
tQ(g )+kg k2 ≤chxiβNt hxi−2mβ +hxiγ (4.10)
x x 2
(cid:0) (cid:1)
and we optimise the RHS over β with
1 ln 2m −1
β (x)= γ− N
M
2m ln(hxi)
(cid:0) (cid:1)!
and since the definition of hxi implies that 2m −1<hxi2m+γ it follows that β (x)<1 for all x. The
N M
proof is completed by substituting β into 4.10
M
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Lemma 4.2. There is a positive constant c
hxi−µ (1−2Nm)γ−µ
σ exp −chxi 1−λ t <k(t,x,x)
tλ  
 
Proof. From the definition of the test functions we recall that g (x)=1, hence
x
|gx(x)| > chxi(2Nm−1)γ (4.11)
tQ(g )+kg k2 t
x x 2
for some constant c. Application of lemma 2.2 yields the inequality
chxi(2Nm−1)γ+µ < Γ(1−λ) +(1−λ)1−λeλ−2 lnδ−1 λ−1
t1−λ α
(cid:18) (cid:19)
and simplyfying we have (cid:0) (cid:1)
(1− N )γ−µ
2m
−chxi 1−λ t<lnδ
and our inequality is obtained on inverting the logarithm and substitution of u(t,x).
5 Heat kernel for thxiγ < 1
Lemma 5.1. There is some constant c>0 such that
Q(gx)+kgxk22 <ct2Nm
Proof. When thxiγ >1 inequality (3.9) reduces to
Q(g )+kg k2 ≤chxiβN thxi−2mβ +1 (5.12)
x x 2
for some c>0. We find that the RHS is optimised over(cid:0)β for (cid:1)
1 2m
β = ln −1 +ln t
M
2m lnhxi N
(cid:18) (cid:18) (cid:19) (cid:19)
Lemma 5.2. There is a positive constant c such that
hxi−µ −µ 2Nm−λ
σ exp −chxi1−λ t 1−λ <k(t,x,x)
tλ
!
Proof. Using g (x)=1 we have
x
|gx(x)| >ct−2mN (5.13)
tQ(g )+kg k2
x x 2
for some constant c>0. Substituting for u(t,x) and an application of lemma 2.2 we have
cσ−1hxiµtλ−2Nm < Γ(1−λ) +(1−λ)1−λeλ−2 lnδ−1 λ−1
α
(cid:18) (cid:19)
(cid:0) (cid:1)
and we solve to complete the proof.
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Proof of theorem 1.1. Follows from lemmas 5.2 and 4.2
Acknowledgements
This research was funded by an EPSRC Ph.D grant. I would like to thank Brian Davies for giving me
this problem and his encouragementsince. I thank Owen Nicholas and Mark Owen for their invaluable
discussions. I am indebted to Anita for all her support.
References
[1] E.B. Davies, Pointwise Lower Bounds On The Heat Kernels of Higher Order Elliptic Operators
Math Proc Camb Phil Soc 125 1999 p105-111
[2] D.W.Robinson A.F.M.ter Elst Local Lower Bounds on Heat Kernels
Positivity 2(1998), 123–151.
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