Reduction of dimension as a consequence of norm-resolvent convergence and applications

0.26 MB·

by  David Krejcirik

REDUCTION OF DIMENSION AS A CONSEQUENCE OF
NORM-RESOLVENT CONVERGENCE AND APPLICATIONS
D.KREJCˇIRˇ´IK,N.RAYMOND,J.ROYER,ANDP.SIEGL
Abstract. This paper is devoted to dimensional reductions via the norm resolvent conver-
gence. We deriveexplicitbounds on theresolvent difference as well as spectral asymptotics.
The efficiency of our abstract tool is demonstrated by its application on seemingly differ-
ent PDE problems from various areas of mathematical physics; all are analysed in a unified
7 mannernow,knownresultsarerecoveredandnewonesestablished.
1
0
2

n 1. Introduction
a
J 1.1. Motivation and context. In this paper we develop an abstract tool for dimensional

0 reductions via the norm resolvent convergence obtained from variational estimates. The results
3 are relevantin particularfor PDE problems, typically Schr¨odinger-typeoperatorsdepending on

an asymptotic parameter having various interpretations (semiclassical limit, shrinking limits,
]
h large coupling limit, etc.). In applications, our resolvent estimates lead to accurate spectral
p asymptotic results for eigenvalues lying in a suitable region of the complex plane. Moreover,
-
h avoidingthetraditionalmin-maxapproach,withitsfundamentallimitationstoself-adjointcases,
t weobtainaneffectiveoperator,thespectrumofwhichdeterminesthespectralasymptotics. The
a
flexibilityofthelatterisillustratedonanon-self-adjointexampleinthesecondpartofthepaper.
m
The powerofour approachis demonstratedby a unified treatmentofdiverseclassicalas well
[
as latest problems occurring in mathematical physics such as:

1
- semiclassical Born-Oppenheimer approximation,
v
- shrinking tubular neighborhoods of hypersurfaces subject to various boundary conditions,
9
1 - domains with very attractive Robin boundary conditions.
8 Inspiteofthevarietyofoperators,asymptoticregimes,andtechniquesconsideredintheprevious
8
literature, all these results are covered in our general abstract and not only asymptotic setting.
0
Our first result (Theorem 1.1) gives a norm resolvent convergence towards a tensorial operator
.
1 in a general self-adjoint setting. A remarkable feature is that only two quantities need to be
0
controlled: the size of a commutator of a “longitudinal operator” with spectral projection on
7
low lying “transverse modes” and the size of the “spectral gap” of a “transverse operator”,
1
: see (1.5) and (1.2), respectively. Although the latter is also very natural it was hardly visible
v
in existing literature due to many seemingly different technical steps as well as various ways
i
X how these quantities enter. As particular cases of the application of Theorem 1.1, we recover,
r in a short manner, known results for quantum waveguides (see for instance [3], [11], [9] or [10])
a
and cast a new light on Born-Oppenheimer type results (see [12], [17], [7] or [16, Sec. 6.2]). To
keep the presentation short we deliberately do not strive for the weakest possible assumptions
inexamples, althoughthe abstractsetting allowsfor many further generalizationsandit clearly
indicates how to proceed.
In the second part of the paper, we prove, in the same spirit as previous results, the norm
convergence result for a non-self-adjoint Robin Laplacian, see Theorem 1.5. It will partially
generalize previous works in the self-adjoint (see [15], [8] and [14]) and in the non-self-adjoint
(see [2]) cases.
As a matter of fact, the crucial step in all the proofs of the paper is an abstract lemma (see
Lemma 1.7) of an independent interest. It provides a norm resolvent estimate from variational
estimates, which is particularly suitable for the analysis of operators defined via sesquilinear
forms.
2010 Mathematics Subject Classification. Primary: 81Q15;Secondary: 35P05, 35P20,58J50,81Q10, 81Q20.
Keywords and phrases. reductionofdimension,normresolventconvergence, Born-Oppenheimerapproxima-
tion,thinlayers,quantum waveguides, effectiveHamiltonian.
1
2 D.KREJCˇIRˇ´IK,N.RAYMOND,J.ROYER,ANDP.SIEGL
1.2. Reduction of dimension in an abstract setting and self-adjoint applications. We
first describe the reduction of dimension for an operator of the form
L S˚S T, T T , (1.1)
s
“ ` “
sPΣ
à
acting on the Hilbert space H H . The norm and inner product in H will be denoted
“ sPΣ s
by and , , respectively; the latter is assumed to be linear in the second argument.
}¨} x¨ ¨y À
HereΣisameasurespaceandT isaself-adjointnon-negativeoperatoronaHilbertspaceH
s s
for all s Σ. Precisedefinitions willbe givenin Section2. A typicalexample is the Schr¨odinger
P
operator
H i~ 2 i 2 V s,t ,
s t
“p´ B q `p´ B q ` p q
acting on L2 R R . We consider a function s γ such that
s t s
p ˆ q ÞÑ
γ inf γ 0. (1.2)
s
“sPΣ ą
Then we denote by Π L H the spectral projection of T on 0,γ , and we set ΠK
s P p sq s r sq s “
IdHs ´Πs. We denote by Π the bounded operator on H such that for Φ P H and s P Σ we
have ΠΦ Π Φ . We similarly define ΠK L H . Our purpose is to compare some spectral
s s s
propeprtiesqof“the operator L with those of thPe sipmpqler operator
Leff ΠLΠ. (1.3)
“
This is an operator on ΠH with domain ΠH Dom L .
In fact, we will first compare L with X p q
L ΠLΠ ΠKLΠK. (1.4)
“ `
Then Leff and LK will be defined axs the restrictions of L to ΠH and ΠKH, respectively, so
that
L Leff LK. x
“ ‘
We will give a sufficient condition for z ρ L to be in ρ L and, in this case, an estimate for
Pxp q p q
the difference of the resolvents. Then, since ΠH and ΠKH reduce L, it is not difficult to check
thatfar from the spectrum of LK the spectrxalpropertiesof L arethe same as those ofLeff, so
we can state a similar statement with L replaced by Leff. In applixcations, we can for instance
prove that the first eigenvalues of L are close to the eigenvalxues of the simpler operator Leff.
We assume that Dom S is invarianxt under Π, that S,Π extends to a bounded operator
p q r s
on H, and we set
a }rS,Πs}LpHq . (1.5)
“ ?γ
For z C, we also define
P
3 6a 3a a
η z a2γ 1 a z 2 z 2,
1
p q“ ?2 ` ?2p ` q| |` γ?2 ` ?2 | |
ˆ ˙
3a 3a a
η z 1 a 2 z ,
2
p q“ ?2p ` q` γ?2 ` ?2 | |
ˆ ˙ (1.6)
3a a 3a a
η z 1 2 z ,
3
p q“ ?2 ` ?2 ` γ?2 ` ?2 | |
ˆ ˙ ˆ ˙
3a a
η z 2 .
4
p q“ γ?2 ` ?2
ˆ ˙
Theorem 1.1. Let z ρ L . If
P p q
1 η z L z ´1 η z 0,
x ´ 1p q}p ´ q }´ 2p qą
then z ρ L and
P p q x
L z ´1 L z ´1
}p ´ q ´p ´ q }
η z L z ´1 L z ´1 η z L z ´1 η z L z ´1 η z .
ď 1p qx}p ´ q }}p ´ q }` 2p q}p ´ q }` 3p q}p ´ q }` 4p q
x x
REDUCTION OF DIMENSION 3
In particular,
η z 1 L z ´1 η z
L z ´1 p 3p q` q}p ´ q }` 4p q.
}p ´ q }ď 1 η z L z ´1 η z
1 2
´ p q}p x´ q }´ p q
In order to compare the resolventofL to the resolventof Leff, this theoremis completedby
x
the following easy estimate:
Proposition 1.2. We have Sp L Sp Leff Sp LK and, for z ρ L such that z
p q “ p q Y p q P p q R
γ, ,
r `8q L z ´1x Leff z ´1Π 1 . x
p ´ q ´p ´ q ď dist z, γ,
In this estimate, it is im››plixcit that Leff z ´1 is com››posed opn trhe`le8ftqbqy theinclusion ΠH H.
› p ´ q › Ñ
Remark 1.3. These results cover a wide range of situations. In Section 3, we will discuss three
paradigmatic applications. The space Σ will be R or a submanifold of Rd, d 2. The set H is
s
ě
fixed, but the Hilbert structure thereon may depend on s. In our examples T is related to
s sPΣ
p q
ananalyticfamilyofself-adjointoperatorswhicharenotnecessarilynon-negative. Nevertheless,
under suitable assumptions, we can reduce ourselves to the non-negative case. Indeed, in our
applications, for all s Σ, T is bounded from below, independently of s Σ. Moreover, the
s
P P
bottom of the spectrum of T will be an isolated simple eigenvalue µ s . Then, we notice that
s 1
p q
inf µ s is well-defined and that T inf µ s is non-negative. We denote by u s a
sPΣ 1 s sPΣ 1 1
p q ´ p q p q
corresponding eigenfunction. We can assume that u s 1 for all s Σ and that u is
1 H 1
} p q} “ P
a smooth function of s. Π is the projection on u s and ΠH can be identified with L2 Σ
s 1
via the map ϕ s ϕ s u1 s . In particular Lpeffq can be seen as an operator on L2 pΣq,
ÞÑ p ÞÑ p q p qq p q
which is what is meant by the “reduction of dimension”. Finally, γ is defined as the bottom
s
µ s inf µ s of the remaining part of the spectrum and
2 sPΣ 1
p q´ p q
γ infµ s infµ s infSp L inf µ s K . (1.7)
2 1 1
“ s p q´ s p qď pp ´sPΣ p qq q
We recall that we assume the spectral gap condition γ 0, see (1.2).
ą
1.3. The Robin Laplacian in a shrinking layer as a non-self-adjoint application. We
now consider a reduction of dimension result in a non-self-adjoint setting, namely the Robin
Laplacian in a shrinking layer. Let d 2. Here, Σ is an orientable smooth (compact or non-
compact) hypersurface in Rd without běoundary. The orientation can be specified by a globally
defined unit normal vector field n : Σ Sd´1. Moreover Σ is endowed with the Riemannian
structure inherited from the EuclideanÑstructure defined on Rd. We assume that Σ admits a
tubular neighborhood, i.e. for ε 0 small enough the map
ą
Θ : s,t s εtn s (1.8)
ε
p qÞÑ ` p q
is injective on Σ 1,1 and defines a diffeomorphism from Σ 1,1 to its image. We set
ˆr´ s ˆp´ q
Ω Σ 1,1 and Ω Θ Ω . (1.9)
ε ε
“ ˆp´ q “ p q
ThenΩ has the geometricalmeaning ofa non-self-intersectinglayerdelimited by the hypersur-
ε
faces
Σ Θ Σ 1 .
˘,ε ε
“ p ˆt˘ uq
Moreover Σ can be identified with Σ via the diffeomorphisms
˘,ε
Σ Σ
Θ˘,ε : s Ñ s ε˘n,εs .
" ÞÑ ˘ p q
Let α : Σ C be a smooth bounded function. We set α α Θ´1 : Σ C and we
Ñ ˘,ε “ ˝ ˘,ε ˘,ε Ñ
consider on L2 Ω the closed operator P (or simply P if no risk of confusion) defined as
ε ε,α ε
p q
the usual Laplace operator on Ω subject to the Robin boundary condition
ε
u
B α u 0, on Σ . (1.10)
n ` ˘,ε “ ˘,ε
B
Remark 1.4. Note that a very special choice of Robin boundary conditions is considered in
thissection. Indeed,the boundary-couplingfunctions consideredonΣ andΣ arethe same
`,ε ´,ε
exceptforaswitchofsign,see(4.1). Morespecifically,α s α s foreverys Σandnisan
˘,ε
p q“ p q P
outwardnormaltoΩ ononeoftheconnectedpartsΣ oftheboundary Ω ,whileitisinward
ε ˘,ε ε
B
pointing on the other boundary. This special choice is motivated by Parity-Time-symmetric
4 D.KREJCˇIRˇ´IK,N.RAYMOND,J.ROYER,ANDP.SIEGL
waveguides [1, 2] as well as by a self-adjoint analogue considered in [14]. It is straightforward
to extend the present procedure to the general situation of two different boundary-coupling
functionsonΣ andΣ ,butthentheeffectiveoperatorwillbeε-dependent(inanalogywith
`,ε ´,ε
the Dirichlet boundary conditions, see Proposition 3.4) or a renormalization would be needed
(cf. [11]).
Our purpose is to prove that, at the limit when ε goes to 0, the operator P converges in a
ε
norm-resolventsense to a Schr¨odinger operator
Leff ∆Σ Veff,
“´ `
onΣ. Here ∆Σ is the Laplace-Beltramioperatoron Σ, andthe potential Veff depends both on
´
the geometry of Σ and on the boundary condition. More precisely we have
Veff α2 2αRe α α κ1 κd´1 . (1.11)
“| | ´ p q´ p `¨¨¨` q
NotethatthesumoftheprincipalcurvaturesisproportionaltothemeancurvatureofΣ. Notice
also that Heff defines an (unbounded) operator on the Hilbert space L2 Σ . In particular Pε
p q
and Heff do not act on the same space.
We denote by Π L L2 Ω the projection on functions which do not depend on t: for
P p p qq
u L2 Ω and s,t Ω we set
P p q p qP
1 1
Πu s,t u s,θ dθ.
p qp q“ 2 p q
ż´1
Then we define ΠK Id Π.
“ ´
Theorem 1.5. Let K be a compact subset of ρ Heff . Then there exists ε 0 and C 0 such
0
p q ą ě
that for z K and ε 0,ε we have z ρ H and
0 ε
P Pp q P p q
}pPε´zq´1´Uε´1pLeff ´zq´1ΠUε}LpL2pΩεqq ďCε.
HereU is aunitarytransformationfrom L2 Ω ,dx toL2 Ω,w x dσdt , wherefor someC 1
ε ε ε
p q p p q q ą
we have
1
ε 0,ε , x Ω, w x C.
0 ε
@ Pp q @ P C ď| p q|ď
As for Theorem 1.1 it is implicit that the resolvent Leff z ´1 is composed on the left by
the inclusionΠL2 Ωε L2 Ωε . Moreoverthe operatoprLeff´onqL2 Σ has beenidentifiedwith
an operator on ΠLp2 ΩqÑ. p q p q
ε
p q
Remark 1.6. In the geometrically trivial situation Σ Rd´1 and special choice Re α 0, a
“ p q “
versionofTheorem1.5waspreviouslyestablishedin[2]. Atthesametime,intheself-adjointcase
Im α 0 and very special geometric setting d 1 (Σ being a curve), a versionof Theorem 1.5
p q“ “
isdueto[14]. Inourgeneralsetting,itisinterestingtoseehowthegeometryenterstheeffective
dynamics, through the mean curvature of Σ, see (1.11).
1.4. From variational estimates to norm resolvent convergence. All the results of this
paper are about estimates of the difference of resolvents of two operators. These estimates will
be deducedfrom the correspondingestimates of the associatedquadraticforms by the following
general lemma:
Lemma 1.7. Let K be a Hilbert space. Let A and A be two closed densely defined operators on
K. AssumethatAisbijectiveandthatthereexistη ,η ,η ,η 0suchthat1 η A´1 η 0
1 2 3 4 1 2
ě ´ } }´ ą
and p
p p
φ Dom A , ψ Dom A˚ ,
@ P p q @ P p q
Aφ,ψ φ,A˚ψ η φ ψ η φ A˚ψ η Aφ ψ η Aφ A˚ψ .
1 2 3 4
|x y´x p y|ď } }} }` } }} }` } }} }` } }} }
Then A is injective with closed range. If moreover A˚ is injective, then A is bijective and we
p p p
have the estimates
η 1 Aˆ´1 η
A´1 p 3` q} }` 4 (1.12)
} }ď 1 η A´1 η
1 2
´ } }´
and
A´1 A´1 η A´1 A´1 pη A´1 η A´1 η .
1 2 3 4
´ ď } }} }` } }` } }`
› ›
›› p ›› p p
REDUCTION OF DIMENSION 5
Since the proof is rather elementary, let us provide it already now.
Proof. Let φ Dom A and consider ψ A´1 ˚φ Dom A˚ . We have
P p q “p q P p q
φ 2 Aφ, A´1 ˚φ φ,A˚ψ Aφ,ψ
|} } ´x p q y|“|x py´x y| p
η A´1 η φ 2 η A´1 η Aφ φ ,
p ďp 1}p }` 2q} } ` 3} }` 4 } }} }
so ´ ¯
φ 2 η Aˆ´1 η φp 2 η 1 Aˆ´1 pη φ Aφ .
1 2 3 4
} } ď } }` } } ` p ` q} }` } }} }
Then if η A´1 η ´1, we get ¯ ´ ¯
1 2
} }` ă
η 1 Aˆ´1 η
3 4
p φ p ` q} }` Aφ . (1.13)
} }ď 1 η A´1 η } }
1 2
´ } }´
In particular, A is injective with closed range. If A˚ is injective, the range of A is dense and
thus A is bijective. In particular, with (1.13), wpe obtain (1.12).
Finally for f,g K, φ A´1f and ψ A´1 ˚g we have
P “ “p q
A´1 A´1 f,g φ,A˚ψ Aφ,ψ ,
x ´ py“x y´x y
and the conclusion follows b`y easy mani˘pulations. (cid:3)
p p
1.5. Organization of the paper. In Section 2, we prove Theorem 1.1. We first define the
operators L, L and Leff, and then we show how Lemma 1.7 can be applied. In Section 3, we
discusssomeapplicationsofTheorem1.1tothesemiclassicalBorn-Oppenheimerapproximation,
the Dirichlet Lxaplacian on a shrinking tubular neighborhood of an hypersurface and the Robin
Laplacian in the large coupling limit. Section 4 is devoted to the proof of Theorem 1.5 about
the non-self-adjoint Robin Laplacian on a shrinking layer.
2. Abstract reduction of dimension
In this section we describe more precisely the setting introduced in Section 1.2 and we prove
Theorem 1.1. The applications will be given in the following section.
2.1. Definition of the effective operator. Let Σ,σ be a measure space. For eachs Σ we
p q P
consideraseparablecomplexHilbertspaceH . Then,onH weconsideraclosedsymmetricnon-
s s
negative sesquilinear form q with dense domain Dom q . We denote by T the corresponding
s s s
p q
self-adjointandnon-negativeoperator,asgivenbytheRepresentationTheorem. Asalreadysaid
inSection1.2,we considerafunction s Σ γ R whoseinfimum ispositive, see(1.2). Then
s
P ÞÑ P
we denote by Πs PLpHsq the spectral projection of Ts on r0,γsq, and we set ΠKs “IdHs ´Πs.
We denote by H the subset of H which consists of all Φ Φ such that the
sPΣ s “ p sqsPΣ
functions s Φ and s Π Φ are measurable on Σ and
s H s s H
ÞÑ} } s ÞÑ} À} s
Φ 2 Φ 2 dσ s .
s H
} } “ } } s p qă`8
żΣ
It is endowed with the Hilbert structure given by this norm. We denote by Π the bounded
operator on H such that for Φ H and s Σ we have ΠΦ Π Φ . We similarly define
s s s
P P p q “
ΠK L H .
P p q
We say that Φ Φ H belongs to Dom Q if Φ belongs to Dom q for all s Σ,
s sPΣ T s s
“ p q P p q p q P
the functions s q Φ and s q Π Φ are measurable on Σ and
s s s s s
ÞÑ p q ÞÑ p q
Q Φ q Φ dσ s .
T s s
p q“ p q p qă`8
żΣ
We consider on H an operator S with dense domain Dom S . We assume that Dom S is
p q p q
invariantunder Π, that S,Π extends to a bounded operatoron H, and we define a as in (1.5).
r s
We assume that
Dom Q Dom S Dom Q
T
p q“ p qX p q
is dense in H, and for Φ Dom Q we set
P p q
Q Φ SΦ 2 Q Φ . (2.1)
T
p q“} } ` p q
We assume that Q defines a closed form on H. The form Q is symmetric and non-negative and
the associated operator is the operator L introduced in (1.1).
6 D.KREJCˇIRˇ´IK,N.RAYMOND,J.ROYER,ANDP.SIEGL
Then we define the operator L (see (1.4)) by its form. For this we need to verify that the
form domain is left invariant both by Π and ΠK.
Lemma 2.1. For all Φ Dom Qxwe have ΠΦ Dom Q and ΠKΦ Dom Q .
P p q P p q P p q
Proof. Let Φ Φ Dom Q . We have Φ Dom S , so by assumption we have ΠΦ
s sPΣ
“ p q P p q P p q P
Dom S . By assumption again, the function s q Π Φ q Π Π Φ is measurable and we
s s s s s s s
p q ÞÑ p q“ p q
have
q Π Φ dσ s supγ Φ 2 dσ s .
s s s s s H
żΣ p q p qď sPΣ żΣ} } s p qă`8
This proves that ΠΦ belongs to Dom Q , and hence to Dom Q . Then the same holds for
T
ΠKΦ Φ ΠΦ. p q p q (cid:3)
“ ´
With this lemma we can set, for Φ,Ψ Dom Q ,
P p q
Q Φ,Ψ Q ΠΦ,ΠΨ Q ΠKΦ,ΠKΨ .
p q“ p q` p q
Lemma 2.2. For all Φ Dom Q we have
P p p q
Q Φ 2Q Φ .
p p qď p q
In particular the form Q is non-negative, closed, and it determines uniquely a self-adjoint oper-
p
ator L on H. Moreover we have Π,L 0 on Dom L .
r s“ p q
p
Proof. We have
x Q Φ Q Φ x Q ΠΦ,ΠKΦ xQ ΠKΦ,ΠΦ .
p q´ p q“ p q` p q
Since the form Q is non-negative we can apply the Cauchy-Schwarzinequality to write
p
1 1
Q ΠΦ,ΠKΦ Q ΠΦ Q ΠKΦ Q ΠΦ Q ΠKΦ Q Φ .
p qď p q p qď 2 p q` p q “ 2 p q
We havethe same estimateafor Q ΠKaΦ,ΠΦ , and the`firstconclusions fo˘llow. We just check the
p
p q
last property about the commutator. Let ψ Dom L . For all φ Dom L we have
P p q P p q
Q φ,Πψ Q Πφ,Πψ Q Πφ,ψ Πφ,Lψ H φ,ΠLψ H.
p q“ p q“ p q“xx y “x x y
This proves that Πψ Dom L with LΠψ ΠLψ and the proof is complete. (cid:3)
p P p q p “ x x
Then, from Q it is easy to define the forms corresponding to the operators Leff and LK:
x x x
Lemma 2.3. Let Qeff be the restriction of Q to ΠDom Q Ran Π Dom Q . Then Qeff
is non-negativepand closed. The associated operator Leff pis qse“lf-adjoipnt,qiXts dompainqis invariant
under Π, and Π,Leff 0 on Dom Leff . Moreover, we have Dom L Ran Π ,L
Dom Leff ,Leffr . s “ p q p p q X p q q “
p p q q
We have similar statements for the restriction Q of Q to ΠKDom Q xRan ΠK Doxm Q
K
and the corresponding operator LK. p q“ p qX p q
Proof. TheclosednessofQeff comesfromtheclosednessofQandthecontinuityofΠ. Theother
properties are proved as for Lemma 2.2. We prove the last assertion. Let ψ Dom Leff . By
P p q
definition of this domain we have Πψ ψ. For φ Dom Q , we have
“ P p q
Q φ,ψ Q Πφ,Πψ Qeff Πφ,Πψ Qeff Πφ,ψ hΠφ,Leffψi hφ,Leffψi.
p q“ p q“ p q“ p pq“ “
This proves that ψ Dom L and Leffψ Lψ. Thus Dom Leff Dom L Ran Π and
p P p q “ p q Ă p qX p q
L Leff on Dom Leff . The reverse inclusion Dom L Ran Π Dom Leff is easy, so the
pro“of is complete.p q x x p qX p q Ă p xq (cid:3)
x x
Finally we have proved that
Dom L Dom L Ran Π Dom L Ran ΠK Dom Leff Dom LK
p q“ p qX p q ‘ p qX p q “ p q‘ p q
and for ϕ Dom`L we have ˘ ` ˘
Px p q x x
Lϕ LeffΠϕ LKΠKϕ.
“ `
x
From the spectral theorem, we deduce the following lemma.
x
Lemma 2.4. We have Sp L Sp Leff Sp LK and, for z ρ L such that z γ, ,
p q“ p qY p q P p q Rr `8q
1
pL ´xzq´1´pLeff ´zq´1Π ď dist z, γ, x.
› › p r `8qq
› x ›
› ›
REDUCTION OF DIMENSION 7
2.2. Comparison of the resolvents. This section is devoted to the proof of the following
theorem that implies Theorem 1.1 via Lemma 1.7.
Theorem 2.5. Let L and L be as above. Let z C and η z ,η z ,η z ,η z as in (1.6).
1 2 3 4
P p q p q p q p q
Then for Φ Dom L and Ψ Dom L˚ we have
P p q P p q
x
Q Φ,Ψ Q Φ,Ψ η z Φ Ψ η z Φ L z¯ Ψ
p q´ p q ď 1p qx} }} }` 2p q} }}p ´ q }
ˇˇˇ p ˇˇˇ`η3pzq}pL ´zqΦ}}Ψ}`η4pzxq}pL ´zqΦ}}pL ´z¯qΨ}.
Theorem 2.5 is a consequence of the following proposition after inserting z and using the
x
triangular inequality.
Proposition 2.6. For all Φ Dom L and Ψ Dom L we have
P p q P p q
1
Q Φ,Ψ Q Φ,Ψ x
γ| p q´ p q|
3a p LΦ LΨ 3a a LΦ LΨ
Φ } } } } a Φ 1 } } Ψ } } .
ď ?2ˆ} }` γ ˙ γ ` ?2ˆ } }`ˆ ` ?2˙ γ ˙˜} }` γ ¸
x x
Proof. Let ν S,Π . We have
“}r s}
Q Φ,Ψ Q Φ,Ψ Q ΠKΦ,ΠΨ Q ΠΦ,ΠKΨ .
p q´ p q“ p q` p q
For the first term we write
p
Q ΠKΦ,ΠΨ SΠKΦ,SΠΨ SΠKΦ, S,Π ΠΨ SΠKΦ,ΠSΠΨ ,
p q“x y“x r s y`x y
so that
Q ΠKΦ,ΠΨ SΠKΦ, S,Π ΠΨ S,ΠK ΠKΦ,ΠSΠΨ .
p q“x r s y`xr s y
We deduce that
Q ΠKΦ,ΠΨ ν SΠKΦ Ψ ν ΠKΦ SΠΨ . (2.2)
| p q|ď } }} }` } }} }
Similarly, we get, by slightly breaking the symmetry,
Q ΠΦ,ΠKΨ ν SΠKΨ Φ ν ΠKΨ SΦ . (2.3)
| p q|ď } }} }` } }} }
We infer that
Q Φ,Ψ Q Φ,Ψ ν SΠKΦ Ψ ν ΠKΦ SΠΨ ν SΠKΨ Φ ν ΠKΨ SΦ . (2.4)
| p q´ p q|ď } }} }` } }} }` } }} }` } }} }
Since Q is non-negative we have
T
p
SΦ 2 Q Φ LΦ Φ . (2.5)
} } ď p qď} }} }
Similarly,
SΠΨ 2 Q Ψ LΨ Ψ . (2.6)
} } ď p qď} }} }
Then we estimate ΠKΦ and SΠKΦ . We have
} } } } p x
ΠKΦ,LΦ Q ΠKΦ,Φ Q ΠKΦ Q ΠKΦ,ΠΦ ,
x y“ p q“ p q` p q
and deduce
Q ΠKΦ LΦ ΠKΦ Q ΠKΦ,ΠΦ .
p qď} }} }`| p q|
From (2.3), we get
Q ΠKΦ LΦ ΠKΦ ν SΠKΦ Φ ν ΠKΦ SΦ .
p qď} }} }` } }} }` } }} }
Moreover,we have
Q ΠKΦ SΠKΦ 2 γ ΠKΦ 2.
p qě} } ` } }
We infer that
SΠKΦ 2 γ ΠKΦ 2
} } ` } }
γ 1 1 ν2 γ ν2
ΠKΦ 2 LΦ 2 SΠKΦ 2 Φ 2 ΠKΦ 2 SΦ 2.
ď 4} } ` γ} } ` 2} } ` 2 } } ` 4} } ` γ } }
Using (2.5) we deduce that
1 1 ν2 ν2 LΦ 2
SΠKΦ 2 γ ΠKΦ 2 LΦ 2 Φ 2 } } Φ 2 ,
2 } } ` } } ď γ} } ` 2 } } ` 2 γ2 `} }
ˆ ˙
` ˘
8 D.KREJCˇIRˇ´IK,N.RAYMOND,J.ROYER,ANDP.SIEGL
and thus
SΠKΦ 2 LΦ 2
} } ΠKΦ 2 2 a2 } } 2a2 Φ 2. (2.7)
γ `} } ďp ` q γ2 ` } }
Let us now consider ΠKΨ and SΠKΨ . We have easily that
} } } }
SΠKΨ 2 γ ΠKΨ 2 Q ΠKΨ Q Ψ,ΠKΨ LΨ ΠKΨ ,
} } ` } } ď p q“ p qď} }} }
and thus
p x
SΠKΨ 2 LΨ 2
} } ΠKΨ 2 } } . (2.8)
γ `} } ď γ2
It remains to combine (2.4), (2.5), (2.6), (2.7), (2.8), andxuse elementary manipulations. (cid:3)
3. Examples of applications
In this section we discuss three applications of Theorem 1.1 and we recall that we are in the
context of Remark 1.3.
3.1. SemiclassicalBorn-Oppenheimerapproximation. Inthisfirstexampleweset Σ,σ
R,ds . We consider a Hilbert space H and set H L2 R,H . Then, for h 0, we cponsidqe“r
T T
p q “ p q ą
onHthe operatorS hD ,whereD i . We alsoconsideranoperatorT onH suchthat
h s s s
“ “´ B
for Φ Φs sPR H we have TΦ s TsΦs, where Ts is a family of operators on HT which
depend“spanaqlyticaPlly on s. Thups thqe o“perator L Lp tqakes the form
h
“
L h2D2 T .
h “ s `
This kind of operators appears in [12, 13] where their spectral and dynamical behaviors are
analyzed. As an example of operator T, the reader can have the Schr¨odinger operator ∆
t
´ `
V s,t in mind, where the electric potential V is assumed to be real-valued. Here the operator
p q
norm of the commutator hDs,Π is controlled by the supremum of su1 s H. Assuming
r s }B p q}
that su1 s H is bounded, we have a a h O h (see (1.5)). Let us also assume, for
}B p q} “ p q “ p q
our convenience, that µ has a unique minimum, non-degenerate and not attained at infinity.
1
Without loss of generality we can assume that this minimum is 0 and is attained at 0. Thus,
here γ just satisfies γ infsPRµ2 s 0.
For k N˚ we set “ p qą
P
λ h sup inf hL ϕ,ϕi. (3.1)
k h
p q“coFdĂimDpoFmqp“Lkh´q1}ϕϕP}“F1
By the min-max principle, the first values of λ h are given by the non-decreasing sequence of
k
isolated eigenvalues of L (counted with multipplicqities) below the essential spectrum. If there
h
is a finite number of such eigenvalues, the rest of the sequence is given by the minimum of the
essential spectrum. We similarly define the sequence λeff,k h corresponding to the operator
Lh,eff. Note that Lh,eff can be identified with the operpator p qq
h2D2 µ s h2 u s 2 .
s ` 1p q` }Bs 1p q}HT
As a consequence of the harmonic approximation(see for instance [4, Chapter 7]or [16, Section
4.3.1]), we get the following asymptotics.
Proposition 3.1. Let k N˚. We have
P
µ2 0
λeff,k h 2k 1 p qh o h , h 0.
p q“p ´ q 2 ` p q Ñ
c
From our abstract analysis, we deduce the following result.
Proposition 3.2. Let c ,C 0. There exist h 0 and C 0 such that for h 0,h and
0 0 0 0
ą ą ą Pp q
z Zh z C0h,C0h : dist z,Sp Lh,eff c0h
P “t Pr´ s p p qqě u
we have z ρ L and
h
P p q
Lh z ´1 Lh,eff z ´1 C.
}p ´ q ´p ´ q }ď
REDUCTION OF DIMENSION 9
Proof. Let hą0 and z PZh. If h is small enough we have C0hăγ so z PρpLh,effqXρpLhKq“
ρ L . Moreover,by the Spectral Theorem,
h
p q
1 1
x pLh´zq´1 ď pLh,eff ´zq´1 ` pLhK´zq´1 ď c0h ` γ C0h.
With the nota›››tioxn (1.6) we››› ha››ve ›› ›› ›› ´
liminf sup 1 η z L z ´1 η z 0.
1,h h 2,h
hÑ0 zPZh ´ p q}p ´ q }´ p q ą
´ ¯
From Theorems 1.1 and Proposition 1.2, we dedxuce that z ρ L ,
h
P p q
L z ´1 h´1,
h
}p ´ q }À
and the estimate on the difference of the resolvents. Here and occasionally in the sequel, we
adopt the notation x y if there is a positive constant C (independent of x and y) such that
x Cy. À (cid:3)
ď
From this norm resolvent convergence result, we recover a result of [13, Section 4.2].
Proposition 3.3. Let k N˚. Then
P
λk h λeff,k h O h2 , h 0.
p q“ p q` p q Ñ
Proof. Let ε 0 be such that λeff,k`1 h λeff,k h 2εh for all h. We set zh λeff,k h εh.
The resolventąLh,eff zh ´1 has k nepgaqt´ive eigepnvqaąlues “ p q`
p ´ q
1 1
... ,
λeff,k h zh ď ď λeff,1 h zh
p q´ p q´
allsmallerthan α hforsomeα 0,andtherestofthespectrumispositive. ByProposition3.2
the resolvent L´ {z ´1 is wellądefined for h small enough and there exists C 0 such that
h h
p ´ q ą
Lh zh ´1 Lh,eff zh ´1 C.
p ´ q ´p ´ q ď
By the min-max principle ap›plied to these two resolvents, w›e obtain that for all j 1,...,k
the j-th eigenvalue of Lh ›zh ´1 is at distance not greate›r than C from 1 λeff,kP`1t´j zhu,
p ´ q {p ´ q
and the rest of the spectrum is greater than C. In particular, for j 1,
´ “
1 1
C.
ˇλkphq´zh ´ λeff,kphq´zhˇď
ˇ ˇ
This gives ˇ ˇ
ˇ ˇ
λk h λeff,k h Cεh λk h λeff,k h εh ,
| p q´ p q|ď | p q´ p q` |
and the conclusion follows for h small enough. (cid:3)
3.2. Shrinking neighborhoods of hypersurfaces. In this paragraphwe consider a subman-
ifold Σ of Rd, d 2,as in Section 1.3. We chooseε 0 and define Θ , Ω and Ω as in (1.8)and
ε ε
ě ą
(1.9). For ϕ H1 Ω , we set
P 0p εq
QDir ϕ ∇ϕ2dx,
Ωεp q“ | |
żΩε
Dir
and we denote by ∆ the associated operator. Then we use the diffeomorphism Θ to see
∆Dir as an operat´or oΩnεL2 Ω . We set, for ψ H1 Ω,dσdt , ε
´ Ωε p q P 0p q
QDir ψ QDir ψ Θ´1 .
ε p q“ Ωεp ˝ ε q
We need a more explicit expressionofQDir in terms of the variables s,t onΩ. For s,t Ω
we have on T Ω T Σ n s R ε p q p qP
ps,tq s
» ˆ p q
dps,tqΘε “pIdTsΣ`εtdsnqbεIdnpsqR.
Hence
dΘεps,tqΘ´ε1 “pIdTsΣ`εtdsnq´1bε´1IdnpsqR.
10 D.KREJCˇIRˇ´IK,N.RAYMOND,J.ROYER,ANDP.SIEGL
We recall that the Weingarten map d n is a self-adjoint operator on T Σ (endowed with
s s
the metric inherited from the Euclidea´n structure on Rd). For ψ H1 Ω,dσdt , x Ω and
ε
P p q P
s,t Θ´1 x we get
p q“ ε p q
∇ ψ Θ´1 x 2 d Θ´1 ˚∇ψ s,t 2
} p ˝ ε qp q}TxΩε “}p x ε q p q}TxΩε
1
Id εtd n ´1∇ ψ s,t 2 ψ s,t 2 .
“}p TsΣ` s q s p q}TsΣ` ε2 |Bt p q|
The eigenvalues of the Weingarten map are the principal curvatures κ ,...,κ . In particular
1 d´1
for s,t Ω we have
p qP
d´1
d Θ εw , where w s,t 1 εtκ s . (3.2)
ps,tq ε ε ε j
| |“ p q“ p ´ p qq
j“1
ź
TheRiemannianstructureonΩisgivenbythepullbackbyΘ oftheEuclideanstructuredefined
ε
on Ω . More explicitly, for s,t Ω the inner product on T Ω is given by
ε ps,tq
p qP
X,Y Tps,tq Ω , gε X,Y dps,tqΘε X ,dps,tqΘε Y Rd.
@ P p q p q“x p q p qy
Then the measure corresponding to the metric g is given by εw dσdt. Thus, if we set
ε ε
G s,t Id εtd n ´2, (3.3)
εp q“p TsΣ` s q
we finally obtain
1
QDir ψ Id εtd n ´1∇ ψ Θ´1 x 2dx ψ Θ´1 x 2dx
ε p q“ |p TsΣ` s q s p ε p qq| ` ε2 |Bt p ε p qq|
żΩε żΩε
1
ε G s,t ∇ ψ,∇ ψ w dσdt ψ 2εw dσdt.
“ x εp q s s yTΣ ε ` ε2 |Bt | ε
żΩ żΩ
The transverse operator T ε is the Dirichlet realization on L2 1,1 ,εw dt of the differ-
s ε
p q pp´ q q
ential operator ε´2w´1 w . We denote by µ s,ε its first eigenvalue and we set µ ε
´ ε Bt εBt 1p q p q “
infsPRµ1 s,ε . We have, by perturbation theory, as ε 0,
p q Ñ
π2 π2
µ s,ε V s O ε , µ ε O 1 ,
1p q“ 4ε2 ` p q` p q p q“ 4ε2 ` p q
where
2
d´1 d´1
1 1
V s κ s 2 κ s .
j j
p q“´2 p q ` 4˜ p q¸
j“1 j“1
ÿ ÿ
We denote by LDir the operator associated to the form QDir and by LDir the corresponding
ε ε ε,eff
effective operatorasdefined in the generalcontextofSection 1.2. It is nothing but the operator
associated with the form H1 Σ ϕ QDir ϕu where u is the positive L2-normalized
p q Q ÞÑ ε p s,εq s,ε
groundstate of the transverse operator (and actually depending on the principal curvatures
analytically). From perturbation theory, we can easily check that the commutator between the
projection on u and S is bounded (and of order ε).
s,ε
Proposition 3.4. Let c ,C 0. There exist ε 0 and C 0 such that for all ε 0,ε and
0 0 0 0
ą ą ą Pp q
z Z z R: z µ ε C , dist z,Sp LDir c
P c0,C0,ε “t P | ´ p q|ď 0 p p ε,effqqě 0u
we have
LDir z ´1 LDir z ´1 Cε.
ε ´ ´ ε,eff ´ ď
› ›
We recover a result of [10›]`(when the˘re is n`o magneti˘c fie›ld).
› ›
Proof. We arein the contextofRemark1.3. The formQ µ ε is non-negative. We denote by
ε
´ p q
L the corresponding non-negative self-adjoint operator and define L as in Lemma 2.2. Given
ε ε
ε 0andz Z wewriteζ forz µ ε . Thus,withthenotationofthe abstractsettingwe
haąve γε ε´P2, ac0ε,C0,εO ε2 , ζ O 1 a´ndphqence η1,ε ζ O ε , η2,ε xζ O ε2 , η3,ε ζ O ε
and η „ζ O ε2“. Mporeqove“r,byptqhe spectral theopremq“, wephqave p q“ p q p q“ p q
4,ε
p q“ p q
L ζ ´1 O 1 .
ε
´ “ p q
› ›
›` x ˘ ›
› ›