Eigenvalues for a Schrodinger operator on a closed Riemannian manifold with holes

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Eigenvalues for a Schrodinger operator on a closed Riemannian manifold with holes

Olivier Lablée

To cite this version:

Olivier Lablée. Eigenvalues for a Schrodinger operator on a closed Riemannian manifold with holes.

2013. �hal-00782066�

Eigenvalues for a Schrödinger operator on a closed Riemannian manifold with holes

Olivier Lablée January 28, 2013

Abstract

In this article we consider a closed Riemannian manifold(M,g)andA a subset ofM. The purpose of this article is the comparison between the eigenvalues(λ_k(M))_k≥1of a Schrödinger operatorP := −^∆g+Von the manifold(M,g)and the eigenvalues(λ_k(M−^A))_k≥1ofPon the manifold (M−A,g)with Dirichlet boundary conditions.

1 Introduction

The behaviour of the spectrum of a Riemannian manifold(M,g)under topo- logical perturbation has been the subject of many research. The most famous exemple is the crushed ice problem[Kac], see also[Ann]. This problem consists to understand the behaviour of Laplacian eigenvalues with Dirichlet boundary on a domain with small holes. This subject was first studied by M. Kac[Kac]in 1974. Then, J. Rauch and M. Taylor[RT]studied the case of Euclidian Laplacian in a compact setMofRⁿ: they showed that the spectrum of∆_Rnis invariant by a topological excision of aMby a compact subsetAwith a Newtonian capac- ity zero. Later, S. Osawa, I. Chavel and E. Feldman[Ca-Fe1],[Ca-Fe2]treated the Riemmannian manifold case. They used complex probalistic techniques based on Brownian motion. In 1995, in a nice article[Cou]G . Courtois stud- ied the case of Laplace Beltrami operator on closed Riemannian manifold. He used very simple techniques of analysis. In[Be-Co]J. Bertrand and B. Colbois explained also the case of Laplace Beltrami operator on compact Riemannian manifold. In this article we focus on the the Schrödinger operator−^∆g+V case on a closed Riemannian manifold.

Assumption. The manifold is closed (ie compact without boundary); the function V is bounded on M andminMV>_0.

In this work we show that under “little” topological excision of a part A from the manifold, the spectrum of−^∆g+VonM−^Ais close of the spectrum onM. More precisely, the “good” parameter for measuring the littleness ofA is a type of electrostatic capacity defined by :

cap(A):=inf

Q(u),u∈^H1(M),^Z

Mu dVg=0,u−^e1∈^H0¹(M−^A)

wheree₁denotes the first eigenfunction of the operator−^∆g+Von the mani- foldM, andQis the following quadradic form :

Q(ϕ):= Z

M|^dϕ|^{2 d}Vg+ Z

MV|^ϕ|^2dVg

andH₀¹(M−^A)is the Sobolev space defined by :

H₀¹(M−^A):={^g∈ ^H1(M), g=0 on a open neighborhood ofA} the closure is for the normk^.kH¹(M),H¹(M)is the usual Sobolev space onM.

Indeed, more cap(A) is small, more the spectrum−^∆g+V on M−^Ais close of the spectrum onMin the following sense :

Theorem. Let(M,g)a closed Riemannian manifold. For all integer k ≥ ^{1, there} exists a constant C_k depending on the manifold(M,g)and on the potential V such that for all subset A of M we have :

0≤^λk(M−^A)−^λk(M)≤^Ck

q cap(A).

The organization of this paper is the following : in the part 2 we start by recall some classicals results in spectral theory, we define our Sobolev space H₀¹(M−^A)and the notion of Schrödinger capacity. In particular, we explain the link between the functionnal Hilbert space H₀¹(M−^A)and Schrödinger capacity cap(A). The last part of this paper is a detailed proof of the main theorem.

2 Spectral problem background

2.1 Schrödinger operator on a Riemannian manifold

We recall here some generality on spectral geometry, for a more detailed survey see for example[Lab]. In Riemannian geometry, the Laplace Beltrami operator is the generalisation of Laplacian∆=

n

∑

j=1

∂²

∂x²_j onRⁿ. For aC²real valued func- tion f on a Riemannian manifold and for a local chartφ : U⊂ ^M→^{Rof the} manifoldM, the Laplace Beltrami operator is given by the local expression :

∆_gf = √¹_g

n

∑

j,k=1

∂

∂xj

√_ggjk∂(f◦^φ−1)

∂xk

whereg=det(g_ij)andg^jk= (g_jk)⁻¹.

The spectrum of this operator is a nice geometric invariant, see Berger, Gauduchon and Mazet[BGM]and[Be-Be]. The spectrum of Laplace Beltrami operator has many applications in geometry topology, physics ,etc ...

For every Riemannian manifold(M,g)with dimensionn≥1 we have the

“natural” Hilbert space L²(M) = L²(M,dVg),Vg is the Riemannian volume form associated to the metricg. ForVa function fromMtoR, we define the Schrödinger operator on the manifold(M,g)by the linear unbounded opera- tor on the set of smooth compact supports real valued functionsCc^∞(M,R)⊂ L²(M)by :−^∆g+V.

2.2 Spectral problem

The spectral problem is the following : find all pairs(λ,u) withλ ∈ ^Rand u∈^L2(M)such that :

−^∆gu+Vu=λu

(withu∈^L2(M)in the non-compact case).

In the case of manifold with boundary, we need boundary conditions on the functionsu, for example the Dirichlet conditions : u= 0 on the boundary ofM, or Neumann conditions : ^∂u_∂n = 0 on the boundary of M. In the case of closed manifolds (compact without boundary) we don’t have conditions.

For our context (the closed case) the natural space to look here is the Sobolev spaceH¹(M,g)define by

H¹(M,g):=C^∞(M) where the closure is for the normk^.kH¹ :k^ukH¹ :=

q

k^uk²L²+k^duk²H¹. An other point of view to define the spaceH¹(M,g)is the following :

H¹(M,g):=ⁿu∈ ^L2(M);du∈ ^L2(M)^o where the derivation is the sense of distribution.

The spaceH¹(M,g)is a Hilbert space for the inner product : h^u,viH¹ :=h^u,viL²+h^du,dviL².

Recall here a classical theorem of spectral theory (see for example[Re-Si]) : Theorem. For the above problems, the operator−^∆g+V is self-adjoint, the spectrum of the operator−^∆g+V consists of a sequence of increasing eigenvalues with finite multiplicity :

λ₁(M)≤^λ2(M)≤ · · · ≤^λk(M)≤ · · · →+^∞.

Moreover, the associate eigenfunctions (ek)_k≥0 is a orthonormal basis of the space L²(M).

Definition. We define the quadradic formQwith domainD(Q):= H¹(M)by :

Q(ϕ):= Z

M|^dϕ|^{2 d}V^g+ Z

MV|^ϕ|^2dV^g.

Recall also (see for example[Co-Hi]) the minimax variational characteriza- tion for eigenvalues : for allk≥1

λ_k(M) = min

E⊂H¹(M)

dim(E)=k

maxϕ∈E

ϕ6=0

R(ϕ)

whereR(ϕ)is the Rayleigh quotient of the functionϕ:

R(ϕ):= ^Q(ϕ) R

Mϕ²dVg.

In our context, a consequence of the minimax principle is :

Proposition 2.1. The first eigenvalue λ₁(M)and e₁ the first eigenfunction of the operator−^∆g+V on the manifold(M,g) satisfyλ₁(M)≥^minMV >_{0and e1}>

0or e₁<_{0in M.}

Proof. It’s clear that Z

M|^de1|^{2 d}V^g+ Z

MV|^e1|^{2 d}V^g≥^min

M Vk^e1k²L²(M)

and on the other hand Z

M|^de1|^2dVg+ Z

MV|^e1|^{2 d}Vg=− Z

M

∆_ge₁e₁dVg+ Z

MV|^e1|^{2 d}Vg

= Z

M −^∆g+V

e₁e₁dVg=λ₁(M)k^e1k²L²(M)

soλ₁(M)≥^minMV. Next, suppose the functione₁changes sign intoM, since e₁∈ ^H1(M), the function f :=|^e1|^{belongs to H1}(M)and|^{d f}|=|^de1|^{(see for} example[GT]), henceR(f) =R(e₁). So, the function f is a first eigenfunction of−^∆g+Von the manifoldMwhich satisfiesf ≥^{0 onM,f} vanish intoMand

−^∆g+V

f =λ₁(M)f ≥^{0 on}M. Using the maximum principle[Pr-We], the function fcan not achieved it minimum in an interior point of the manifoldM, hence f does not vanish onM, so we obtain a contradiction.

3 Proof of the main theorem

3.1 Somes usefull spaces

We define on the spaceH¹(M)the⋆_{-norm by :} k^uk²⋆:=

Z

M|^du|^2dVg+ Z

MV|^u|^{2 d}Vg

so, without difficulty we have :

Proposition. The application k^.k⋆ is a norm on the space H¹(M); moreover this norm is equivalent to the Sobolev norm k^.kH¹(M). In particular H¹(M),k^.k⋆ is a Banach space.

Now, for a compact subset Aof the manifold M the usual Sobolev space H₀¹(M−^A)is defined by the closure of test functions space on M−^{Afor the} normk^.kH¹(M):

H₀¹(M−^A):=D(M−^A).

What happens when the set Ais not compact ? For example if Ais a dense and countable subset of points of the manifold M, the space of test functions D(M−^A)is reduced to{⁰}. Therefore we cannot define the spaceH₀¹(M−^A). In this case, we propose a definition ofH¹₀(M−^A)for any subsetAofM.

Definition. We define the Sobolev spacesH¹₀(M−^A)andH¹₀(M−^A)by : H¹0(M−^A):=ⁿg∈ ^H1(M),g=0 on a open neighborhood ofAo

; H₀¹(M−^A):=H¹₀(M−^A)

where the closure is for the normk^.kH¹(M). We have the :

Proposition. If the set A is compact, the previous definition of the space H¹₀(M−^A) coincides with the usal ones.

Proof. Let f ∈ ^H₀¹(M−^A), then by definition : for allε ≥ 0 there exists g ∈ H¹₀(M−^A)such that k^f−^gkH¹(M) ≤ ε. So, we will show that we can write g as a limit of sequence from the space D(M−A) and conclude. Since g ∈ H¹₀(M−^A)there exists an open setU ⊃ ^{Asuch thatg}_|U = 0. Consider two open setsU₁andU₂of the manifoldMsuch that :

A⊂^U1, M−^U⊂^U2,U₁∩^U2=^∅; and consider also a functionϕ∈ D(M)such that :

ϕ_|U1 =0, ϕ_|U2 =1.

Of course, the function ϕ belongs to the space D(M−^A). Next, since g ∈ H¹₀(M−^A) ⊂ ^H1(M)and as the set of smooth functionsC^∞(M)is dense in H¹(M): there exists a sequence(gn)_ninC^∞(M)such that lim

n→+∞gn=gfor the normk^.kH¹(M). Therefore we claim that : lim

n→+∞ϕgn =gfor the normk^.kH¹(M). Indeed, start by, for all integern:

k^ϕgn−^gk²H¹(M)≤ k^gn−^gk²H¹(M−U)+k^ϕgn−^gk²H¹(U)

≤ k^gn−^gk²H¹(M)+k^ϕgn−^gk²H¹(U). Next, we observe that, for all integern:

k^ϕgn−^gk²H¹(U)=k^ϕgnk²H¹(U)

= Z

U|^ϕgn|^{2 d}V^g+ Z

U|^dϕgn+ϕdgn|^2dV^g

≤ Z

U|^ϕgn|^{2 d}V^g+ Z

U|^dϕgn|^{2 d}V^g+ Z

U|^ϕdgn|^2dV^g+2^Z

U|^dϕgnϕdgn|^dV^g

≤ k^ϕk²∞k^gnk²L²(U)+k^dϕk²L^∞(M)k^gnk²L²(U)

+k^ϕk²∞k^dgnk²L²(U)+2k^dϕk∞k^ϕk∞ Z

U|^gndgn| ^dVg

≤ k^ϕk²∞k^gnk²L²(U)+k^dϕk²∞k^gnk²L²(U)

+k^ϕk²∞k^dgnk²L²(U)+2k^dϕk∞k^ϕkL^∞(M)k^gnkL²(U)k^dgnkL²(U),

by Cauchy-Schwarz inequality.

Finally we get for all integern: k^ϕgn−^gk²H¹(U)≤ k^gnk²H¹(U)

2k^ϕk²∞+k^dϕk²∞+2k^dϕk∞k^ϕk∞

. As a consequence, we have for all integern:

k^ϕgn−^gk²H¹(M)≤ k^gn−^gk²H¹(M−U)

+k^gnk²H¹(U)

2k^ϕk²∞+k^dϕk²∞+2k^dϕk∞k^ϕk∞

.

Now, it suffices to note that k^gnk²_H¹_(U) = k^gn−^gk²_H¹_(U) ≤ k^gn−^gk²_H¹_(M) (sinceg=0 on the open setU) and we have finally :

k^ϕgn−^gk²H¹(M)≤ k^gn−^gk²H¹(M)

1+2k^ϕk²∞+k^dϕk²∞+2k^dϕk∞k^ϕk∞

. The sequence (ϕgn)_n belong toD(M−^A)^N,and since lim

n→+∞gn = g for the norm k^.kH¹(M) the previous inequality implies lim

n→+∞ϕgn = g for the norm k^.kH¹(M).

So we have shown that every function f ∈ ^H₀¹(M−^A)is a limit (for the normk^.kH¹(M)) of a sequence ofD(M−^A).

Let us also denote the spacesH⋆¹(M)andS_A(M)by : H⋆¹(M):=

f ∈^H1(M),^Z

Mf dV^g=0

; and

SA(M):=ⁿu∈ ^H⋆1(M),u−^e1∈ ^H0¹(M−^A)^o. In the definition of the spaceH⋆¹(M)the conditionR

Mf dV^g = 0 is analog to a boundary condition. We observe that the spaceH¹⋆(M)is a Hilbert space for the norm :

k^uk⋆:= Z

M|^du|^{2 d}Vg+ Z

MV|^u|^{2 d}Vg; andSA(M)is just an affine closed subset ofH¹(M).

3.2 Schrödinger capacity

Next, we introduce the Schrödinger capacity of the setA;

Definition. Let us consider the Schrödinger capacity cap(A)of the set Ade- fined by

cap(A):=inf _Z

M|^du|^2dV^g+ Z

MV|^u|^2dV^g,u∈^SA(M)

.

Let us remark that : there exists an unique functionuA ∈^SA(M)such that cap(A) =

Z

M|^duA|^2dVg+ Z

MV|^uA|^2dVg.

Indeed : here the capacity cap(A)is just the distance between the function 0 and the closed spaceSA(M). This distance is equal tok^uAk⋆whereuAis the orthogonal projection of 0 onSA(M):

cap(A) =d⋆(0,S_A(M)):=inf{k^uk⋆,u∈^SA(M)}=k^uAk⋆.

In the following lemma we give the relationships between the capacity cap(A), the functionsuA,e₁and the Sobolev spacesH₀¹(M−^A), H¹(M).

Lemma. For all subset A of the manifold M, the following properties are equivalent : (i) cap(A) =0;

(ii) uA=0;

(iii) e₁∈ ^H₀¹(M−^A); (iv) H₀¹(M−^A) =H¹(M).

Proof. It is clear from the formula (3.1) that(i)⇔ (ii) ⇔ (iii). Next, suppose the property (iii) holds : so there exists a sequence (vn)n ∈ H¹₀(M−^A)^N such that lim

n→+∞vn = e₁ for the norm k^.kH¹(M). So, for all smooth function ϕ∈ C^∞(M)we have lim

n→+∞(ϕvn)/e₁ =ϕfor the normk^.kH¹(M), indeed for all integern:

ϕvn

e₁ −^ϕ

2 H¹(M)

= Z

M

ϕvn

e₁ −^ϕ

2

dV^g+ Z

M

d

ϕvn

e₁

−^dϕ

2

dV^g. First, we have for all integern:

Z M

ϕvn

e₁ −^ϕ

2

dV^g= Z

M

1

|^e1|²|^ϕ(vn−^e1)|^{2 d}V^g

≤ 1 e₁

2

∞k^ϕk²∞k^vn−^e1k²_L²_(M) so, since lim

n→+∞vn=e₁for the normk^.kH¹(M)we have

n→lim+∞ Z

M

ϕvn

e₁ −^ϕ

2

dV^g=0.

On the other hand, for all integern: Z

M

d

ϕvn

e₁

−^dϕ

2

dV^g= Z

M

d(ϕvn)e₁−^ϕvnde1

e²₁ −^dϕ

2

dV^g

= Z

M

1 e²₁

!

d(ϕ)vne₁+ϕd(vn)e₁−^ϕvnd(e₁)−^d(ϕ)e²₁

2 dV^g

≤ 1 e₁

2

∞

dϕvne₁−^dϕe21+ϕdvne₁−^ϕvnde1

2 L²(M)

≤ 1 e₁

2

∞

dϕvne₁−^dϕe2_1L2(M)+k^ϕdvne₁−^ϕvnde₁k_L²_(M) 2

≤ 1 e₁

2

∞

hk^dϕk∞k^e1k∞k^vn−^e1k_L²_(M)+

k^ϕk∞k^e1(dvn−^de1) +e₁de₁−^vnde₁k_L²_(M)ⁱ²

≤ 1 e₁

2

∞

hk^dϕk∞k^e1k∞k^vn−^e1k_L²_(M)+

k^ϕk∞k^e1k∞k^dvn−^de1k_L²_(M)+k^ϕk∞k^de1k∞k^e1−^vnk_L²_(M)^i2; so, since lim

n→+∞vn =e₁for the normk^.kH¹(M)we have

n→lim+∞ Z

M

d

ϕvn

e₁

−^dϕ

2

dV^g=0.

Therefore, for all function ϕ ∈ C^∞(M)we have lim

n→+∞ ϕvn

e₁ = ϕfor the norm k^.kH¹(M).

Next, by density ofC^∞(M)in H¹(M): for all function f ∈ ^H1(M)we have

n→lim+∞

f vn

e₁ = f . Since the sequence_{f v}

n

e₁

n∈ H¹₀(M−^A)^Nwe get finally that f belongs to spaceH¹₀(M−^A). Finally, it is easy to see that(iv)⇒(iii).

An obvious consequence of this lemma is the following result :

Proposition. The spectrum of−^∆g+V on the manifold(M,g)and on the manifold (M−^A,g)are equal if and only if cap(A) =0.

3.3 The Poincaré inequality

Now, let introduce the Poincaré inequality :

Theorem. If λ₁(M) denotes the first eigenvalue of the operator −^∆g+V on the manifold(M,g), the following inequality

k^uAk²L²(M)≤ ^cap(A) λ₁(M) holds for all subset A of M.

Proof. The case cap(A) =0 is an obvious consequence of the lemma in section 3.2. Suppose here that cap(A) >_{0, then}k^uAkL²(M)> 0. The first eigenvalue λ₁(M)of the operator−^∆g+Von the manifold(M,g)is given by :

λ₁(M) = min

E⊂H¹(M)

dim(E)=1

maxϕ∈E

ϕ6=0

R

M|^dϕ|²+V|^ϕ|^2dVg

R

M|^ϕ|^2dVg

= min

ϕ∈H¹(M)

ϕ6=0

R

M|^dϕ|²+V|^ϕ|^{2 d}V^g R

M|^ϕ|^{2 d}Vg

Sinceu_Abelongs to the spaceH¹(M)we getλ₁(M)≤ _ku^cap(A)

Ak²_L2(M)

.

3.4 The main theorem

Recall our main result :

Theorem. Let(M,g)a compact Riemannian manifold. For all integer k ≥^{1, there} exists a constant C_kdepending on the manifold of(M,g)and the potential V such that for all subset A of M we have :

0≤^λk(M−^A)−^λk(M)≤^Ck

q cap(A).

Remark. We can easily adapt the proof for a compact Riemannian manifold with boundary.

Proof. Let us denote by(ek)_k≥1an orthonormal basis of the spaceL²(M)with eigenfunctions of the operator−^∆g+Von the manifold(M,g). For all integer k≥1, we consider the sets

Fk:=span{^e1,e₂, . . . ,ek} and

E_k :=

f

1−^u_e^A

1

, f ∈ ^Fk

.

First, observe thatE_k ⊂ ^H1₀(M−^A). For allj ∈ {^{1, . . . ,k}}we introduce also the functionsφ_j :=e_j

1−^u_e^A₁ ∈^Ek.

•^{Step 1}: we compute the L²-inner product φi,φj

L²(M)for all pairs(i,j) ∈ {1, . . . ,k}²:

φi,φj

L²(M)= Z

Meiej

1−^u_e^A

1

₂ dV^g

=δ_i,j−² Z

M

e_ie_j

e₁ uAdVg+ Z

Meiej

u²_A e²₁ dVg. Thus, for all pair(i,j)∈ {^{1, . . . ,k}}^{2we get :}

φ_i,φ_j

L²(M)−^δi,j

≤²

Z M

e_ie_j e₁ u_A

dV^g+

Z M

e_ie_ju²_A e²₁

dV^g, hence, by Cauchy-Schwarz inequality we obtain

φ_i,φ_j

L²(M)−^δi,j

≤^{2 max}

1≤i,j≤k

e_ie_j e²₁

_∞k^uAkL²(M)+ max

1≤i,j≤k

e_ie_j e²₁

_∞k^uAk²L²(M)

≤^{2 max}

1≤i,j≤k

eiej

e_{1 ∞}

qvol(M)k^uAkL²(M)+ max

1≤i,j≤k

eiej

e²₁

_∞k^uAk²L²(M)

hence by Poincaré inequality we have

φi,φj

L²(M)−^δi,j

≤^Bk,M

qcap(A) +cap(A)

whereB_k=B_k(e₁,e₂, ...,e_k,λ₁(M),M)≥0, and since the eigenfunctionse₁,e₂, ...,e_k and the eigenvalueλ₁(M)depends only on(M,g)andV, for all integerkthe constantBkdepends only on(M,g)andV, ie :Bk=Bk(M,V).

Therefore, there existsεk ∈]0, 1[(depends on the constantBk) such that for all A⊂ ^{Mwe have :}

cap(A)≤^εk⇒^dim(E_k) =kand∀^j∈ {^{1, ...,k}}^, φ_j

²

L²(M)−¹≤^Dk

qcap(A) where (and for the same reasons as in the study ofB_k) for all integerk, the con- stantD_kdepends only onMandV, ieD_k=D_k(M,V).

•^{Step 2}: Let a functionφ= f

1−^u_e^A₁∈^Ek, with f ∈^Fk. Without loss gener- ality we can assume thatk^fkL²(M)=1, indeed : we haveR(φ) =R

φ kfk_L2(M)

and in our context we intererest in the Rayleigh quotient ofφ(see the end of the final step of the proof).

Setv_A := ^u_e^A₁, we have : Z

M|^dφ|^2dV^g= Z

M|^{d f}−^d(f vA)|^{2 d}V^g

= Z

M|^{d f}|^2dV^g+ Z

M|^{d f v}A+f dv_A|^2dV^g−² Z

Md f d(f v_A)dV^g

= Z

M|^{d f}|^{2 d}Vg+ Z

M|^{d f v}A|^{2 d}Vg+ Z

M|^{f dv}A|^2dVg

+2^Z

Md f dvAf vAdVg−² Z

M|^{d f}|^2vAdVg−² Z

Md f dvAf dVg

= Z

M|^{d f}|^{2 d}V^g+ Z

M|^{d f v}A|^{2 d}V^g+ Z

M|^{f dv}A|^2dV^g

−2^Z

M|^{d f}|^2vAdVg−2^Z

Md f dv_Af(1−^vA) dVg. Recall we havedvA= ^duAe1−_e2^uAde1

1 , and : Z

MV|^φ|^2dVg= Z

MV|^f|^{2 d}Vg−² Z

MV|^f|^2vAdVg+ Z

MV|^vAf|^2dVg

hence Z

M|^dφ|^2dVg+ Z

MV|^φ|^2dVg= Z

M|^{d f}|^{2 d}Vg+ Z

MV|^f|^2dVg

| {z }

:=A(f)

+ Z

M|^{d f v}A|^{2 d}Vg

| {z }

:=B(f)

+ Z

M|^{f dv}A|^2dVg+ Z

MV|^vAf|^{2 d}Vg

| {z }

:=C(f)

−²





 Z

M|^{d f}|^2vAdVg+ Z

MV|^f|^2vAdVg

| {z }

:=D(f)







−2^Z

Md f dv_Af(1−^vA)dVg

| {z }

:=E(f)

.

_{Study of A}(f) := R

M|^{d f}|^{2 d}Vg+R

MV|^f|^2dVg ≥ ^{0 : since f} ∈ ^Fk we can write f =

k

∑

i=1

α_ie_iwhere(α_i)_1≤i≤k∈ ^{Rkand with}

k

∑

i=1

α²_i = 1 (sincek^fkL²(M)= 1), thus we get

A(f) =

* k

∑

j=1

α_jde_j,

∑

i=1

α_ide_i +

L²(M)

+

*√ V

k

∑

j=1

α_je_j,√ V

k

∑

i=1

α_ie_i +

L²(M)

=

∑

i,j

αiαj

dej,dei

L²(M)+ Z

MVejeidV^g

=

∑

i,j

αiαj

−^ej,∆_gei

L²(M)+ Z

MVejeidV^g

=

∑

i,j

α_iα_j

e_j, −^∆g+V e_i

L²(M)

=

∑

i,j

α_iα_jλ_i(M)e_j,e_i

L²(M)=

k

∑

i=1

α²_iλ_i(M)≤^λk(M).

Hence, for all integerk, and for all function f ∈ ^Fksuch thatk^fkL²(M)=1 we have

0≤^A(f)≤^λk(M). _{Study ofB}(f):=R

M|^d(f)vA|^{2 d}V^{g: herev}A = ^u_e^A₁ anddvA = ^duAe1−_e2^uAde1

1 ,

so we getB≤ k^{d f}k²∞k^vAk²L²(M)and, with the Poincaré inequality : k^vAk²L²(M)≤

1 e₁

2

∞k^uAk²L²(M)≤ 1 e₁

2

∞

cap(A) λ₁(M)

hence, for all integerk, and for all function f ∈ ^Fksuch thatk^fkL²(M) =1 we have

0≤^B(f)≤^Ekcap(A)

whereE_k = E_k(e₁,λ₁(M))> 0, moreover since the eigenfunctione₁and the eigenvalueλ₁(M)depends only on(M,g)andV, for all integerkthe constant E_kdepends only on(M,g)andV, ie :E_k=E_k(M,V).

_{Study ofC}(f): hereC(f)is equal to^Z

M|^{f dv}A|^2dV^g

| {z } +

:=C₁(f) Z

MV|^vAf|^{2 d}V^g

| {z }

:=C₂(f)

. Let

us observe firstC₁(f):

C₁(f)≤ k^fk²∞k^dvAk²L²(M)

and

k^dvAk²L²(M)= Z

M

duAe₁−^uAde₁ e²₁

2

dV^g

≤ 1 e₁

2

∞ Z

M|^duAe₁−^uAde₁|^{2 d}V^g

≤ 1 e₁

2

∞

_Z

M|^duAe₁|^{2 d}Vg+2^Z

M|^duAde₁e₁uA| ^dVg+ Z

M|^de1uA|^{2 d}Vg

≤ 1 e₁

2

∞

k^duAk²L²(M)k^e1k²∞+2k^de1k∞k^e1k∞k^duAkL²(M)k^uAkL²(M)+k^de1k²∞k^uAk²L²(M)

. Next we have also :

C₂(f) = Z

MV|^vAf|^2dVg≤ k^fk²∞ Z

MV|^vA|^2dVg

≤ k^fk²∞

1 e₁

2

∞ Z

MV|^uA|^{2 d}Vg. Hence we get :

C(f)≤ k^fk²∞

1 e₁

2

∞

hk^duAk²L²(M)k^e1k²∞

+2k^de1k∞k^e1k∞k^duAkL²(M)k^uAkL²(M)+k^de1k²∞k^uAk²L²(M)

i

+k^fk²∞

1 e₁

2

∞ Z

MV|^uA|^2dVg

≤ k^fk²∞

1 e₁

2

∞

hk^duAk²L²(M)k^e1k²∞+2k^de1k∞k^e1k∞k^duAkL²(M)k^uAkL²(M)+k^de1k²∞k^uAk²L²(M)

+ Z

M|^duA|^2dVg+ Z

MV|^uA|^2dVg

≤ k^fk²∞

1 e₁

2

∞

hk^duAk²L²(M)+k^Vk∞k^uAk²L²(M)

+2k^de1k∞k^e1k∞k^duAkL²(M)k^uAkL²(M)+k^de1k²∞k^uAk²L²(M)

i; so, sincek^duAk²L²(M)≤ ^cap(A)andk^uAk²L²(M)≤ ^cap(A)_λ1(M) we get for all integer k, and for all function f ∈ ^Fksuch thatk^fkL²(M)=1 :

0≤^C(f)≤^Fkcap(A)

where Fk = Fk(f,e₁,λ₁(M)) > 0. Here, for k fixed, the constantFk depends also on f, and f depends on the functions f₁,f₂,· · ·^,fk (which are depends only on M andV) and on the scalarsα₁,α₂,· · ·^,αk; since

∑

i=1

α²_i = 1, all the (α_i)_1≤i≤kare bounded inR, so finally, for all integer kthe constant F_k can be bounded by a constant (we denotes also by F_k = F_k(M,V)) which depends only onMandV.

_{Study of}|^D(f)|^{: we have}

|^D|= Z

M|^{d f}|^2vAdVg+ Z

MV|^f|^2vAdVg

≤ k^{d f}k²∞

1 e₁

_∞

Z M

uA

e₁ dVg+

V|^f|² e₁

_∞

Z

M|^uA| ^dVg

≤^max k^{d f}k²∞

1 e_{1 ∞},

V|^f|² e₁

_∞

!Z

M|^uA| ^dV^g

≤^max k^{d f}k²∞

1 e₁

_∞,

V|^f|² e₁

_∞

!q

Vol(M)k^uAkL²(M)

≤^max k^{d f}k²∞

1 e₁

_∞,

V|^f|² e₁

_∞

!q Vol(M)

scap(A) λ₁(M)^. Hence, for all integerk, and for all function f ∈^Fksuch thatk^fkL²(M)=1 :

|^D(f)| ≤^Gk

qcap(A)

where (and for the same reasons as in the study ofF, see the constantF_k) for all integerk, the constantG_kdepends only onMandV, ieG_k=G_k(M,V). _{Study of}|^E(f)|: recall thatE(f) =R

Md f dvAf(1−^vA) dVg, hence

|^E(f)| ≤ Z

M|^{d f dv}A| |^f| ^dVg+ Z

M|^{d f dv}A| |^{f v}A| ^dVg. For the first termR

M|^{d f dv}A| |^f|^dV^{gwe have :} Z

M|^{d f dv}A| |^f| ^dV^g≤ k^fk∞k^{d f}k∞

qVol(M)k^dvAkL²(M);

we have see in the study ofC(f)that k^dvAk²L²

≤ 1 e₁

2

∞

k^duAk²L²(M)k^e1k²∞+2k^de1k∞k^e1k∞k^duAkL²(M)k^uAkL²(M)+k^de1k²∞k^uAk²L²(M)

so withK:=k^fk∞k^{d f}k∞

pVol(M)_e¹₁

∞we get Z

M|^{d f dv}A| |^f| ^dVg

≤^Kqk^duAk²L²(M)k^e1k²∞+2k^de1k∞k^e1k∞k^duAkL²(M)k^uAkL²(M)+k^de1k²∞k^uAk²L²(M)

≤^K vu

utcap(A)k^e1k²∞+2k^de1k∞k^e1k∞

qcap(A)

scap(A)

λ₁(M) +k^de1k²∞

cap(A) λ₁(M)

≤^Hk

qcap(A)