D Le Peutrec
email: dorian.lepeutrec@math.u-psud.fr

F Nier

C Viterbo
email: viterbo@math.polytechnique.fr

Precise Arrhenius law for p-forms: The Witten Laplacian and Morse-Barannikov complex

Keywords: 57N65, 58J32, 58J37, 58J50, 81Q10, 81Q20 Exponentially small eigenvalues, Witten Laplacians, Morse-Barannikov complex

Accurate asymptotic expressions are given for the exponentially small eigenvalues of Witten Laplacians acting on p-forms. The key ingredient, which replaces explicit formulas for global quasimodes in the case p = 0, is Barannikov's presentation of Morse theory in [Bar].

Introduction and main statements 1.Presentation

The Brownian motion of a particle, at position x(t) (in R d for this rapid presentation), and experiencing a gradient field -2∇f (x), can be modelled 1 by the Smoluchowski stochastic differential equation (see a.e. [Nel]):

dx = -2∇f (x)dt - √ 2hdW , x(t) = x 0 ∈ R d . ( 1
)
Local minima of the energy profile, f , are stable steady states when h = 0 and become metastable states when h > 0 and small. If h > 0 is thought as a temperature, the lifetime such a metastable states U 0 is exponentially large in term of 1/h. Its inverse τ U 0 (h) follows an Arrhenius law τ U 0 (h) ∝ e -E act (U 0 ) h , where the activation energy E act (U 0 ) equals 2(f (U 1 )f (U 0 )) and U 1 is a proper saddle point, of the energy profile f , associated with U 0 . Those inverse lifetime are actually the exponentially small eigenvalues of the Feller semigroup generator, associated with (1),

-2∂ x f.∂ x -h∆ x = 1 h (-h∂ x + 2∂ x f ).(h∂ x ) on R d , (2)
defined on L 2 (R d , e -2f (x) h dx) , while e -2f (x) h dx is the invariant measure associated with (1). The analysis of these activation energies, or exponentially small eigenvalues in terms of h, has motivated various mathematical studies within the probabilistic approach and simulated annealing techniques in the 80's (see for instance [HKS] [FrWe]). More recently several works have been devoted to the accurate computation of the prefactors, P U 0 (h) in τ U 0 (h) = P U 0 (h)e -E act (U 0 ) h , with a probabilistic and potential theory approach in [BEGK] [BGK], or with PDE and spectral techniques in [START_REF] Helffer | Hypoelliptic estimates and spectral theory for Fokker-Planck operators and Witten Laplacians[END_REF] [HKN][HeNi1][Lep1][Lep2][Lep3][HHS2]. After conjugating with e -f h and multiplying by h, the operator (2) becomes a Witten Laplacian acting of functions (0-forms)

(-h∂ x + ∂ x f ).(h∂ x + ∂ x f ) = -h 2 ∆ + |∇ x | 2 -h 2 ∆ x = d * f,h d f,h = ∆ (0) f,h , (3)
with d f,h = e -f h (hd)e f h = hd + df ∧ and d * f,h = hd * + i ∇f . On a general configuration space, that is a manifold, the Witten Laplacian, acting on the space of all smooth differential forms

∆ f,h = (d f,h + d * f,h ) 2 = d * f,h d f,h + d f,h d * f,h (4)
is decomposed as the direct sum ∆

f,h = ⊕ d p=0 ∆ (p) f,h with ∆ (p)
f,h acting on pforms. It provides the geometrically intrisic writing, depending on the metric g and the Morse function f , and exhibits the relationship with other structures. In his celebrated article [Wit], Witten showed that this deformation of Hodge theory allows one to recover analytically the Morse inequalities for the function f . The number of eigenvalues of ∆ (p) f,h lying in [0, h 3/2 ) equals, for h > 0 small enough, the number m p of critical points of f with index p, while conjugating the differential with e f h provides an isomorphism in cohomology between the de Rham chain complex (Ω * (M), d) and the chain complex (Ω * f,h , d f,h ), where Ω p f,h is the space generated by the eigenmodes with eigenvalue less than h 3/2 for ∆ p f,h , and d f,h the Witten differential. Shortly after Witten's article, it was proved in [START_REF] Helffer | Puits multiples en limite semi-classique IV -Etude du complexe de Witten[END_REF] that the O(h 3/2 ) eigenvalues of ∆ for p = 0 were already known from [FrWe] [HKS]. The accurate determination, in the case of functions (p = 0), of the prefactors,

P (0) k (h) in λ (0) k = P (0) k (h)e -C (0) k h
, came later, motivated by probabilistic questions in [BEGK] [BGK], or by the analysis of the Kramers-Fokker-Planck operators in [HerNi]. The accurate computation of the small eigenvalues of ∆ (p) f,h is made difficult by the interactions due to the tunneling effect between the m p quantum wells, with a hierarchy of weakly resonant tunneling quantum wells, according to the terminology of [HeSj2][HeSj3]. This hierarchy, which orders the exponentially small quantities, can be solved by considering the interaction via the deformed differential d f,h with the eigenmodes of ∆ . When p = 0, it is simply understood within the probabilistic approach by ordering the exit times, following in some sense the intuition of Arrhenius law. It actually amounts to elementary topological arguments by considering how the number of connected components of the sublevel set f λ = {x, f (x) < λ} varies as λ crosses a critical value. Moreover, the analysis carried out in [START_REF] Helffer | Hypoelliptic estimates and spectral theory for Fokker-Planck operators and Witten Laplacians[END_REF] [HKN][HeNi1][Lep1][Lep2][Lep3], relied on the important remark that the eigenvalues of ∆ (0) f,h are the squares of the singular values of the differential d (0) f,h . The Fan inequalities (see [Sim]) for singular values then allow to propagate relative errors (i.e. small errors relative to various exponentially small quantities). For all these reasons, the study of exponentially small eigenvalues of ∆ (p) f,h for a general p, is a natural question which is also encountered in geometry (see for instance [Zha] [BiLe][Bis2] and references therein) or in statistical physics (see [TTK]). A first attempt was done in [START_REF] Peutrec | Small eigenvalues of the Witten Laplacian acting on p-forms on a surface[END_REF], extending the result for p = 0 to p = 0, 1, 2 on surfaces, with simple duality and chain complex arguments. For a general p, the global quasimodes of the form χ U 0 (x)e -f (x)-f (U 0 ) h used for the case p = 0 in [START_REF] Helffer | Hypoelliptic estimates and spectral theory for Fokker-Planck operators and Witten Laplacians[END_REF] [HKN][HeNi1][Lep1][Lep2] [START_REF] Peutrec | Small eigenvalues of the Neumann realization of the semiclassical Witten Laplacian[END_REF] and which propagate the information through weakly resonant quantum wells, are missing.

The solution comes from the use of Barannikov's version of the Morse complex from [Bar] which fits exactly the handling of global quasimodes for Witten Laplacian. There are two reasons for this: 1) this new chain complex has nice restriction properties which are implemented by boundary Witten Laplacians;

2) a side result coming from this presentation of Morse theory allows to replace the analytical computations with χ U 0 (x)e -f (x)-f (U 0 ) h in the case p = 0, by a subtle repeated use of Stokes' formula.

We conclude this introduction by emphasizing that the accurate analysis of the tunnel effect required for the computation of exponentially small eigenvalues, goes far beyond the instantonic picture (see [START_REF] Bott | Morse theory indomitable[END_REF]), which sticks in some sense to the intuition of classical mechanics. However, it is remarkable that discriminating between so small quantities (exponentially small quantities e -C k /h as h → 0) is made possible by global topological arguments.

Assumptions and result

Hypothesis 1. We shall work on an oriented compact riemannian manifold (M, g) and f will be an excellent Morse function : f is smooth has non degenerate critical points and these have distinct critical values. Moreover, homology and cohomology will always be with real coefficients.

Barannikov's simple Morse complex, allows to partition the set of critical points, U = {x ∈ M, ∇f (x) = 0} (resp. the set of critical points with index p,

U (p) = {x ∈ M, ∇f (x) = 0 , sign(Hess f)(x) = (d -p, p)})
, into upper, lower and homological critical points:

U = U U ⊔ U L ⊔ U H (5) resp. U (p) = U (p) U ⊔ U (p) L ⊔ U (p) H . (6)
Homological critical points in U (p)

H are associated with the kernel ker(∆

(p) f,h ) ∼ ker ∆ (p)
Hodge and their number is the p-th Betti number β p = dim H p (M, R) . The boundary operator ∂ B of Barannikov's chain complex, defined on ⊕ U ∈U RU, associates with any

U ′ ∈ U (p) U an element U ∈ U (p-1) L such that f (U) < f (U ′ ),
and vanishes on all other critical points U ′ ∈ U H ∪ U L . Details are given in Section 2. The second assumption avoids technical (nevertheless interesting) questions about multiplicities of non zero exponentially small eigenvalues.

Hypothesis 2. The values f (U ′ )-f (U) obtained for U ′ ∈ U U and ∂ B U ′ = U are all distinct.
Here is our main result Theorem 1.1. Assume Hypotheses 1 and 2 hold. Let U H , U L , U U respectively denote the sets of homological, lower and upper critical points. For h 0 small enough and 0 < h < h 0 , there exists a mapping j from

U := U H ∪ U L ∪U U onto σ(∆ f,h )∩[0, h 3 2 [ and the restriction j p := j U (p) is onto σ(∆ (p) f,h )∩ [0, h 3 2
[ and one to one provided the eigenvalues of σ(∆ (p) f,h ) are counted with multiplicities. Moreover, the map j satisfies the following properties:

1. For U (p) in U (p) H , j(U (p) ) = 0 . 2. For U (p) in U (p) L , let U (p+1) denote the element of U (p+1) U s.t. ∂ B (U (p+1) ) = U (p) . Then, there exists a homological constant κ(U p+1 ) ∈ R * such that j(U (p) ) = κ 2 (U (p+1) ) h π |λ (p+1) 1 • • • λ (p+1) p+1 | |λ (p) 1 • • • λ (p) p | |Hess f (U (p) )| 1 2 |Hess f (U (p+1) )| 1 2 × e -2 f (U (p+1) )-f (U (p) ) h (1 + O(h)) , where λ (ℓ) 1 , . . . , λ (ℓ)
ℓ denote the negative eigenvalues of Hess f (U (ℓ) ), for ℓ ∈ {p, p + 1}.

Finally , for U

(p) in U (p) U , the equality j(U (p) ) = j(∂ B (U (p) )) = j(U (p-1) ) holds with U (p-1) := ∂ B (U (p) ) ∈ U (p-1) L , i.e. j(U (p) ) = κ 2 (U (p) ) h π |λ (p) 1 • • • λ (p) p | |λ (p-1) 1 • • • λ (p-1) p-1 | |Hess f (U (p-1) )| 1 2 |Hess f (U (p) )| 1 2 × e -2 f (U (p )-f (U (p-1) ) h (1 + O(h)) , where λ (ℓ) 1 , . . . , λ (ℓ) ℓ denote the negative eigenvalues of Hess f (U (ℓ) ), for ℓ ∈ {p -1, p}.
A relative version of this result, implemented with boundary Witten Laplacians, is given in Subsection 4.5 at the end of this paper.

Remark 1.2. Although we are not able to prove it in general, there is a strong indication that the "homological" constant κ p (U 0 ) equals ±1 . This is indeed the case for p = 0 as shown in [HKN][HeNi1][Lep3]. By duality it is also true when p = d . Finally in the case of surfaces treated in [START_REF] Peutrec | Small eigenvalues of the Witten Laplacian acting on p-forms on a surface[END_REF], a combination of these results says that it is true for p = 0, 1, 2 . A general proof requires a better understanding of the topological aspects of Morse theory and of Barannikov's construction. Surely, this constant is completely determined by the structure of the homology groups of the sublevel sets H * ({f < λ}), λ ∈ [-∞, +∞] . It does not depend on h, on the riemannian metric g or on the Morse function f (as long our generic assumptions are fulfilled), contrary to the other factors. This is a reason to use the attribute "homological" for this, up to now unknown, constant.

Barannikov's simple complex and Morse theory

In this section, we adapt the approach of Barannikov, using notations and definitions better suited to the treatment of Witten Laplacians.

Sublevel sets and bases of Morse theory

Remember that we work on a riemannian compact oriented manifold (M, g), endowed with an excellent Morse function according to Hypothesis 1. With such an assumption we may identify a critical point U with the corresponding critical value c = f (U) . For any index p, 0

≤ p ≤ d = dim M, the set of crit- ical points of f with index p is U (p) = U (p) k , 1 ≤ k ≤ m p .
It is equivalently represented by a vertical line with a m p points with heights c

(p) k = f (U (p) k ), 1 ≤ k ≤ m p .
The vector space spanned by these points is denoted by

C (p) (f ) and we set C(f ) = ⊕ d p=0 C p (f ) .
We shall construct explicitly a differential on C(f ) quasi-isomorphic to the Morse chain complex associated with f . For λ ∈ [-∞, +∞], f λ denotes the sublevel set {x ∈ M, f (x) < λ} while f λ denotes the upper level set {x ∈ M, f (x) > λ} and f µ λ = {x ∈ M, λ < x < µ} . Let us recall a few elementary facts related with Morse theory known from [Mil][Bot][Lau1][Lau2]. We refer to [Hat][Mas][BoTu][Ful] for basic material in homological algebra. When λ ∈ (min f, max f ) is not a critical value, f λ and f λ are boundary manifolds, while f λ = M, f λ = ∅ for λ > max f and f λ = ∅, f λ = M for λ < min f . When there is no critical value between λ 1 and λ 2 ,the natural inclusion of f λ 1 in f λ 2 (resp. f λ 2 in f λ 1 ) induces a homotopy equivalence and therefore induces an isomorphism of their homology groups:

∀p ∈ {0, . . . , d} , H p (f λ 1 ) = H p (f λ 2 ) and H p (f λ 1 ) = H p (f λ 2 ) .
With the help of the five lemma, this holds also for the relative homology groups H * (f µ , f λ 1 ) and H * (f µ , f λ 2 ), and

H * (f λ 1 , f µ ) H * (f λ 2 , f µ ) for µ > λ 1 , λ 2 ,
as is easily seen using the long exact sequences

H * +1 (f λ ) i * / / H * +1 (f µ ) j * / / H * +1 (f µ , f λ ) ∂ / / H * (f λ ) , H * +1 (f µ ) i * / / H * +1 (f λ ) j * / / H * +1 (f λ , f µ ) ∂ / / H * (f µ ) .
when µ > λ are not critical values. Passing a critical point with index p and critical value c, the pair (f c+ε , f c-ε ) is homologous to the pair (D p , ∂D p ), associated with the p-cell e p = D p \ ∂D p (see [Mil]). This gives using excision

0 / / H p (f c-ε ) / / H p (f c+ε ) / / H p (f c+ε , f c-ε ) / / H p-1 (f c-ε ) / / H p-1 (f c+ε ) / / 0 , (7) with dim H p (f c+ε , f c-ε ) = 1 and ensures that for k = p, p -1 we have the equality of H k (f c±ε ) .
This yields two mutually exclusive cases

H p (f c-ε ) ∼ / / H p (f c+ε ) and 0 / / H p (f c+ε , f c-ε ) / / H p-1 (f c-ε ) / / H p-1 (f c+ε ) → 0 , ( 8
) or 0 / / H p (f c-ε ) / / H p (f c+ε ) → H p (f c+ε , f c-ε ) / / 0 and H p-1 (f c-ε ) ∼ / / H p-1 (f c+ε ) . (9)
The Poincaré duality takes a nice form owing to Theorem 3.43 [Hat] (M is oriented), with the excision argument

H * (f µ , f λ ) = H * (f µ λ , {f = λ}): For two non critical values -∞ ≤ λ < µ ≤ +∞, the cohomology group H k (f µ , f λ ) is isomorphic to H d-k (f λ , f µ ) .
In [Spa]-p296, this is called Alexander duality and proved, without excision, via coverings and Mayer-Vietoris techniques. With f λ = (-f ) -λ , this is often summarized by changing f into -f and inverting indexes p and dp . Thus, the dual version of (7) for a critical value with index p is

0 H d-p-1 (f c-ε ) o o H d-p-1 (f c+ε ) o o H d-p (f c-ε , f c+ε ) o o H d-p (f c-ε ) o o H d-p (f c+ε ) O O 0 O O .
For this, use the excision property

H k (f c+ε , f c-ε ) = H k ({c -ε ≤ f < c + ε} , {f = c -ε}) while noticing that according to Poincaré duality H k ({f = c -ε}) is isomor- phic to H d-1-k ({f = c -ε}) .
Hence passing a critical value with upper level sets leads to the two exclusive cases

H d-p (f c+ε ) ∼ / / H d-p (f c-ε ) and 0 / / H d-p (f c-ε , f c+ε ) / / H d-p-1 (f c+ε ) / / H d-p-1 (f c-ε ) → 0 , ( 10
) or 0 / / H d-p (f c+ε ) / / H d-p (f c-ε ) → H p (f c-ε , f c+ε ) / / 0 and H d-p-1 (f c+ε ) ∼ / / H d-p-1 (f c+ε ) . ( 11
)
2.2 Classification of critical points

Partition

The critical points are divided in three classes, and we prove that these classes make a partition of the set of critical points, satisfying a number of additional properties.

(X, A, B) = (f c+ε , f c-ε , ∅, ) and (X ′ , B ′ , A ′ ) = (M, f c-ε , ∅, )
with the mapping i ∞,c+ε :

f c+ε → M = f +∞ : H * (f c-ε ) / / Id H * (f c+ε ) / / i ∞,c+ε * H * (f c+ε , f c-ε ) ∂ / / i ∞,c+ε * H * -1 (f c-ε ) Id H * (f c-ε ) / / H * (M) / / H * (M, f c-ε ) ∂ ′ / / H * -1 (f c-ε )
. Now, assume that both the mappings

H * (f c+ε , f c-ε ) → H * (M, f c-ε ) and H * (f c+ε ) → H * (f c+ε , f c-ε ) both vanish. This implies that ∂ is injective while i ∞,c+ε = 0 . This contradicts the commutativity ∂ = ∂ ′ • i ∞,c+ε .

Upper critical points

We first give other characterizations for upper critical points with index p, which will be used further. An additional property about the rank of i λ * :

H * (f λ ) → H * (M) is given.
Proposition 2.3. A critical value c with index p is an upper critical value, iff one of the following conditions is satisfied:

1. The critical value satisfies the condition (8), namely:

H p (f c-ε ) ∼ / / H p (f c+ε ) and 0 / / H p (f c+ε , f c-ε ) / / H p-1 (f c-ε ) / / H p-1 (f c+ε ) → 0 . 2. The mapping ∂ : H * +1 (f c+ε , f c-ε ) → H * (f c-ε ) is one to one. 3. There exists λ ∈] -∞, c) such that the mapping H * (f c+ε , f λ ) → H * (f c+ε , f c-ε ) vanishes. 4. The exists λ ∈] -∞, c) such that the mapping ∂ : H * +1 (f c+ε , f c-ε ) → H * (f c+ε , f λ ) is one to one.
Proof. The condition 1) is just the explicit form of the definition of an upper critical value, in view of the long exact sequence (7).

Condition 2) is obtained by considering the right-hand side of the long exact sequence

H * +1 (f c-ε ) / / H * +1 (f c+ε ) / / H * +1 (f c+ε , f c-ε ) ∂ / / H * (f c-ε ) .
Similarly condition 4) is equivalent to condition 3). Consider condition 3): It is necessary (take λ = -∞). The middle square of the commutative diagram (12) with the embedding

ϕ = i λ : (X, A, B) = (f c+ε , f c-ε , f -∞ = ∅) → (X ′ , A ′ , B ′ ) = (f c+ε , f c-ε , f λ ): H * (f c-ε ) / / i λ * H * (f c+ε ) / / i λ * H * (f c+ε , f c-ε ) ∂ / / Id H * -1 (f c-ε ) i λ * H * (f c-ε , f λ ) / / H * (f c+ε , f λ ) 0 / / H * (f c+ε , f c-ε ) ∂ / / H * -1 (f c-ε , f λ ) ,
provides the sufficiency.

Consider for -∞ ≤ λ < µ ≤ +∞ which are not critical values, the embeddings i λ : f λ → M and i µ :

f µ → M and i µ,λ : (f λ , M) → (f µ , M) . Then the commutative diagram H * (f λ ) i λ * / / i µ,λ * H * (M) / / Id H * (M, f λ ) ∂ / / i µ,λ * H * -1 (f λ ) i µ,λ * H * (f µ ) i µ * / / H * (M) / / H * (M, f µ ) ∂ / / H * -1 (f µ ) implies i λ * = i µ * • i µ,λ * (λ < µ) (13) Im i λ * ⊂ Im i µ * , rank i λ * ≤ rank i µ * . ( 14
)
Proposition 2.4. When c is an upper critical value, the ranges of i c+ε * :

H * (f c+ε ) → H * (M) and i c-ε * : H * (f c-ε ) → H * (M) are the same.
Proof. The condition 1) of Proposition 2.3 ensures that for any k ∈ {0, . . . , d}, the mapping

i c+ε,c-ε * : H k (f c-ε ) → H k (f c+ε ) is onto. Therefore i c+ε * and i c-ε * = i c+ε • i c+ε,c-ε *
have the same range.

Lower critical points

With

H * (M, f c-ε ) = H * (f +∞ , f c+ε ) and the duality H * (f µ , f λ ) ≃ H d- * (f λ , f µ ),
says that the lower critical values are the one for which the mapping

H * (f c-ε ) = H * (f c-ε , f +∞ ) / / H * (f c-ε , f c+ε )
vanishes. It is therefore the dual notion to the one of upper critical values and all the dual properties of the ones of the upper critical values will hold for lower critical points. We shall use the following characterization.

Proposition 2.5. A critical value c with index p is a lower critical value, iff one of the following conditions is satisfied:

1. The critical value satisfies the condition (10), namely:

H d-p (f c+ε ) ∼ / / H d-p (f c-ε ) and 0 / / H d-p (f c-ε , f c+ε ) / / H d-p-1 (f c+ε ) / / H d-p-1 (f c-ε ) → 0 .
2. The mapping ∂ :

H * +1 (M, f c+ε ) → H * (f c+ε , f c-ε ) is onto.
3. There exists λ ∈ (c, +∞] such that the mapping

H * (f c+ε , f c-ε ) → H * (f λ , f c-ε ) vanishes.
4. There exists λ ∈ (c, +∞] such that the mapping ∂ :

H * +1 (f λ , f c+ε ) → H * (f c+ε , f c-ε ) is onto.
Proof. The condition 1) is the dual statement of (8) which is equivalent to the dual notion of upper critical values.

The equivalence with the condition 2) is contained in the long exact sequence

H * +1 (M, f c-ε ) / / H * +1 (M, f c+ε ) ∂ / / H * (f c+ε , f c-ε ) / / H * (M, f c-ε ) .
Similarly the condition 3) and 4) are equivalent. Consider the condition 3): It is necessary (take λ = +∞). If there exists λ ∈ (c, +∞] such that the composition of the embeddings i λ,c+ε : (

f c+ε , f c-ε ) → (f λ , f c-ε ) and i λ : (f λ , f c-ε ) → (M = f +∞ , f c-ε ) implies that the composed map H * (f c+ε , f c-ε ) i λ,c+ε * =0 / / H * (f λ , f c-ε ) i λ * / / H * (M, f c-ε ) ,
which equals i c+ε * , vanishes.

We next prove that the lower critical values share the same property as the upper critical values, concerning the rank of i λ * : H * (f λ ) → H * (M) . Proposition 2.6. When c is a lower critical value, the ranges of i c+ε * :

H * (f c+ε ) → H * (M) and i c-ε * : H * (f c-ε ) → H * (M) are the same.
Proof. Assume that c is a critical value with index p . Then for any k = p the map i c+ε,c-ε *

: H k (f c-ε ) → H k (f c+ε ) is onto and the range of i c-ε * = i c+ε * • i c+ε,c-ε * and i c+ε
* , when restricted to H k , are equal. For the case k = p, we start from the exact sequence

0 / / H p (f c-ε ) i c+ε,c-ε * / / H p (f c+ε ) / / H p (f c+ε , f c-ε ) ∂ / / H p-1 (f c-ε ) / / 0 ,
where the ∂-arrow vanishes, because c cannot be an upper critical value.

Hence the range of i c+ε,c-ε * is an hyperplane of H p (f c+ε ) and the equality

i c-ε * = i c+ε * • i c+ε,c-ε * implies that i c+ε * and i c-ε * have the same range iff 1 + dim ker(i c+ε * Im i c+ε,c-ε * ) -dim ker(i c+ε * ) = 0 . We write for simplicity i = i c+ε,c-ε .
We have to prove that there exists α ∈ H p (f c+ε ) such that i c+ε * (α) = 0 and α ∈ Im i * . The functoriality of the relative homology gives

H * (f c-ε ) i c-ε * / / i * H * (M) j c-ε * / / Id H * (M, f c-ε ) ∂ c-ε / / i * H * -1 (f c-ε ) i * H * (f c+ε ) i c+ε * / / H * (M) j c+ε * / / H * (M, f c+ε ) ∂ c+ε / / H * -1 (f c+ε ) . ( 15
)
The condition 2) of Proposition 2.5 and the long exact sequence of relative homologies for the triple (M, c+ε , f c-ε ), provide the exact sequence

0 / / H p+1 (M, f c-ε ) i * / / H p+1 (M, f c+ε ) ∂ / / H p (f c+ε , f c-ε ) → 0 . Let α 0 ∈ H p+1 (M, f c+ε ) be such that ∂α 0 = 0 . By the second line of the above commutative diagram (15), ∂ c+ε α 0 ∈ H p (f c+ε ) belongs to ker i c+ε * . Assume ∂ c+ε α 0 ∈ Im i * and take β ∈ H p (f c-ε ) such that ∂ c+ε α 0 = i * β .
The commutative diagramm (15) then implies

i c-ε * β = (Id • i c-ε * )β = (i c+ε * • i * )β = i c+ε * (∂ c+ε α) = 0 . Hence β ∈ ker i c-ε * = Im ∂ c-ε . Hence there exists γ ∈ H p+1 (M, f c-ε ) such that ∂ c+ε α 0 = (i * • ∂ c-ε )γ = ∂ c+ε i * γ . The cycle α 0 -i * γ belongs to ker ∂ c+ε = Im j c+ε * . Hence there exists δ ∈ H p+1 (M) such that α 0 -i * γ = (j c+ε * • Id)δ = (i * • j c-ε * )δ .
We finally get

α 0 = i * (γ + j c-ε * δ) ∈ Im i * = ker ∂ . But this contradicts the first assumption ∂α 0 = 0 . Thus we have found α = ∂ c+ε α 0 ∈ H p (f c+ε ) such that i c+ε (α) = (i c+ε • ∂ c+ε )α 0 = 0 and α ∈ Im i * .
This ends the proof.

Properties of homological critical values

The attribute "homological" is justified by the following result. Let

C (p)
H be the set of homological critical values with index p and set

C H = ∪ d p=0 C (p) H . Remember the mappings i λ * : H * (f λ ) → H * (M) .
Theorem 2.7. For every p ∈ {0, . . . , d}, there is a one to one mapping

α (p) : C (p)
H → H p (M) such that:

• The range of α (p) is a basis of H p (M); • For every c ∈ C (p) H , the quotient Im i c+ε * Im i c-ε * is the one-dimensional space spanned by class of α (p) (c) . Finally the cardinal of C (p) H is the p th Betti number of M .
We need the following result, where homological critical values differ from the lower and upper critical values.

Proposition 2.8. Assume that c is an homological critical value (resp. point) according to the Definition 2.1. Then the mapping

H * (f c+ε ) → H * (M, f c-ε ) is non zero. Moreover the mappings i c±ε * : H * (f c±ε ) → H * (M) satisfy Im i c-ε * ⊂ Im i c+ε * , rank i c-ε * = rank i c+ε * -1 .
Proof. By definition, homological critical value is neither a lower critical value nor an upper critical value. Therefore the mappings

H * (f c+ε , f c-ε ) → H(M, f c-ε ) and H * (f c+ε ) → H * (f c+ε , f c-ε ) are non zero. Since H * (f c+ε , f c-ε
) is one dimensional, the second one is onto and the composed map

H * (f c+ε ) σ / / H * (M, f c-ε )
is non zero. Consider now the second statement. We have already checked in (13)( 14) the relations

i c-ε * = i c+ε * • i c+ε,c-ε * and Im i c-ε * ⊂ Im i c+ε * .
From the long exact sequence (when c is a critical value with index p)

0 / / H p (f c-ε ) i c+ε,c-ε * / / H p (f c+ε ) / / H p-1 (f c+ε , f c-ε ) / / 0 , we know that the codimension of Im i c+ε,c-ε * is at most one. Thus rank i c+ε * -1 ≤ rank i c-ε * ≤ rank i c+ε * ,
and it suffices to find α ∈ Im i c+ε * which does not belong to Im i c-ε * . We use the diagram

H * (f c+ε ) i c+ε * σ ' ' O O O O O O O O O O O H * (f c-ε ) i c-ε * / / H * (M) j c-ε * / / H * (M, f c-ε ) . We know that there exists α 0 ∈ H * (f c+ε ) such that (j c-ε * •i c+ε * )(α 0 ) = σ(α 0 ) = 0 . Take α = i c+ε * (α 0 ) . It belongs to Im i c+ε *
and not in ker

j c-ε * = Im i c-ε * .
Proof of Theorem 2.7: Fix the degree p, 0 ≤ p ≤ d, and consider i λ * :

H p (f λ ) → H p (M) . Start from λ = max f + ε for which Im i λ * = Im Id = H p (M)
and decrease λ down to min fε for which Im i λ * = {0} . According to Morse theory, Proposition 2.4 and Proposition 2.6, the range of i λ * does not change except when λ passes an homological critical value with index p . For such a critical value, Proposition 2.8 says that the rank of i λ * is exactly decreased by 1 . This yields the result.

The Morse

-Barannikov chain complex 2.3.1 Definition Remember that C(f ) = ⊕ d p=0 C (p) (f )
is the vector space spanned by the critical points (identified with the critical values and the same notation c will be used for the two objects). The following definition will be proved to define a chain complex structure on C(f ) of which the homology groups are isomorphic to the H p (M) .

Definition 2.9. On C(f ) consider the linear mapping ∂ B defined by:

• When c is a lower critical point or an homological critical point, ∂ B c = 0 .

• When c is an upper critical point, take for c ′ , according to the condition 3) of Proposition 2.3, the supremum of the

λ ′ s in ] -∞, c) such that the mapping H * (f c+ε , f λ ) → H * (f c+ε , f c-ε ) vanishes and set ∂ B c = c ′ .
Theorem 2.10. The mapping

∂ B : C(f ) → C(f ) sends C (p) (f ) into C (p-1) (f ) and satisfies ∂ B • ∂ B = 0 . Moreover the homology groups H * (C(f )) are isomorphic to H * (M) and a basis of H * (M) is indexed by the set C H (f ) of homological critical points.
Proof. It suffices to prove that when c is an upper critical point with index p, the point c ′ is a lower critical point with index p -1 . Assume that the mapping

H * (f c+ε , f c ′ -ε ) → H * (f c+ε , f c-ε ) vanishes while the mapping σ : H * (f c+ε , f c ′ +ε ) → H * (f c+ε , f c-ε ) is non zero. Consider the commutative diagram H * (f c+ε , f c ′ -ε ) ϕ + / / 0 H * (f c+ε , f c ′ +ε ) ∂ + / / σ v v l l l l l l l l l l l l l H * -1 (f c ′ +ε , f c ′ -ε ) H * (f c+ε , f c-ε )
where the first line is the long exact sequence for the triple

f c ′ -ε ⊂ f c ′ +ε ⊂ f c+ε . Since σ is non zero while σ • ϕ + = 0, ϕ + cannot be onto and ∂ + is non zero. We have found λ = c + ε such that the mapping ∂ : H * (f λ , f c ′ +ε ) → H * -1 (f c ′ +ε , f c ′ -ε ) is onto. By the characterization 4) of Proposition 2.5, c ′ is a lower critical point.
Clearly it has the index p -1 when c has the index p .

Therefore

∂ B • ∂ B = 0. Before we conclude, we check that if ∂(c) = c ′ , then c is the infimum of the λ ′ s such that ∂ : H * (f λ , f c ′ +ε ) → H * (f c ′ +ε , f c ′ -ε ) is onto.
We have to prove that the map

∂ -: H * (f c-ε , f c ′ +ε ) → H * (f c ′ +ε , f c ′ -ε ) van- ishes. Consider the diagram H * (f c-ε , f c ′ -ε ) ϕ - / / i c+ε,c-ε * H * (f c-ε , f c ′ +ε ) ∂ - / / i c+ε,c-ε * H * -1 (f c ′ +ε , f c ′ -ε ) Id H * (f c+ε , f c ′ -ε ) ϕ + / / 0 H * (f c+ε , f c ′ +ε ) ∂ + / / σ v v l l l l l l l l l l l l l H * -1 (f c ′ +ε , f c ′ -ε ) H * (f c+ε , f c-ε ) .
The maps σ and ∂ + have one dimensional ranges and their kernels have the codimension 1. Due to σ • ϕ + = 0, we know ker ∂ + = Im ϕ + ⊂ ker σ . With the same dimension, this yields ker

∂ + = ker σ . If ∂ -does not vanish, there exists u such that ∂ + (i c+ε,c-ε + u) = ∂ -u = 0 . Hence we get (σ • i c+ε,c-ε * )u = 0, which contradicts the fact that σ • i c+ε,c-ε * = 0 as a part of the long exact sequence for the triple f c ′ +ε ⊂ f c-ε ⊂ f c+ε . Hence c is the infimum of the λ's such that ∂ : H * (f λ , f c ′ +ε ) → H * (f c ′ +ε , f c ′ -ε ) is onto.
Now assume that c ′ be a lower critical point. By the characterization 4) of Proposition 2.5, the infimum of the λ's in (c, +∞], such that ∂ :

H * (f λ , f c ′ +ε ) → H * (f c ′ +ε , f c ′ -ε ) is onto, exists. Call it c.
By the dual argument of the previous one, c is an upper critical point and c ′ is the supremum of the λ's such that the mapping

H * (f c+ε , f λ ) → H * (f c+ε , f c-ε ) vanishes. Hence c is an upper critical point such that ∂ B (c) = c ′ .
We have finally proved that the range of ∂ B : C(f ) → C(f ) contains all the lower critical points. The other statements are now straightforward consequences of Theorem 2.7.

Remark 2.11. This result provides another proof of Morse inequalities, for excellent Morse functions, without making use of homotopy arguments to reduce the problem to self-indexed Morse functions (see [Mil] [Bot][Lau2]).

H c ′ 1 H H c ′ 2 c ′ 3 H H c 1 c 3 c 2 H Figure 1
Example with a compact surface with genius 2 where f is the height function. The homological critical points are labelled by H while the pairing of other critical points follows

∂ B c k = c ′ k .
Proposition 2.12. When c is an upper critical point with index p such that ∂c = c ′ , then the following commutative diagram holds:

0 H p (f c ′ +ε , f c ′ -ε ) i c-ε,c ′ +ε * 0 0 / / H p+1 (f c+ε , f c-ε ) ∂ / / H p (f c-ε , f c ′ -ε ) i c+ε,c-ε * / / j * H p (f c+ε , f c ′ -ε ) / / j * 0 0 / / H p (f c-ε , f c ′ +ε ) i c+ε,c-ε * / / H p (f c+ε , f c ′ +ε ) / / O O o o 0 0 0 . In particular, if H p+1 (f c+ε , f c-ε ) = R[e p+1 ] and H p (f c ′ +ε , f c ′ -ε ) = R[e p ],
then there exists κ ∈ R * such that ∂e p+1 and κe p are homologous in

f c-ε relatively to f c ′ -ε : [∂e p+1 ] = k[e p ] in H p (f c-ε , f c ′ -ε ) .
Proof. From the definition of ∂ B (c) = c ′ , we know that the mapping ∂ -:

H p+1 (f c+ε , f c-ε → H p (f c+ε , f c ′ -ε
) is one to one while the mapping ∂ + :

H p+1 (f c+ε , f c-ε → H p (f c+ε , f c ′ +ε ) vanishes.
Put in the long exact sequences associated with the two triples (f c ′ -ε , f c-ε , f c+ε ) and (f c ′ +ε , f c-ε , f c+ε ), this provides the two lines of the diagram.

Similarly, the relation

∂ B (c) = c ′ implies that the mapping ∂ -: H p+1 (f c+ε , f c ′ +ε ) → H p (f c ′ +ε , f c ′ -ε ) vanishes (or equivalently the mapping H p (f c ′ +ε , f c ′ -ε ) → H p (f c+ε , f c ′ -ε ) vanishes) and the mapping ∂ -: H p+1 (f c-ε , f c ′ +ε ) → H p (f c ′ +ε , f c ′ -ε ) vanishes.
Inserted in the long exact sequences associated with the triples

(f c ′ -ε , f c ′ +ε , f c+ε ) and (f c ′ -ε , f c ′ +ε , f c-ε ), this provides the two columns of the diagram. The diagram implies that the two mappings i c-ε,c ′ +ε * : H p (f c ′ +ε , f c ′ -ε ) → H p (f c-ε , f c ′ -ε ) and ∂ : H p+1 (f c+ε , f c-ε ) → H p (f c-ε , f c ′ -ε
) have the same one dimensional range. This ends the proof.

[e p ] [∂e p+1 ] c c ′ 3 f = c -ε c ′ f = c ′ -ε Figure 2
One example on a surface for which [∂e p+1 ] = 1[e p ], p = 1 .

Restriction

In the previous construction the manifold M equals f +∞ while the homology group

H * (f λ ) equals H * (f λ , ∅) = H * (f λ , f -∞ ) .
All the construction can be done with sublevel sets f a and f b with -∞ ≤ a < b ≤ +∞ which are not critical values. For λ ∈] -∞, +∞[ which is not a critical value, consider C(f, λ), the chain subcomplex of C(f ) generated by critical values (points) below level λ . Since ∂ B preserves C(f, λ), we can introduce the quotient C(f, λ, µ) = C(f, λ)/C(f, µ) when µ < λ . And there are relative homology groups H * (C(f, λ), C(f, µ)) for ∂ B , which will be denoted by

H * (C(f, λ), C(f, µ)) .
All the previous definitions and proofs can be translated to the restricted and relative homologies, after replacing H * (M, f λ ) by H * (f b , f λ ) and H * (f λ ) by H * (f λ , f b ), when a < λ < b are not critical values. This observation gives at once. Theorem 2.13. For any a, b, -∞ ≤ a < b ≤ +∞, which are not critical values, the relative homology groups

H * (C(f, b), C(f, a)) are isomorphic to H * (f b , f a ) and the following diagram H * (f a ) i b,a * / / ≃ H * (f b ) j * / / ≃ H * (f b , f a ) ∂ / / ≃ H * -1 (f a ) ≃ H * (C(f, a)) / / H * (C(f, b)) / / H * (C(f, b), C(f, a)) ∂ B / / H * -1 (C(f, a))
is commutative.

Since we have a good basis of the chain complex (C(f ), ∂ B ), where the image by ∂ B of a generator is either 0 or another generator, we have a nice identification

H * (C(f, b), C(f, a)) .

Relative Witten chain complex

The Witten Laplacian is a deformation of the Hodge Laplacian, related with de Rham cohomology, which allows to give within a semiclassical asymptotic framework, an analytic proof of the Morse inequalities (see [START_REF] Witten | Supersymmetry and Morse inequalities[END_REF][START_REF] Cycon | Schrödinger operators with application to quantum mechanics and global geometry[END_REF][START_REF] Helffer | Puits multiples en limite semi-classique IV -Etude du complexe de Witten[END_REF]). The accurate computations of its exponentially small eigenvalues has connections with various topics going from stochastic analysis ( [FrWe, BEGK, BGK]) with kinetic theory ( [START_REF] Hérau | Isotropic hypoellipticity and trend to the equilibrium for the Fokker-Planck equation with high degree potential[END_REF][START_REF] Helffer | Hypoelliptic estimates and spectral theory for Fokker-Planck operators and Witten Laplacians[END_REF][START_REF] Hérau | Semiclassical analysis for the Kramers-Fokker-Planck equation[END_REF][START_REF] Hérau | Tunnel effect for Kramers-Fokker-Planck type operators[END_REF][START_REF] Hérau | Tunnel effect and symmetries for Kramers Fokker-Planck type operators[END_REF]), the computation of geometric invariants ([BiLe] [Bis]), differential topology ( [START_REF] Milnor | Morse theory[END_REF][START_REF] Bott | Lectures on Morse theory, old and new[END_REF][START_REF] Laudenbach | On the Thom-Smale complex[END_REF]) The case of manifold with boundaries has been considered in [START_REF] Chang | A cohomology complex for manifolds with boundary[END_REF][START_REF] Helffer | Quantitative analysis of metastability in reversible diffusion processes via a Witten complex approach: the case with boundary[END_REF][START_REF] Koldan | Semiclassical Asymptotics on Manifolds with Boundary[END_REF][START_REF] Peutrec | Local WKB construction for Witten Laplacians on manifolds with boundary[END_REF][START_REF] Peutrec | Small eigenvalues of the Neumann realization of the semiclassical Witten Laplacian[END_REF] with a spectral approach and more recently in [START_REF] Laudenbach | A Morse complex on manifolds with boundary[END_REF] with a pure topological point of view partly inspired by those previous works. We shall consider here directly the case with boundary, which is of interest here, and recall a few basic facts. We want to specify the realization of the Witten Laplacian on the manifold f b a with boundaries {f = a} and {f = b}, when -∞ ≤ a < b ≤ +∞ are not critical values, which is associated with the relative homology (after de Rham duality) H * (f b , f a ) .

Functional analysis

We recall that T *

x M = ⊕ d p=0 p T * x M is the exterior algebra on the cotangent fiber T *

x M, T * M is the corresponding fiber bundle and F (M; T * M) denotes the space of sections of class F on M (F stands for C ∞ , L p or W m,r ). The notation F (f b a ; T * M) is the set of restrictions to f b a of elements in F (M; T * M) . The spaces T *

x M and L 2 (M; T * M) and L 2 (M; T * M) are endowed with their natural scalar products inherited from the riemannian metric g . A shorter notation for the Sobolev spaces will be

ΛW m,r = W m,r (M; T * M) , Λ p W m,r (f b a ) = W m,r (f b a ; p T * M) .
After the introduction of the Hodge-⋆ operator, the scalar product of two p-forms equals

ω 1 | ω 2 Λ p L 2 = Ω ω 1 ∧ ⋆ω 2 ,
The notation t and n are specific to the case with boundary and useful for the analysis of boundary Hodge and Witten Laplacians (see [Sch][HeNi1][Lep3]).

Here is their specific meaning: On the boundary ∂Ω of a regular domain Ω, decompose the tangent vectors X i ∈ T σ Ω, σ ∈ ∂Ω, as X i = X T i + x ⊥ i n σ where n σ is a normalized outgoing vector normal to ∂Ω and set for ω ∈ C ∞ (Ω;

T * Ω) (tω) σ (X 1 , . . . , X p ) = ω σ (X T 1 , . . . , X T p ) , ∀σ ∈ ∂Ω , nω = ω ∂Ω -tω ∈ C ∞ (∂Ω; p T * Ω) .
Note that the tω and nω have a natural extension to a neighborhood of ∂Ω when the metric is fixed. After the right choice of coordinates, with x d parametrizing normal curves to ∂Ω, tω is the part with no dx d while nω takes the form dx d ∧ ω ′ . On C ∞ differential forms, they satisfy various relations with the Hodge-⋆ operator, the differential d and the codifferential d * (see [Sch][HeNi1][Lep3] for details),

⋆d * ,(p-1) = (-1) p d (d-p) ⋆ , ⋆d (p) = (-1) p+1 d * ,(d-p-1) ⋆ , ( 16
) ⋆ n = t ⋆ , ⋆ t = n ⋆ , (17)
t d = d t , n d * = d * n , (18)
and the Stokes' formula,

∀ω ∈ C ∞ (Ω; p T * Ω), Ω dω = ∂Ω tω .
When there is no boundary, the Hodge-de Rham theory makes the relation between the spectral theory of the Hodge Laplacian and de Rham duality of homology and cohomology groups (see [Ful]). For boundary manifold relative and absolute (co-)homology groups can be considered. By excision, remember that H * (f b , f a ) = H * (f b a , {f = a}) . We briefly recall why these relative homology group are naturally associated with specific boundary conditions for the Hodge and Witten Laplacians. We refer the reader to [Gue] for a review and [Tay] for a complete but different presentation relying on the isometric doubling of the boundary manifold. When γ is a cycle in f b a relative to {f = a}, there is a natural (i.e. independent of the representant Note that the differential d preserves the class C ∞ T while the codifferential preserves C ∞ N . Better commutations relations appear after considering the Hodge Laplacian (see further). The de Rham cohomology group H p (f b , f a ) is then given by ker

lying in {a ≤ f } of γ) integration γ ω of forms ω ∈ C ∞ (f b a ; T * M) such that tω {f =a} = 0 .
d (p) /(Im d p-1 ∩ C ∞ T N (f b a , p T * M)), when d is defined on C ∞ T N (f b a , T * M)
. The Witten deformation consists in introducing a small parameter h → 0 and to set

d f,h = e -f h (hd)e f h = hd + df ∧ , d * f,h = e f h (hd * )e f h = hd * + i ∇f .
The Witten Laplacian is defined as a differential operator in

f b a = {a < f < b} by ∆ f,h = (d f,h +d * f,h ) 2 = d f,h d * f,h +d * f,h d f,h = h 2 (d+d * ) 2 +|∇f | 2 +h(L ∇f +L * ∇f ) .
On the boundaries {f = a} and {f = b}, the boundary conditions have to be completed with f -dependent additional boundary conditions in order to get a self-adjoint realization which is elliptic up to the boundary (see [Sch]).

An additional property required here, is the commutation of the resolvent with d f,h and d * f,h . We follow the scheme of [START_REF] Chang | A cohomology complex for manifolds with boundary[END_REF][START_REF] Helffer | Quantitative analysis of metastability in reversible diffusion processes via a Witten complex approach: the case with boundary[END_REF][START_REF] Peutrec | Small eigenvalues of the Neumann realization of the semiclassical Witten Laplacian[END_REF] where the "Dirichlet problem" and the "Neumann problem" have been considered separately. Here the "Dirichlet" boundary conditions occurs on {f = a} while the "Neumann" boundary condition appears on {f = b} . Consider

in ΛW 1,2 T N = C ∞ T N (f b a ; T * M) W 1,2
the quadratic form given by

D T N (ω, η) = d f,h ω d f,h η + d * f,h ω d * f,h η . D T N (ω) = D T N (ω, ω) = d f,h ω 2 L 2 + d * f,h ω 2 L 2 .
Since {f = a} and {f = b} are disjoint, the main arguments are local (Sobolev trace theorem, Lopatinski-Schapiro conditions for the ellipticity up to the boundary and finally playing with (16)(17)( 18)), we can combine without repeating the proofs the results of [START_REF] Helffer | Quantitative analysis of metastability in reversible diffusion processes via a Witten complex approach: the case with boundary[END_REF] and [START_REF] Peutrec | Small eigenvalues of the Neumann realization of the semiclassical Witten Laplacian[END_REF] in order to state the following result. Note that due to the boundaries of the domain Ω, we avoid to consider the closure of the differential operators d f,h and d * f,h in ΛL 2 which are not very explicit.

Proposition 3.1. The non-negative quadratic form ω → D T N (ω) is closed on ΛW 1,2

T N . The associated (self-adjoint) Friedrichs extension is denoted by ∆ T N f,h . Its domain is

D(∆ T N f,h ) = ω ∈ ΛW 2,2 (f b a ); tω f =a = 0 , td * f,h ω f =a = 0 , nω f =b = 0 , nd f,h ω f =b = 0 ,
and acts as

∀ω ∈ D(∆ T N f,h ), ∆ T N f,h ω = ∆ f,h ω . The operator ∆ T N
f,h has a compact resolvent and a discrete spectrum. Moreover the commutations

(z -∆ T N f,h ) -1 • d f,h ω = d f,h • (z -∆ f,h ) -1 ω , (z -∆ T N f,h ) -1 • d * f,h ω = d * f,h • (z -∆ f,h ) -1 ω , 1 E (∆ T N f,h ) • d f,h ω = d f,h • 1 E (∆ T N f,h )ω , and 1 E (∆ T N f,h ) • d * f,h ω = d * f,h • 1 E (∆ T N f,h )ω ,
hold for all z ∈ C \ R, all Borel set E in R and all ω ∈ ΛW 1,2 T N .

Remark 3.2.

• The introduction of ∆ T N f,h , as a Friedrichs extenstion of a non negative closed quadratic form defined on ΛW 1,2

T N , ensures that it is a non negative self-adjoint operator. Requiring ∆ T N f,h u ∈ ΛL 2 for u ∈ D(∆ T N f,h ) forces the additional boundary conditions, after integration by part.

• The commutation relations do not result simply of the commutatition of the differential operators

∆ f,h •d f,h = d f,h •∆ f,h valid in the interior f b a . Indeed ∆ T N
f,h can be applied only to elements of D(∆ T N f,h ) fulfilling the boundary conditions while d f,h do not preserve these boundary conditions even for C ∞ -forms up to the boundaries. For details and complete proofs, we refer again the reader to [HeNi1][Lep3] and [ChLi].

• We can also define analogously the self-adjoint operator ∆ N T f,h , with domain D(∆ N T f,h ), by switching the above conditions on n and t. Here is the main result of this section.

Theorem 3.3. There are two operators L ± ∈ L(ΛL 2 ; ΛW 1,2 T N ), commuting for all Borel sets E in R with 1 E (∆ T N f,h ), such that every u ∈ ΛL 2 admits the orthogonal decomposition

u = 1 {0} (∆ T N f,h )u + d f,h L -u + d * f,h L + u . ( 20
)
When F M denotes the finite dimensional space Im

1 [0,M ] (∆ T N f,h ) and β M = d f,h F M , its adjoint is β * M = d * f,h F M
and F M admits the orthogonal decomposition

F M = ker ∆ T N f,h ⊥ ⊕ Im β M ⊥ ⊕ Im β * M . After setting F (p) M = Im 1 [0,M ] (∆ T N,(p) f,h
), the two finite dimensional chain complexes

0 / / F (0) M . . . F (p-1) M β (p-1) M / / o o F (p) M β (p) M / / β (p-1) * M o o F (p+1) M . . . F (d) M / / β (p) * M o o 0 o o (21)
are dual to each other and ker

β (p) M /Im β (p-1) M is diffeomorphic to H p (f b , f a ) .
For any p ∈ {0, . . . , d}, the spectrum of σ(∆ T N,(p) f,h

)∩(0, M] lying in (0, M] and counted with multiplicities, the set of λ 2 (counted with multiplicities) when λ ranges over the singular values, counted with multiplicities, of β

T N f,h , the set of C ∞ (f b a ; T * M) ∩ D(∆ T N f,h )) is dense in D(∆ T N f,h ) because 1 + ∆ T N f,h : D(∆ T N f,h ) → ΛL 2 is an isomorphism. For ω ∈ C ∞ (f b a ; T * M) ∩ D(∆ T N f,h )), we have n(d f,h ω) f =b = 0 and t(d f,h ω) f =a = 0 because ω ∈ D(∆ T N f,h ) . Moreover ω ∈ D(∆ T N f,h ) also says tω f =a = 0 and nω f =b = 0 . But since te ± f h ω = e ± f h tω and ne ± f h ω = e ± f h nω, the commutations (18) imply t(d f,h ω) f =a = 0 and n(d * f,h ω) f =b = 0 .
This ends the proof.

Lemma 3.5. The relation

d f,h θ 1 , θ 2 ΛL 2 = θ 1 , d * f,h θ 2 ΛL 2 holds for all θ 1 , θ 2 ∈ ΛW 1,2
T N . Proof. Since both quantities are continous and

C ∞ T N (f b a ; T * M) is dense in ΛW 1,2 T N , it suffices to consider θ 1 , θ 2 ∈ C ∞ T N (f b a ; T * M) . After writing ω 1 = e f h θ 1 and ω 2 = e -f h θ 2 in C ∞ T N (f b a ; T * M), our identity amounts to dω 1 , ω 2 ΛL 2 = ω 1 , d * ω 2 ΛL 2 .
But the Stokes' formula with the relations between d, ⋆ and ∧ gives

f =a t(ω 1 ∧ ⋆ω 2 ) + f =b t(ω 1 ∧ ⋆ω 2 ) = dω 1 | ω 2 ΛL 2 + ω 1 | d * ω 2 .
With the help of (17), write

t(ω 1 ∧ ⋆ω 2 ) = t [(tω 1 ) ∧ ⋆(tω 2 ) + (tω 1 ) ∧ ⋆(nω 2 ) + (nω 1 ) ∧ ⋆(tω 2 ) +(nω 1 ) ∧ ⋆(nω 2 )] = t [(tω 1 ) ∧ n(⋆ω 2 ) + (tω 1 ) ∧ t(⋆ω 2 ) + (nω 1 ) ∧ t(⋆ω 2 ) +(nω 1 ) ∧ n(⋆ω 2 )]
notice that t((nu 1 ) ∧ (tu 2 )) = t((tu 1 ) ∧ (nu 2 )) = 0 . This leads to

t(ω 1 ∧ ⋆ω 2 ) = t [(tω 1 ) ∧ ⋆(nω 2 ) + (nω 1 ) ∧ ⋆(tω 2 )] ,
where both terms vanishes on {f = a}∪{f = b} when ω

1 , ω 2 ∈ C ∞ T N (f b a ; T * M) .
Proof of Theorem 3.3. The operator ∆ T N f,h is a self-adjoint operator with a compact resolvent. Therefore it is invertible when restricted to ker(∆ T N f,h ) ⊥ = Im 1 (0,+∞) (∆ * f,h ) . Take

L + = d f,h (∆ T N f,h ) -1 1 (0,+∞) (∆ T N f,h ) and L -= d * f,h (∆ T N f,h ) -1 1 (0,+∞) (∆ T N f,h ) , Since (∆ T N f,h ) -1 1 (0,+∞) (∆ T N f,h ) ∈ L(ΛL 2 , D(∆ T N f,h )), L ± ∈ L(ΛL 2 , ΛW 1,2 T N ) ⊂ L(ΛL 2 , ΛL 2 ) according to Lemma 3.4. The commutation of L ± with 1 E (∆ f,h
) is then a consequence of the commutation stated in Proposition 3.1. These commutations of Proposition 3.1 also imply

L + ω = (∆ T N f,h ) -1 1 (0,+∞) (∆ T N f,h )d f,h ω and L -ω = (∆ T N f,h ) -1 1 (0,+∞) (∆ T N f,h )d * f,h ω , when ω ∈ ΛW 1,2 T N so that L ± ∈ L(ΛW 1,2 T N ; D(∆ T N f,h ))
. By using again Lemma 3.4,

d f,h • L -and d * f,h • L + belong to L(ΛW 1,2 T N ) . Consider now the decomposition ω = 1 {0} (∆ T N f,h )ω + d f,h L -ω + d * f,h L + ω when ω ∈ ΛW 1,2 T N . All the terms belong to ΛW 1,2 T N while d f,h (d f,h L -ω) = d f,h (1 {0} (∆ T N f,h )ω) = 0 and d * f,h (d * f,h L + ω) = d * f,h (1 {0} (∆ T N f,h )ω) = 0 .
Therefore Lemma 3.5 implies that the decomposition is orthogonal when ω ∈ ΛW 1,2 T N , and this extends by continuity to ω ∈ ΛL 2 . Proposition 3.1 ensures that d f,h and d * f,h send F M into itself and Lemma 3.5 with

F M ⊂ D(∆ T N f,h ) ⊂ ΛW 1,2 T N implies β * M = (1 [0,M ] (∆ T N f,h )d f,h 1 [0,M ] (∆ T N f,h )) * = (1 [0,M ] (∆ T N f,h )d * f,h 1 [0,M ] (∆ T N f,h )) .
The orthogonal decomposition follows from the result for ω ∈ ΛL 2 . The chain complex structure comes from

d f,h • d f,h = d * f,h • d * f,h = 0.
The space ker β

(p) M β (p-1) M is isomorphic to ker(∆ (p) f,h ), which is contained like F M in C ∞ T N (f b a ; p T * M)
kerβ (p) M /Im β (p-1) M ∼ ker d (p) /Im d (p-1) = H p (f a , f b )
by Hodge-de Rham theory (see for instance [Ful] for the usual boundaryless case and [Gue][Tay] for the case with boundary). The result concerned with the spectrum of ∆ T N,(p) f,h is a direct consequence of the orthogonal decomposition of F M with the chain complex structure (21). To be more specific, the decomposition of ∆

T N,(p) f,h F (p) M
according to Remark 3.6. Note that the duality between the two chain complexes associated with β M and β * M and their homology groups, is another version of the topological duality f → -f . Actually changing f to -f and p-forms with dp-forms with the Hodge-⋆ operator, interchanges d f,h and d * f,h .

F (p) M = ker ∆ T N,(p) f,h ⊥ ⊕ Im β (p-1) M ⊥ ⊕ Im β * ,(p+1) M writes ∆ T N,(p) f,h F (p) M = 0 ⊥ ⊕ β (p-1) M β (p-1), * M ⊥ ⊕ β (p), * M β (

Adapting Helffer-Sjöstrand analysis

We still work in f b a and we introduce like in [START_REF] Helffer | Puits multiples en limite semi-classique IV -Etude du complexe de Witten[END_REF] the Agmon distance d Ag associated with the degenerate metric |∇f | 2 g, where g is the initial Riemannian metric on M . This distance satisfies

d Ag (x, y) ≥ |f (x) -f (y)|
with equality when an integral curve of ∇f joins x and y .

Before the following crucial theorem, let us introduce two definitions which will be very useful in the sequel. The first one recalls Helffer-Sjöstrand notation Õ, very convenient when handling exponentially small quantities.

Definition 3.7. For two quantities A(h), estimated with a norm |A(h)|, and B(h) ≥ 0, parametrized by h ∈ (0, h 0 ), the notation A(h) = Õ(B(h)) means:

∀ε > 0, ∃C ε > 0, ∀h ∈ (0, h 0 ), |A(h)| ≤ C ε B(h)e ε h .
Definition 3.8. Let U ∈ M be a critical point of f with index p and let Φ(x) := d Ag (x, U). A local coordinate system y 1 , . . . , y d around U is said to be an adapted Morse coordinate system for f if y 1 , . . . , y d is centered at U, dy 1 , . . . , dy d is an orthonormal positively oriented basis of T * U M, and if, in these coordinates, the following Morse decompositions for f and Φ,

f (y) = f (U) + 1 2 d j=1 λ j y 2 j , Φ(y) = 1 2 d j=1 |λ j |y 2 j ,
hold locally around U, with λ j < 0 for j ≤ p and λ j > 0 for j > p.

Let us notice that such a coordinate system always exists, according to [START_REF] Helffer | Puits multiples en limite semi-classique IV -Etude du complexe de Witten[END_REF] pp. 272-281. Moreover, in such a coordinate system, the stable and unstable manifolds of -∇f , respectively denoted by C St and C Unst are locally parametrized by

C St = {y ; y 1 = • • • = y p = 0} and C Unst = {y ; y p+1 = • • • = y d = 0} . ( 22
)
Theorem 3.9. Let p belong to {0, . . . , d} and denote by

U (p) = U (p) 1 , . . . , U (p) mp
the set critical points of f in f b a . There exists h 0 > 0 such that, for all h ∈ (0, h 0 ], the spectral subspace

F (p) = 1 [0,Ch 3/2 ] (∆ T N (p) f,h
) is spanned by m p normalized vector v j , 1 ≤ j m p , which satisfy, for x ∈ M and α ∈ N d ,

|∂ α x v j | = Õ(e - d Ag (x,U (p) j ) h ) , ( 23
) |∂ α x d f,h v j | = Õ e - β + j (x) h , |∂ α x d * f,h v j | = Õ e - β - j (x) h (24) with β + j (x) = min U ∈ U (p+1) ∪ U (p) \ U (p) j d Ag (U j , U) + d Ag (U, x) and β - j (x) = min U ∈ U (p-1) ∪ U (p) \ U (p) j d Ag (U j , U) + d Ag (U, x) .
The eigenvalues of ∆

T N (p) f,h lying in [0, h 3/2 ] are O(e -C h
) . When the metric g is Euclidean in some adapted Morse coordinates for f in B(U

(p) j , 2η), with f (y) = f (U j ) + 1 2 d j=1 λ j y 2 j , then the form v j satisfies v j = |λ 1 . . . λ d | 1/4 (πh) -d/4 e - d j=1 |λ j |y 2 j 2h dy 1 ∧ . . . ∧ dy p + O e -Cη h in C ∞ (B(U (p) j , η)) .
In the general case of a Riemannian metric, there exists, for η small enough, some adapted Morse coordinates for f in B(U

(p) j , 2η), with f (y) = f (U j ) + 1 2 d j=1 λ j y 2
j , and such that the form v j satisfies

e d j=1 |λ j |y 2 j 2h v j = ω 0 (x) + O(h 1-d/4 ) in C ∞ (B(U (p) j , η)) , with ω 0 = |λ 1 . . . λ d | 1/4 (πh) d/4 dy 1 ∧ . . . ∧ dy p along C Unst ∩ B(U (p) j , η) ,
and

ω 0 = (-1) p(d-p) |λ 1 . . . λ d | 1/4 (πh) d/4 ⋆ (dy p+1 ∧ . . . ∧ dy d ) along C St ∩ B(U (p) j , η) .
We shall need an integration by part formula adapted from Lemma 4.3.3 in [START_REF] Helffer | Quantitative analysis of metastability in reversible diffusion processes via a Witten complex approach: the case with boundary[END_REF] and Lemma 4.3 in [START_REF] Peutrec | Small eigenvalues of the Neumann realization of the semiclassical Witten Laplacian[END_REF].

Lemma 3.10. Let Ω be a regular domain of f b a with boundary of three disjoint pieces ∂Ω = {f = a} ⊔ {f = b} ⊔ Γ . Consider the self-adjoint realization ∆ T N D f,h of ∆ f,h given by the form

D(ω, ω ′ ) = d f,h ω , d f,h ω ′ + d * f,h ω , d * f,h ω ′
with the form domain

ΛW 1,2 T N D = ω ∈ ΛW 1,2 (Ω) ; tω f =a = 0 , nω f =b = 0 , ω Γ = 0 ,
and the operator domain

D(∆ T N D f,h ) =    ω ∈ ΛW 2,2 (Ω) ; tω f =a = 0 , td * f,h ω f =a = 0 , nω f =b = 0 , nd f,h ω f =b = 0 , ω Γ = 0    .
Let ϕ be any Lipschitz function. Then for all ω ∈ ΛW 1,2

T N D we have the integration by part formula

Re D(ω, e 2ϕ h ω) = h 2 de ϕ h ω 2 + h 2 d * e ϕ h ω 2 + (|∇f | 2 -|∇ϕ| 2 + hL ∇f + hL * ∇f )e ϕ h ω , e ϕ h ω + h f =b - f =a ω , ω ΛT * σ Ω e 2ϕ(σ) h ∂f ∂n (σ) dσ , ( 25
)
where ∂f ∂n is the exterior normal derivative. Moreover when

ω ∈ D(∆ T N D f,h ) then D(ω, e 2ϕ h ω) = Re e 2ϕ h ∆ T N D
f,h ω, ω . Proof of Theorem 3.9: First of all, applying the integration by part ( 25) with ϕ = 0 and the local harmonic approximation around critical points like in [CFKS], one obtains that the number of eigenvalues in [0, Ch 3/2 ] is m p , with no other eigenvalues in (Ch 3/2 , h/C], when C is chosen large enough. The boundary term in ( 25) is non negative because ∂f ∂n is non positive (resp. non negative) on {f = a} (resp. {f = b}). The IMS localization formula -h 2 ∆ =j χ j (h 2 ∆)χ jh 2 j |∇χ j | with one χ ja (resp. χ j b ) localizing around {f = a} (resp. {f = b}) shows that eigenfunctions associated with the O(h 3/2 ) eigenvalues have asymptotically no mass around {f = a} ∪ {f = b} . Contrary to [START_REF] Helffer | Quantitative analysis of metastability in reversible diffusion processes via a Witten complex approach: the case with boundary[END_REF] and [START_REF] Peutrec | Small eigenvalues of the Neumann realization of the semiclassical Witten Laplacian[END_REF], there are no generalized critical points at the boundary and the assumption that f restricted to the boundary is a Morse function is not necessary We construct now a global quasimode φ h j associated with the critical point U (p) j . Following [HeSj4][HeSj2], consider, for a small constant γ > 0, the domain

Ω j = f b a \ ∪ k =j B(U (p) k , γ) , with ∂Ω j = {f = a} ∪ {f = b} ∪ Γ and Γ = ∪ k =j ∂B(U (p)
k , γ), and take the self-adjoint realization ∆

T N D(p) f,h
of Lemma 3.10 acting on p-forms. It admits a single eigenvalue µ h j which is O(h 3/2 ) (with the rest of the spectrum in [h/C, +∞)) and take φ h j a normalized eigenvector associated with µ h j :

φ h j ΛL 2 = 1 , ∆ T N D(p) f,h φ h j = µ h j φ h j .
Applying Lemma 3.10, in the spirit of [DiSj] pp. 49-55, with

ω = φ h j , ϕ ε (x) = (1 -ε)d Ag x, B(U (p) j , ε) , |∇ϕ ε | ≤ (1 -ε)|∇f | , gives e ϕε h φ h j ΛW 1,2 = O ε ( 1 h ) ,
where the subscript ε recalls that the factor of 1 h depends on the parameter ε > 0 . By elliptic regularity up to the boundary of ∆

T N D(p) f,h , φ h j is C ∞ in Ω j . The differential operator e ϕ h ∆ f,h e -ϕ h equals h 2 (dd * + d * d) + |∇f | 2 -|∇ϕ| 2 + h(L ∇ϕ -L * ∇ϕ + L ∇f + L * ∇f )
where the last part is a first order differential operator. With the boundary conditions, the form u h j = e ϕε h φ h j satisfies the system

       (dd * + d * d)u h j = r j tu h j f =a = 0 td * u h j f =a = ̺ j,a nu h j f =b = 0 ndu h j f =b = ̺ j,b u h j Γ = 0 where r j ΛL 2 , ̺ j, * ΛW 1/2,2 are O( 1 h 3 )
. This provides a O( 1 h 3 ) estimate for u h j ΛW 2,2 and bootstraping gives

∀α ∈ N d , ∀x ∈ Ω j , |∂ α x φ h j (x)| = Õ(e -ϕε(x) h
) .

Since this holds for all ε > 0, the definition of Õ provides the result with ε = 0. It means the following estimate holds, with ϕ(x)

:= ϕ 0 (x) = d Ag (x, U (p) j )
:

∀α ∈ N d , ∀x ∈ Ω j , |∂ α x φ h j (x)| = Õ(e -ϕ(x) h ) . The differential d f,h φ h j solves in Ω j the differential equation ∆ f,h (d f,h φ h j ) = d f,h (∆ f,h φ h j ) = µ h j d f,h φ h j .
The same argument as the one for Lemma 3.4 leads to the fact that d f,h φ h j satisfies the same boundary conditions as φ h j on {f = a} ∪{f = b} . Consider now the domain Ω

′ j = Ω j \∪ U ∈U (p+1) B(U, γ), note V j = U (p+1) ∪ U (p) \ U (p) j
, and work with the associated ∆

T N D(p+1) f,h . The form u ′ h j = χ γ d f,h φ h j , where χ γ ∈ C ∞ (Ω ′ j ) vanishes in ∪ U ∈V j B(U, 2γ) and equals 1 outside ∪ U ∈V j B(U, 3γ), belongs to D(∆ T N D(p+1) f,h
) and solves

∆ T DN (p+1) f,h (u ′ h j ) = µ h j u ′ h j + r ′ j ,
with supp r ′ j ⊂ ∪ U ∈V j B(U, 3γ) and

|∂ α x r ′ j (x)| = Õ(e -min U ∈V j d Ag (U,U (p) j )+cγ h ) (with c > 0) . With our choice of Ω ′ j , ∆ T N D(p+1) f,h
has no eigenvalue in [0, h/C] and the same analysis as above leads to

∀α ∈ N d , ∀x ∈ Ω ′ j , |∂ α x d f,h φ h j (x)| = Õ e -min U ∈V j d Ag (U (p) j ,U )-d Ag (U,x)+cγ h ,
where the previous estimates extend the result to all Ω j . After changing

U (p+1) into U (p-1) , a similar result holds for d * f,h φ h j . A simple computation now gives µ h j = D(φ h j , φ h j ) = Õ(e -2 min U ∈V j d Ag (U,U j )+cγ h ) . Now let us work with ∆ T N (p) f,h on f b a . Consider the cut-off θ j,γ ∈ C ∞ (Ω j ) which vanishes in ∪ U ∈U (p) \ U (p) j B(U, 2γ) and equals 1 in ∪ U ∈U (p) \ U (p) j B(U, 3γ), and set ψ h j = θ j,γ φ h j .
These m p vectors belong to D(∆

T (p) f,h
) and satisfy

∆ T N (p) f,h ψ h j = µ h j ψ h j + r j with µ h j = O(e -C h ) , |r j (x)| = O(e -d Ag (x,U j ) h ) , supp r j ⊂ ∪ U ∈U (p) \ U (p) j B(U, 3γ) , and
ψ h j , ψ h k 1≤j,k≤mp = Id + O(e -C h ) . while ∆ T N (p) f,h
has only m p eigenvalues in [0, Ch 3/2 ]. The Proposition 4.1 in [Hel] implies

ψ h j -1 [0,Ch 3/2 ] (∆ T N (p) f,h )ψ h j = O(e -C h ) ,
and we set

u h j = 1 [0,Ch 3/2 ] (∆ T N (p) f,h
)ψ h j . The min-max principle applied with the ψ h j 's also implies that the eigenvalues of ∆

T N (p) f,h
in [0, Ch 3/2 ] are actually exponentially small. With the integration contour

C h = z ∈ C, |z| = h 3/2 , write u h j -ψ h j = 1 2iπ C h (z -µ h j ) -1 (z -∆ T N (p) f,h
) -1 r j dz .

The resolvent estimates of Proposition 2.2.5 in [START_REF] Helffer | Puits multiples en limite semi-classique II -Interaction moléculaire-Symétries-Perturbations[END_REF] can be carried over to our boundary problem thanks to Lemma 3.10 and elliptic regularity up to the boundary. With the estimates and support condition on r j , they lead to

∀α ∈ N d , ∀x ∈ f b a , |∂ α x ω z (x)| = Õ e -min U ∈U (p) \ { U (p) j } d Ag (U j ,U )+d Ag (U,x)+cγ h = Õ e -d Ag (U j ,x)+cγ h when ω z = [(z -∆ T N (p) f,h
) -1 r j ] and z ∈ C h . With the estimates on ψ j = θ j,h φ h j , this leads to

∀α ∈ N d , ∀x ∈ f b a , |∂ α x u h j | = Õ(e -d Ag (x,U j )+cγ h
) ,

and we take

v h j = u h j -1 u h j = (1 + O(e -C/h ))u h j . The estimates for d f,h v h j (resp. d * f,h v h j ) are obtained after writing the equation for d f,h (u h j -ψ h j ) (resp. d * f,h (u h j -ψ h j )
) and using the resolvent estimates for 34

∆ T N (p+1) f,h (resp. ∆ T N (p-1) f,h
) . Finally, these estimates hold for any γ > 0, the definition of Õ provides the same result with γ = 0. The rest of the proof of Theorem 3.9 is a direct consequence of Theorem 2.5 of [START_REF] Helffer | Puits multiples en limite semi-classique IV -Etude du complexe de Witten[END_REF] and therein related WKB construction.

An important remark

It is clear that the results stated for f b a hold when a = -∞ or b = +∞, that is when one boundary is empty. Another variation on it consists in deforming homotopically {f = a} (resp. {f = b}) while preserving the sign conditions ∂f ∂n < 0 (resp. ∂f ∂n > 0).

4 Barannikov-Morse chain complex and construction of accurate global quasimodes

Properties of quasimodes associated with lower and upper critical points

Consider the operator ∆ T N f,h defined on f b a and set

F (p) = Im 1 [0,h 3/2 ] (∆ T N (p) f,h ) for p ∈ {0, . . . , d}, F = ⊕ p F (p) , β = d f,h F and β * = d * f,h F .
According to Section 3, dim F (p) = m p and the chain complex associated with β

0 / / F (0) . . . F (p-1) β (p-1) / / o o F (p) β (p) / / β (p-1) * o o F (p+1) . . . F (d) / / β (p) * o o 0 o o has the homology group H * (f b a , {f = a}) dual to H * (f b , f a ) = H * (f b a , {f = a})
. Remember also that F (p) admits the orthogonal decompositions

F (p) = ker ∆ T N (p) f,h ⊥ ⊕ Im β (p-1) ⊥ ⊕ Im β (p) * = ker β (p) ⊥ ⊕ Im β (p) * = Im β (p-1) ⊥ ⊕ ker β (p-1) *
and that F (p) admits an almost orthonormal basis v U , U ∈ U (p) fulfilling the properties of Theorem 3.9. The orthogonal projection on any subspace G of the above orthogonal decomposition will be denoted by Π G .

Proposition 4.1. Assume that U ∈ U (p) not an upper critical points in f b a , then v U is almost orthogonal to Im β:

Π Im β (p-1) v U = O(e -Cη h ) ,
where the constant C η depends on the small radius η fixed by the geometry in Theorem 3.9.

Proof.

When U ∈ U (p) with f (U) = c U is not an upper critical point, it means that the mapping H p (f c U +ε ) → H p (f c U +ε , f c U -ε ) does not vanish. Since H p (f c U +ε , f c U -ε
) is one dimensional and is generated by e p U the unstable manifold for -∇f leaving U and restricted to

f c U -ε , there exists a cycle C p U in f c U +ε , or a cycle in M supported in f c U +ε , of which the restriction to f c U +ε c U -ε is e p U .
We choose ε = ε η > 0 so that e p U is contained in the ball B(U, η) of Theorem 3.9. In adapted Morse coordinates, e p U equals up to the orientation

y p+1 = . . . = y d = 0, 1 2 p j=1 |λ j |y 2 j < ε η , while e f -c U h v U equals |λ 1 . . . λ d | 1/4 (πh) -d/4 e - p j=1 |λ j |y 2 j h (ω 0 (y) + hω ′ (y, h)) + O e -Cη h with ω ′ bounded in C ∞ (B(U, η)) and ω 0 = dy 1 ∧ . . . ∧ dy p along {y p+1 = • • • = y d = 0} ∩ B(U, η) . Decompose v U according to v U = v ′ U + v ′′ U = Π Im β (p-1) v U + Π ker β (p-1) * v U , v ′ U = U ′ ∈U (p) t U ′ v U ′ , v ′′ U = U ′ ∈U (p) s U ′ v U ′ . The decomposition v U = v ′ U + v ′′ U is orthogonal v ′ U 2 + v ′′ U 2 = 1 .
Meanwhile, the exponential decay estimates of (v U ′ ) U ′ ∈U (p) stated in Theorem 3.9 provide the almost orthogonality

U ′ |t U ′ | 2 ≤ 1 + O(e -C/h ) , U ′ |s U ′ | 2 ≤ 1 + O(e -C/h ) , t U + s U = 1 + O(e -C/h ) and t U ′ + s U ′ = O(e -C/h ) for U ′ = U . All the v U ′ have Õ(1) estimates in C ∞ (f b a p T * M
) and the support conditions on e p U and C p U give

e p U e f -c U h v U = C p U e f -c U h v U + O(e -Cη h ) . By using v ′ U = d f,h ω we get e p U e f -c U h v U = h C p U d e f -c U h ω + C p U e f -c U h v ′′ U ,
and finally with ∂C p U = 0,

e p U e f -c U h v U = U ′ ∈U (p) s U ′ C p U e f -c U h v U ′ = U ′ ∈U (p) s U ′ e p U + C p U \e p U e f -c U h v U ′ . With s U ′ = O(1), e f -c U h = O(e -Cη h ) on C p U -e p U and |v U ′ | = Õ(1)
the second integral gives an exponentially small term. When U ′ = U the exponential decay estimate of Theorem 3.9 imply that s

U ′ e p U e f -c U h v U ′ is O(e -C/h
) . We have proved

e p U e f -c U h v U = s U e p U e f -c U h v U + O(e -Cη h ) .
But a direct calculation in the Morse coordinates gives

e p U e f -c U h v U = (1 + O(h)) p j=1 |λ j |y 2 j <2εη |λ 1 . . . λ d | 1/4 (πh) d/4 e - p j=1 |λ j |y 2 j h dy 1 . . . dy p = |λ p+1 . . . λ d | 1/4 |λ 1 . . . λ p | 1/4 (πh) (2p-d)/4 (1 + O(h)) .
This proves s U = 1+O(e -Cη h ) while all the other coefficients are O(e -Cη h ) .

By duality f → -f , other results can be deduced.

Proposition 4.2. When U ∈ U (p) is not a lower critical point in f b a , v U is almost orthogonal to Im β (p) * : Π Im β (p) * v U = O(e -Cη h ) . When U ∈ U (p) is an homological point in f b a , v U is exponentially close to ker ∆ (p) f,h : v U -Π ker ∆ T N(p) f,h v U = O(e -Cη h ) .
Finally, when U ∈ U (p) is an upper (resp. a lower) critical point, v U is exponentially close to Im β (p-1) (resp. Im β (p) * ):

v U -Π Im β (p-1) v U = O(e -Cη h ) (resp. v U -Π Im β (p) * v U = O(e -Cη h )) .
Proof. The first statement is dual to the one of Proposition 4.1. For the second one it suffices to notice that homological critical points are neither upper nor lower critical points. For the last one it suffices to notice that the number of homological critical points equals the dimension of ker ∆

T N (p) f,h . Hence the set of Π ker ∆ T N(p) f,h v U ′ when U ′ ranges over the homological critical points, is an almost orthonormal basis of ker ∆ T N (p) f,h . If U is an upper critical point, Π ker β (p) v U = v U +O(e -Cη h ) is almost orthogonal to the v U ′ and therefore to ker ∆ T N (p) f,h . We deduce v U = Π Im β (p-1) v U + Π ker ∆ T N(p) f,h v U + O(e -Cη h ) = Π Im β (p-1) v U + O(e -Cη h ) .

Construction of accurate global quasimodes

We now define the global quasimodes for ∆ f,h on M which will be used in our computations.

• When U is an homological critical point, take simply

ω U = Π ker ∆ f,h v U ,
where v U is the form defined in Theorem 3.9 with a = -∞ and b = +∞ .

• When U is an upper critical point, take

ω U = Π Im β v U
where v U is the form defined in Theorem 3.9 with a = -∞ and b = +∞ .

• When U ∈ U (p) is a lower critical point there exist U ∈ U (p+1) with f (U 1 ) = c 1 such that ∂U 1 = U . In f c 1 -ε , U becomes an homological critical point. We take

ω U = 1 [0,h 3/2 ] (∆ f,h )χ ε ṽU
where ṽU is now the form defined in Theorem 3.9 with a = -∞ and b = c 1ε while ∆ f,h is the operator defined on all M . The function χ ε vanishes in f c 1 -3 2 ε and equals 1 in f c 1 -2ε . The value of the parameter ε will be specified further according to η .

Computation of the matrix of d f,h

We work with the basis (ω U ) U ∈U constructed before and we will denote by U H , U L , U U the sets of homological, lower and upper critical points of f , and

U (p) H , U (p) L , U (p)
U their respective intersection with U (p) , the set of critical points of f with index p.

Proposition 4.3. When U 0 belongs to U (p) U ∪ U (p) H , then for any U ′ ∈ U (p+1) , ω U ′ | d f,h ω U 0 = 0 . ( 26
)
When U 0 belongs to U (p)

L , let U 1 denote the upper critical point with index p + 1 s.t. ∂ B (U 1 ) = U 0 . Then there exists a real constant C > 0 and a homological constant κ = κ(U 1 ) = 0 such that for U ′ ∈ U (p+1) :

If U ′ = U 1 , ω U ′ | d f,h ω U 0 = O(e -f (U 1 )-f (U 0 )+C h ) , (27)
If U ′ = U 1 , ω U 1 | d f,h ω U 0 = ±κA(h)e -f (U 1 )-f (U 0 ) h (1 + O(h)) . ( 28
)
Moreover the prefactor A(h) is given by the formula

A(h) = ( h π ) 1 2 |λ 1 1 • • • λ 1 p+1 | 1 4 |λ 1 p+2 • • • λ 1 d | 1 4 |λ 0 p+1 • • • λ 0 d | 1 4 |λ 0 1 • • • λ 0 p | 1 4 , ( 29
)
where

λ ℓ 1 < • • • < λ ℓ p+ℓ < 0 < λ ℓ p+ℓ+1 < • • • < λ ℓ d denote the eigenvalues of Hess f (U ℓ ), for ℓ ∈ {0, 1}.
The rest of this section is devoted to the proof of this proposition. We are going to prove the relations ( 26) and ( 27), then, in order to prove ( 28) and ( 29), we will in a first time work with a metric which is locally Euclidean around the critical points of f before showing that it remains valid for a general Riemannian metric.

Proof of equations ( 26) and (27). When U 0 is an upper critical point or a homological critical point, then the definition of ω U 0 says p) is a lower critical point with critical value c 0 . Let U 1 ∈ U (p+1) be the upper critical point with critical value c 1 such that

d f,h ω U 0 = βω U 0 = 0 , which yields equation (26). Let us now compute ω U ′ , d f,h ω U 0 for U ′ ∈ U (p+1) when U 0 ∈ U (
∂ B U 1 = U 0 . The commutation d f,h 1 [0,h 2 ] (∆ f,h ) = 1 [0,h 2 ] (∆ f,h )d f,h gives ω U ′ , d f,h ω U 0 = ω U ′ , d f,h 1 [0,h 2 ] (∆ (p) f,h )χ ε ṽU 0 = ω U ′ , d f,h χ ε ṽU 0 = v U ′ , d f,h χ ε ṽU 0 + ω U ′ -v U ′ , d f,h χ ε ṽU 0 Since d f,h ṽU 0 = 0 in supp ∇χ ε , we have d f,h χ ε ṽU 0 = hdχ ε ∧ ṽU 0 . Since dχ ε is supported in f c 1 -3 2 ε c 1 -2ε and |ṽ U 0 (x)| = Õ(e -d Ag (x,U 0 ) h ) = O(e -c 1 -c 0 -Cε h ) for x ∈ supp ∇χ ε , the remainder term ω U ′ -v U ′ , d f,h χ ε ṽU 0 is bounded by ω U ′ -v U ′ O(e -c 1 -c 0 -Cε h ) .
When U ′ is not a lower critical point, the relation

ω U ′ -v U ′ = O(e -C h
) comes from Proposition 4.2. When U ′ is a lower critical point, simply note that both terms of the r.h.s. in

ω U ′ -v U ′ ≤ ω U ′ -χ ε ṽU ′ + χ ε ṽU ′ -v U ′ (30)
are O(e -C h ) . Actually, the estimate for the second term is obtained after comparing v U ′ and ṽU ′ with the single eigenmode of a Dirichlet realization of ∆ (p+1) f,h in B(U ′ , η 0 ), again by following [Hel]. Hence have proved

ω U ′ , d f,h ω U 0 = v U ′ , hdχ ε ∧ ṽU 0 + O(e -c 1 -c 0 +C h ) when ε > 0 is chosen small enough. If U ′ = U 1 the exponential decay of v U ′ , |v U ′ (x)| = Õ(e -d Ag (x,U ′ ) h ) = O(e -|c 1 -f (U ′ )|-Cε h ) for x ∈ supp ∇χ ε , leads to ω U ′ , d f,h ω U 0 = O(e -c 1 -c 0 +C h ) ,
and equation ( 27) is proved.

Proof of Proposition 4.3 when the metric is Euclidean in some adapted Morse coordinates

Let us check equations ( 28)-( 29), when the metric is Euclidean in some local adapted Morse coordinates for f (around each critical point).

Note first that we have already proved, for a general metric, the following result,

∀U ′ ∈ U (p+1) , ω U ′ , d f,h ω U 0 = v U ′ , hdχ ε ∧ ṽU 0 + O(e -c 1 -c 0 +C h ) ,
where C is a positive constant. According to the choice of χ ε , the first term of the right-hand side vanishes when ∂ B U 1 = U 0 . Thus, we can focus on the term

v U 1 , hdχ ε ∧ ṽU 0 , when U 1 ∈ U (p+1) satisfies ∂ B (U 1 ) = U 0 .
In the ball B(U 1 , η), we use the above adapted Morse coordinates (y ′ , y ′′ ) with y ′ = (y 1 , . . . , y p+1 ), y ′′ = (y p+2 , . . . , y d ), and f (y

) -c 1 = 1 2 d j=1 λ 1 j y 2 j . The parameter ε > 0 is chosen according to η > 0 so that f c 1 -3 2 ε c 1 -2ε ∩ B(U 1 , η) = ∅ . More precisely one takes C 1 , C 2 > 1 and ε = ε η such that f c 1 -3 2 ε c 1 -2ε ∩ |y ′′ | < η C 1 ⊂ η C 2 < |y ′ | < 2η C 2 , |y ′′ | < η C 1 ⊂ B(U 1 , η) .
Lemma A.2.2 of [START_REF] Helffer | Puits multiples en limite semi-classique IV -Etude du complexe de Witten[END_REF] says that d Ag (x, y) = |f (x)f (y)| if and only if there is generalized integral curve of ∇f going from x to y . Hence in f c 1 -3 2 ε c 1 -2ε the only points such that d Ag (U 1 , y) = c 1f (y) are the points lying on the unstable manifold for -∇f . Hence there exists a constant C η > 0 such that

∀y ∈ f c 1 -3 2 ε c 1 -2ε \ |y ′′ | < η C 1 , d Ag (U 1 , y) ≥ c 1 -f (y) + C η .
By combining this with the exponential decay estimates for ṽU 0 we deduce

ω U 1 , hdχ ε ∧ ṽU 0 = |y ′′ |≤ η C 1 v U 1 , hdχ ε ∧ ṽU 0 ΛT * y M + O(e -c 1 -c 0 +Cη h ) .
With the above inclusion and the approximation of v U 1 in B(U 1 , η) stated in Theorem 3.9 we get

ω U 1 , hdχ ε ∧ ṽU 0 = K h U 1 |y ′′ |≤ η C 1 e - d j=1 |λ 1 j |y 2 j 2h dy 1 ∧ . . . ∧ dy p+1 , hdχ ε ∧ ṽU 0 ΛT * y M + O(e -c 1 -c 0 +Cη h ) , with K h U 1 = |λ 1 1 ...λ 1 d | 1/4 (πh) d/4 . With an Euclidean metric, inserting e -f -c 1 h × e f -c 0 h in the bracket implies that e c 1 -c 0 h K h U 1 ω U 1 , hdχ ε ∧ ṽU 0 equals |y ′′ |≤ η C 1 e - d j=p+2 |λ 1 j |y 2 j h dy 1 ∧ . . . ∧ dy p+1 , hdχ ε ∧ e f -c 0 h ṽU 0 ΛT * y M + O(e -Cη h ) = ± |y ′′ |≤ η C 1 e - d j=p+2 |λ 1 j |y 2 j h dy p+2 ∧ . . . ∧ dy d ∧ (hdχ ε ) ∧ e f -c 0 h ṽU 0 + O(e -Cη h ) .
But our assumption says that d e

h K h U 1 ω 1 , hdχ ε ∧ ṽU 0 equals ±h |y ′′ |≤ η C 1 |y ′ |= 2η C 2 e - d j=p+2 |λ 1 j |y 2 j h dy p+2 ∧ . . . ∧ dy d ∧ e f -c 0 h ṽU 0 + O(e -Cη h ) ,
and by introducing for every fixed y ′′ such that

|y ′′ | ≤ η C 1 the cycle C y ′′ supported by (y ′ , y ′′ ), |y ′ | = 2η C 2 and homotopic to ∂e p+1 U 1 we get e c 1 -c 0 h K h U 1 ω U 1 , hdχ ε ∧ ṽU 0 = ±h |y ′′ |≤ η C 1 e - d j=p+2 |λ 1 j |y 2 j h C y ′′ e f -c 0 h ṽU 0 +O(e -Cη h ) .
For any y ′′ , the cycle C y ′′ is homologous to ∂e p+1 U 1 and according to Proposition 2.12 to κ[e p U 0 ] in f c 1 -ε relatively to f c 0 -γη , with γ η > 0 small enough. Owing to d e f h ṽU 0 = 0 in f c 1 -ε and the exponential decay estimate of ṽU 0 stated in Theorem 3.9 for ṽU 0 , we obtain

e c 1 -c 0 h K h U 1 ω U 1 , hdχ ε ∧ṽ U 0 = ±κh |y ′′ |≤ η C 1 e - d j=p+2 |λ 1 j |y 2 j h e p U 0 e f -c 0 h ṽU 0 +O(e -Cη h ) .
Using again Theorem 3.9 with ṽU 0 , f (U 0 ) = c 0 , and decomposition of f around U 0 , f (z)c 0 = 1 2 d j=1 λ 0 j z 2 j in some (adapted) Morse coordinates (z ′ , z ′′ ) = (z 1 , . . . , z p , z p+1 , . . . , z d ), we get,

e c 1 -c 0 h K h U 1 K h U 0 ω U 1 , hdχ ε ∧ ṽU 0 = ±κh |y ′′ |≤ η C 1 e - d j=p+2 |λ 1 j |y 2 j h e p U 0 e d j=1 λ 0 j z 2 j 2h e - d j=1 |λ 0 j |z 2 j 2h + O(e -Cη h ) = ±κh |y ′′ |≤ η C 1 e - d j=p+2 |λ 1 j |y 2 j h e p U 0 e - p j=1 |λ 0 j |z 2 j h + O(e -Cη h ) , with K h U 0 = |λ 0 1 ...λ 0 d | 1/4
(πh) d/4 . Now, writing successively two Laplace methods, we obtain

e c 1 -c 0 h K h U 1 K h U 0 ω U 1 , hdχ ε ∧ ṽU 0 = ±κh (πh) d-1 2 |λ 1 p+2 • • • λ 1 d | 1 2 |λ 0 1 • • • λ 0 p | 1 2 (1 + O(h)) ,
which leads, finally, to the following formula:

ω U 1 , hdχ ε ∧ ṽU 0 = ±κ( h π ) 1 |λ 1 1 • • • λ 1 p+1 | 1 4 |λ 1 p+2 • • • λ 1 d | 1 4 |λ 0 p+1 • • • λ 0 d | 1 4 |λ 0 1 • • • λ 0 p | 1 4 e -c 1 -c 0 h (1 + O(h)) .
The picture below summarizes the scheme of the calculation and use of Stokes' formula, for d = 3 and p = 1.

∂e p+1 U 1 C y ′′ κ× f = f (U 0 ) f = f (U 1 ) [e p U 0 ]

Proof of Proposition 4.3 for a general Riemannian metric

As in the previous subsection, we look at the term v U 1 , hdχ ε ∧ ṽU 0 , where U 1 ∈ U (p+1) satisfies ∂ B (U 1 ) = U 0 , and we use some adapted Morse coordinates (y ′ , y ′′ ) = (y 1 , . . . , y p+1 , y p+2 , . . . , y d ) in the ball B(U 1 , η). Let us recall that the function f has the following decomposition in these coordinates:

f (y) -c 1 = 1 2 d j=1 λ 1 j y 2 j . Again, one takes C 1 , C 2 > 1 and ε = ε η such that f c 1 -3 2 ε c 1 -2ε ∩ |y ′′ | < η C 1 ⊂ η C 2 < |y ′ | < 2η C 2 , |y ′′ | < η C 1 ⊂ B(U 1 , η) ,
and we have the existence of C η > 0 s.t.

v U 1 , hdχ ε ∧ ṽU 0 = |y ′′ |≤ η C 1 v U 1 , hdχ ε ∧ ṽU 0 ΛT * y M + O(e -c 1 -c 0 +Cη h ) . Choose η > 0 small enough such that U 1 is the only critical point of f in f -1 ([c 1 -2η, c 1 + 2η]).
Now, let us introduce the metric g 1 ,

g 1 (y) = χ(y)g (y) + (1 -χ(y))g(y) , where 0 ≤ χ ≤ 1 is a smooth cut-off function such that χ = 1 in B(U 1 , η), χ = 0 outside B(U 1 , 3 2
η), g is the usual metric on M, and g e is the Euclidean metric g e = d i=1 (dy i ) 2 .

For g and g 1 , let ṽU 1 and ṽ1 U 1 denote respectively the forms defined in Theorem 3.9 with a = c 1 -2η and b

= c 1 + 2η. Since U 1 is the only critical point of f in f -1 ([c 1 -2η, c 1 + 2η]), this means that ṽi
U 1 is a normalized form in the one dimensional kernel of ∆ T N,(p+1) g,f,h (resp. ∆ T N,(p+1) g 1 ,f,h

), the Witten Laplacian corresponding to the metric g (resp. g 1 ). Note also that the boundary conditions are strictly the same for both ṽU 1 and ṽ1 U 1 , since the metrics g and g 1 coincide near the boundary. In particular, ṽU 1 and ṽ1 The following lemma gives some useful relations between ṽU 1 and ṽ1 U 1 , especially the second one which will be crucial in the sequel.

f -c 1 h ṽU 1 -α 1 e f -c 1 h ṽ1 U 1 = h d e f -c 1 h ω . (33)
Moreover, the form ⋆ṽ U 1 (resp. ⋆ 1 ṽ1 U 1 ) belongs to ker(∆ N T,(d-p-1) g,-f,h

) (resp.

ker(∆ N T,(d-p-1) g 1 ,-f,h

)), and there exists another constant α ′ 1 = 0 s.t.

⋆ṽ U 1 -α ′ 1 ⋆ 1 ṽ1 U 1 ∈ Ran d N T -f,h .
According to the definition of d -f,h , it means that there exists ω ′ in D(∆

N T,(d-p-2) -f,h ) s.t. ⋆ e -f -c 1 h ṽU 1 -α ′ 1 ⋆ 1 e -f -c 1 h ṽ1 U 1 = h d e -f -c 1 h ω ′ . (34)
In order to show that α 1 = 1 + O(h), let us integrate equation (33) along the unstable manifold C Unst , which is a (p + 1)-cycle in f c 1 +2η c 1 -2η relatively to {f = c 1 -2η}. Using Stokes' formula, we obtain

C Unst e f -c 1 h ṽU 1 -α 1 ṽ1 U 1 = h C Unst d e f -c 1 h ω = 0 .
Consequently, the constant α 1 is given by

α 1 = C Unst e f -c 1 h ṽU 1 C Unst e f -c 1 h ṽ1 U 1 = 1 + O(h) ,
where the last equality comes from a Laplace method applied to each integral, after introducing the first order WKB approximations of ṽU 1 and ṽ1 U 1 recalled from [START_REF] Helffer | Puits multiples en limite semi-classique IV -Etude du complexe de Witten[END_REF] in the last statements of Theorem 3.9. In order to obtain the same estimate for α ′ 1 , we make the same computation with equation 34 along the stable manifold C St , which is a (dp -1)-cycle in f c 1 +2η c 1 -2η relatively to {f = c 1 + 2η}. This gives, according again to the last statements of Theorem 3.9,

α ′ 1 = C St e -f -c 1 h ⋆ ṽU 1 C St e -f -c 1 h ⋆ 1 ṽ1 U 1 = 1 + O(h) .
But, looking at the second equation of Lemma 4.4 and using our exponential decay estimates, the second term of the r.h.s. is also (up to an exponentially error term),

h 2 ∂({|y ′′ |≤ η C 1 }) d e -f -c 1 h ω ′ ∧ (e f -c 0 h ṽU 0 ) + O(e -C h ) ,
where the integral term is 0 owing to Stokes' formula.

Equation 35 can then be rewritten Since the metric g 1 is Euclidean in the ball B(U 1 , η), the Morse decomposition of f combined with Theorem 3.9 gives (πh) d/4 . We can then follow the same proof as the one used in the locally Euclidean case in order to obtain the wanted result. The only difference arises in the fact that the WKB expansion of ṽU 0 , given in the last statement of Theorem 3.9, contains higher order correcting terms, because the full WKB expansion of ṽU 0 depends on the metric, and this produces a relative O(h) error term.

B = K h U 1 |y ′′ |≤ η C 1 |y ′ |= 2η C 2 e -

A relative version of Theorem 1.1

The analysis for Theorem 1.1 is done on M = f +∞ -∞ . All the constructions and the good restriction properties of Morse-Barannikov chain complex, when considering H * (f b , f a ), -∞ ≤ a < b ≤ +∞, have their counterpart with the Witten Laplacians ∆ T N f,h defined on f b a in Section 3. Hence all the proof of Theorem 1.1 is still valid for ∆ T N f,h on f b a except that some end points of the relation ∂ B U (p+1) = U (p) disappear when they lie in f a or f b . We state without more detail the spectral result for ∆ T N f,h on f b a . Theorem 4.5. With the same assumptions as in Theorem 1.1 and when a, b are not critical values of f , -∞ ≤ a < b ≤ +∞, the exponentially small eigenvalues of ∆ T N f,h , defined on f b a according to Proposition 3.1, are given by a mapping j b a : U ∩ f b a → σ(∆ T N f,h ) derived from the mapping j of Theorem 1.1 by:

• j b a (U) = j(U)(1+O(h)) = 0 if U ∈ U L ∩f b a and U = ∂ B U ′ with U ′ ∈ f b a ; • j b a (U) = j(U)(1+O(h)) = 0 if U ∈ U U ∩f b a and ∂ B U = U ′ with U ′ ∈ f b a ; • j b a (U) = 0 else.

  are actually exponentially small, λ (p) k = O(e -C (p) k h ) without specifying the C (p) k 's. The values of the activation energies C (0) k

  and via the deformed codifferential d * f,h with the eigenmodes of ∆ (p-1) f,h

  Proposition 2.14. The relative homology group H * (f b , f a ) has a basis made of critical values (resp. points) c ∈ (a, b) satisfying one of the following conditions 1. c is an homological critical value (resp. point) in M ; 2. or c is an upper critical value (resp. point) such that ∂c = c ′ is below a . 3. or c is a lower critical value (resp. point) in M, that is ∂c ′ = c in C(f ), but c ′ is above b ; Remark 2.15. What the theorem says is that the homological critical points for C(f, b, a), that should be denoted C H (f, b, a) are not the points of C H (f ) with critcal value in [a, b], but the union of those, together with the upper critical values in [a, b] such that ∂c is below a, and the lower critical values c such that ∂c ′ = c with c ′ above b.

  The dual condition along {f = b}, nω {f =b} = 0, simply means tω {f =b} = ω and ensures that such form are determined by integration along chains lying in {f ≤ b} . The class F = C ∞ T N , C ∞ T and C ∞ N for C ∞ forms fulfilling respectively both conditions, the first one or the second one, among tω {f =a} = 0 , nω {f =b} = 0 . (19)

  When ω belongs to D(∆ T N f,h ), d f,h ω and d * f,h ω belong to ΛW 1,2 T N . Proof. The differential operators d f,h and d * f,h are continuous from D(∆ T N f,h ) ⊂ ΛW 2,2 into ΛW1,2 . By the elliptic regularity up to the boundary of ∆

  operators d f,h restricted to C ∞ T N (f a b , T * M) . Since ω → e -fh α is an isomorphism between the two spaces C ∞ T N (f a b , T * M) respectively defined for ∆ f,h or d f,h and ∆ 0,h = h 2 ∆ Hodge or d, we obtain

  p+2 ∧ . . . ∧ dy d ) = 0 . Hence the integrand is nothing but hd χ ε e - d j=p+2 |λ 1 j |y 2 j h dy p+2 ∧ . . . ∧ dy d ∧ e f -c 0 h ṽU 0 . By Stokes' formula the quantity e c 1 -c 0

Figure 3

 3 Figure 3 The arrows show the use of Skokes' formula. The dotted part of [e p U 0 ] shows the part of [e p U 0 ] lying below f (U 0 )γ η .

  , ⋆ṽ U 1 and ⋆ 1 ṽ1 U 1 belong to the same domainD(∆ N T,(d-p-1) -f,h ) := D(∆ N T,(d-p-1) g,-f,h ) = D(∆ N T,(d-p-1) g 1 ,-f,h ) (we refer to Remark 3.2 for the meaning of ∆ N T -f,h )). Since v U 1 = ṽU 1 + O(e -Cη h ) holds in f -1 ([c 1 -2η, c 1 + 2η]) ∩ supp dχ ε , it suffices to estimate v U 1 , hdχ ε ∧ ṽU 0 = |y ′′ |≤ η C 1 ṽU 1 , hdχ ε ∧ ṽU 0 ΛT * y M + O(e -c 1 -c 0 +Cη h ) .

∼

  f -c 1 h ṽU 1 = h d e -f -c 1 h ω ′ + (1 + O(h)) ⋆ 1 e -f -The ṽU 1 (resp. ṽ1 U 1 ) is in the one dimensional kernel of ∆ Ker d T N f,h / Ran d T N f,h ∼ Ker ∆ T N,(p+1) g 1 ,f,h, with the middle set independent of the metrics, implies the existence of a constantα 1 = 0 s.t. ṽU 1α 1 ṽ1 U 1 ∈ Ran d T N f,h. This means the existence of ω in D(∆

  e

h

  -f -c 1 h ṽU 1 ) ∧ (hdχ ε ) ∧ (e f -c 0 h ṽU 0 ) .By our assumption, ṽU 1 is in f -1 ([c 1 -2η, c 1 + 2η]) solution of ∆ we have on this domain the following equality:d ⋆(e -f -c 1 h ṽU 1 ) = (-1) d-p e -f -c 1 h d * g,f,h ṽU 1 = 0 .Keeping in mind the relation d(ef -c 0 h ṽU 0 ) = 0 in supp dχ ε , this implies ⋆(e -f -c 1 h ṽU 1 ) ∧ (hdχ) ∧ (e f -c 0 h ṽU 0 ) = hd χ ε ⋆(e -f -c 1 h ṽU 1 ) ṽU 0 ) + O(e -C h ) .

h

  ṽU 0 ) + O(e -C h ) ,and we can focus on the integral part of the r.h.s.,

  p+2 ∧ . . . ∧ dy d ∧ (e f -c 0 h ṽU 0 ) + O(e -Cη h ) ,

Acknowledgements : The authors wish to thank François Laudenbach from many insightful discussions concerning Morse theory. These played an essential role in the elaboration of this work.

End of the proof of Theorem 1.1

We recall firstly some notations of Section 4.1. The operator ∆ f,h is defined on M (i.e. on f +∞ -∞ ) and

According to Section 3, F (p) admits an almost orthonormal basis, v U , U ∈ U (p) , fulfilling the properties of Theorem 3.9. Consider now the family of quasimodes, ω U , U ∈ d p=0 U (p) , constructed in Section 4.2. For any p in {0, . . . , d} and U ∈ U (p) , the p-form ω U belongs to F (p) and, as already mentioned, ω U satisfies the relation ω U -v U = O(e -Cη h ) (see Proposition 4.2 for upper or homological critical points and (30) for lower critical points). The family ω U , U ∈ U (p) is then an almost orthonormal basis of F (p) and, thanks to Proposition 4.3, Theorem 2.3 of [START_REF] Peutrec | Small singular values of an extracted matrix of a Witten complex[END_REF] applies. For h 0 small enough and h ∈ (0, h 0 ], we obtain an accurate writing of the non zero eigenvalues of

More precisely, when restricted to F (p) , the non zero eigenvalues of d * f,h d f,h are the quantities

, and

In particular, d * f,h d f,h : F → F has exactly card U U = card U L non zero eigenvalues and these eigenvalues are distinct. This provides all the exponentially small non zero eigenvalues of ∆ f,h according to the last statement of Theorem 3.3. This ends the proof of Theorem 1.1.