CUBO A Mathematical Journal Vol.11, No¯ 04, (49–57). September 2009 Small singular values of an extracted matrix of a Witten complex. D. Le Peutrec IRMAR, UMR-CNRS 6625, Université de Rennes 1, campus de Beaulieu, 35042 Rennes Cedex, France. email: dorian.lepeutrec@univ-rennes1.fr ABSTRACT It is shown how rather tricky induction processes, used for the accurate computation of exponentially small eigenvalues of Witten Laplacians, essentially amount to some Gaussian elimination after the proper rewriting. RESUMEN Se muestra cómo un proceso de inducción bastante truculento se usa para el cálculo preciso de los valores propios pequeños de los laplacianos de Witten utilizando esen- cialmente cantidades de eliminaciones gausianas después de una reescritura correcta. Key words and phrases: Induction process, Witten Laplacian, exponentially small eigenvalues, Gaussian elimination. Math. Subj. Class.: 81Q10, 15A18, 34L15, 58J10, 58J37. 1 Introduction and motivations The accurate computation of exponentially small eigenvalues of Witten Laplacians on 0-forms, or generators associated with reversible diffusion processes, relies on some rather tricky induction 50 D. Le Peutrec CUBO 11, 4 (2009) process. In [2] [3], the induction scheme is modelled after the probabilistic picture of exit times. After [7] [8] [16] it appeared that this induction scheme could be extracted from its spectral analysis or probabilistic framework as a pure problem of finite dimensional linear algebra. The aim of this short text is to show that all these previous and rather involved inductions essentially amount, after a proper rewriting, to some Gaussian elimination. We first recall that the Witten Laplacian writes ∆ (0) f,h = d (0),∗ f,h d (0) f,h = −h 2 ∆ + |∇f(x)|2 − h∆f(x) (1.1) on functions and more generally on differential forms with arbitrary degree, ∆f,h = ( d ∗ f,h + df,h )2 with df,h = e − f h (hd)e f h . (1.2) Since it has a square structure, the eigenvalues of ∆ (0) f,h (resp. ∆f,h) are the squares of the singular values of d (0) f,h (resp. (d ∗ f,h +df,h)). Remember that in the study of Witten Laplacians d denotes the exterior differential on a Riemannian manifold, d∗ the codifferential, h > 0 is a small parameter considered in the limit h → 0 and f is a Morse function. In the case when the manifold is Rn with the Euclidean metric, recall that the Witten Laplacian on functions (1.1) in L2(Rn,dx) is unitary equivalent to the following operator −h(−2∇f(x).∇ + h∆) in L2(Rn,e−2f/hdx). This last operator fits better with the probabilistic presentation ( [4] [17] [19]) and the simulated annealing framework ( [14]). The main purpose is the accurate computation of the smallest non zero eigenvalue of these operators among a finite collection of exponentially small eigenvalues, i.e. of order e− Ck h as h → 0. The inverse of this eigenvalue can be interpreted as the longest lifetime of metastable states. The issue is the suitable control of errors (which are in absolute values larger than the final result) at every step of the induction process. A usual Gramm-Schmidt type orthonormalisation process as it is used in the semiclassical multiple wells problem ( [6] [9]) does not allow such a control. As this was pointed out in [7] [16], working with singular values rather than with eigenvalues of the square operator allows to use the Fan inequalities ( [5] [18]) in their simplest form. These multiplicative inequalities propagate the control of the small relative errors on the singular values through the induction process. At the moment, this approach has been applied systematically only in the case of Witten Laplacian acting on functions. Some cases with higher order Witten Laplacians can be considered. The only condition is the construction of global quasimodes, which is not completely elucidated for the moment except in the case of 0-forms. Besides the simplification of previous proofs, this text aims at providing an abstract and general result to be referred to in the next future. CUBO 11, 4 (2009) Small singular values of an extracted matrix of a Witten complex. 51 2 Result Let F (0) and F (1) be two complex Hilbert spaces respectively of dimension m0 < +∞ and m1 < +∞ . Let 〈 | 〉 denote the scalar product on F (0) or F (1) (without distinction), and let ‖ψ‖ and ‖A‖ = supψ 6=0 ‖Aψ‖ ‖ψ‖ denote the norms of the vector ψ and of the linear application A associated with this scalar product. Let moreover h0 and ε0 be two positive numbers. Consider a linear application B(h) depending on h ∈ (0, h0]: B(h) : F (0) −→ F (1), and set A0(h) = B ∗ (h)B(h) ≥ 0 . Let A1(h) = B(h)B ∗ (h) ≥ 0 , and note the intertwining relation: B(h)A0(h) = A1(h)B(h) . Definition 1. For a number (resp. a linear operator) g(h), the notation g(h) = Oε(e− α h ) means that, for all ε ∈ (0,ε0], there exists a constant Cε > 0 such that: ∀h ∈ (0, h0] , |g(h)| ≤ Cεe− α h (resp. ‖g(h)‖ ≤ Cεe− α h ) . Assumption 2.1. Assume that there exist two bases (of F (0) and F (1) respectively) depending on (ε,h) ∈ (0,ε0] × (0, h0] and a positive number α independent of (ε,h) ∈ (0,ε0] × (0, h0] such that: ψ (0) k = ψ (0) k (ε,h) (k ∈ {1, . . . ,m0}) , 〈 ψ (0) k | ψ (0) k′ 〉 = δkk′ + Oε(e− α h ) , ψ (1) j = ψ (1) j (ε,h) (j ∈ {1, . . . ,m1}) , 〈 ψ (1) j | ψ (1) j′ 〉 = δjj′ + Oε(e− α h ) . Assumption 2.2. Assume furthermore that there exist an injective map j : {1, . . . ,m0} → {1, . . . ,m1} , a decreasing sequence (αk)k∈{1,...,m0} of real numbers, and a positive number d (in- dependent of (ε,h) ∈ (0,ε0] × (0, h0]) such that: ∀ε ∈ (0,ε0], ∃Cε > 1,∀k ∈ {1, . . . ,m0} , ∀h ∈ (0,h0], C−1ε e− αk+dε h ≤ ∣∣∣ 〈 ψ (1) j(k) | B(h)ψ(0)k 〉∣∣∣ ≤ Cεe− αk−dε h ∀h ∈ (0,h0],∀j′ 6= j(k), ∣∣∣ 〈 ψ (1) j′ | B(h)ψ (0) k 〉∣∣∣ ≤ Cεe− αk+α h . Theorem 2.3. There exist positive numbers h′ 0 ≤ h0 and ε′0 ≤ ε0 such that, under Assump- tions 2.1 and 2.2, the eigenvalues 0 ≤ λ1(h) ≤ · · · ≤ λm0 (h) of A0(h) satisfy: 0 < λ1(h) < · · · < λm0 (h) , ∀k ∈ {1, . . . ,m0} , λk(h) = ∣∣∣ 〈 ψ (1) j(k) | B(h)ψ(0)k 〉∣∣∣ 2 (1 + Oε(e− η h )) , where η > 0 is a real number independent of (ε, h) ∈ (0, ε′ 0 ] × (0, h′ 0 ]. 52 D. Le Peutrec CUBO 11, 4 (2009) Remark 2.4. More generally, vanishing eigenvalues can be included. It suffices to allow the value +∞ for the first values α1 = · · · = αℓ = +∞ and αm0 < · · · < αℓ+1 ∈ R , for some given ℓ ∈ {1, . . . ,m0}. In this last case, the eigenvalues of A0(h) satisfy: λ1 = · · · = λℓ = 0 and 0 < λℓ+1 < · · · < λm0 , while the above estimates hold for the non-zero eigenvalues. This theorem, or a modified form of this theorem according to Remark 2.4, can be applied to simplify the final proof done in [7] for the case of the Witten Laplacian acting on 0-forms on a Riemannian manifold without boundary or the one in [8] for some Dirichlet realization in the case with a boundary. This final part of the analysis in [7] [8] has been reconsidered in [16], without giving all the possible simplifications. The reader can also find in [16] various illustrations in practical cases of this approach . Once the quasimodes satisfying Assumptions 2.1 and 2.2 are constructed, Theorem 2.3 can be applied as soon as we work with a self-adjoint operator with a square structure. The application to Witten Laplacians on 0-forms with alternative boundary conditions is in progress. Some examples of Witten Laplacians acting on p-forms for which quasimodes are constructed can be treated with this result and Theorem 2.3 may be useful for a future generalization. While working with Witten Laplacians on 0-forms, the quasimodes ψ (0) k ’s are constructed globally after truncating e− f h , while the ψ (1) j ’s are introduced locally via a WKB approximation around saddle points of f, U (1) j(k) . Note that the discussion in [7] [16] about sending U (1) j(1) to infinity when λ1 = 0 is replaced by consedering α1 = +∞ (according to Remark 2.4) with an arbitrary additionnal ψ (1) j(1) . The application to some non self-adjoint Fokker-Planck operators with a distorted square strusture (see [1] [8] [11] [15]) seems more delicate (see Remark 3.6). 3 Proof Let us begin by fixing the positive numbers ε′ 0 and η. We first choose ε′ 0 small enough such that: α ′ = α − dε′ 0 > 0 and α′′ = min k>k′ {αk′ − αk − 2dε′0} > 0 . Then, we set: η = min{α, α′ ,α′′} = min{α′ ,α′′} . To prove Theorem 2.3, it will be more convenient to work with matrices. Let us give a definition and an easy application which will be very useful. CUBO 11, 4 (2009) Small singular values of an extracted matrix of a Witten complex. 53 Definition 2. A square matrix V (h) is said quasi-unitary if there exists an unitary matrix U such that: V (h) = U + Oε(e−η/h). Lemma 3.1. The product of quasi-unitary matrices is a quasi-unitary matrix. Furthermore, to prove Theorem 2.3, we need a particular case of Fan inequalities that we recall here (we refer the reader to [18] for a proof). Lemma 3.2. Let B and C be respectively a compact and a bounded linear operator on a Hilbert space H. The inequalities µn(BC) ≤ ‖C‖µn(B) µn(CB) ≤ ‖C‖µn(B) , where µn(B) is the n-th singular value of B, hold for all n ≤ dim H. We apply this lemma with H = H0 ⊥ ⊕ H1, while identifying B : H0 → H1 with J1BΠ0 ∈ L(H), where Π0 is the orthogonal projection H → H0 and J1 the embedding H1 → H. Corollary 3.3. Let H0, H1 be two Hilbert spaces. Let B be a compact linear operator from H0 to H1. Assume that C ∈ L(H1) and D ∈ L(H0) are two invertible operators with: max { ‖C‖ , ∥∥C−1 ∥∥ , ‖D‖ , ∥∥D−1 ∥∥} ≤ 1 + ρ, for some ρ > −1. Then the inequality (1 + ρ) −2 µn(B) ≤ µn(CBD) ≤ (1 + ρ)2µn(B) holds for all n ≤ min( dim H0, dim H1). Remark 3.4. We will apply this corollary in the particular case when C and D depend on h ∈ (0, h ′ 0 ] and are quasi-unitary: C(h) = U + Oε(e− η h ) and D(h) = V + Oε(e− η h ) , where U and V are unitary matrices and ρ = Oε(e− η h ). We obtain the equivalent relations: µn(CBD) = µn(B)(1 + Oε(e− η h )) , µn(B) = µn(CBD)(1 + Oε(e− η h )) . (3.1) From A0(h) = B ∗ (h)B(h), we deduce that the eigenvalues of A0(h) are the squares of the singular values of B(h): ∀k ∈ {1, . . . ,m0} , λk(h) = µ2m0+1−k(B(h)) ( µ1(B(h)) = ‖B(h)‖) . 54 D. Le Peutrec CUBO 11, 4 (2009) In order to apply Corollary 3.3, it will be easier to work with the singular values of B(h) than with the eigenvalues of A0(h). Choose now two arbitrary orthonormal bases B(0) and B(1) (of F (0) and F (1) respectively). We make the identifications: B(h) = Mat B(0), B(1) (B(h)) , B ∗ (h) = ( Mat B(0), B(1) (B(h))) ∗ . Let be B′(h) = (〈 ψ (1) j | B(h)ψ (0) k 〉) j,k = ( b ′ j k ) j,k . For i ∈ {1, . . . , ml} and l ∈ {0, 1}, we set Cl = MatB(l) ( ψ (l) 1 . . . ψ (l) ml ) , where ψ (l) i is written as a column vector in B(l). These change-of-coordinates matrices give B′(h) = C ∗ 1 B(h)C0 . Remark 3.5. By Assumption 2.1, the matrices C0 and C∗1 are quasi-unitary and Assumption 2.2 implies, for h′ 0 small enough: ∀ 1 ≤ k′ < k ≤ m0 , b′j(k′) k′ = b′j(k) k.Oε(e− η h ) , (3.2) ∀ 1 ≤ k ≤ m0 , ∀j 6= j(k) , b′j k = b′j(k) k.Oε(e− η h ) . (3.3) We now simplify B′(h) by Gaussian elimination in the following order: Step 0: By permuting the rows, that is by left-multiplying with permutation matrices which are unitary, put the coefficients b′ j(k) k (for k in {1, . . . ,m0}) on the k-th row and k-th column. The new matrix has the form: B ′′ (h) =    b ′′ 1 1 = b ′ j(1) 1 b ′ j(2) 2 .Oε(e− η h ) . . . b ′ j(m0) m0 .Oε(e− η h ) ... b′′ 2 2 = b ′ j(2) 2 ... b ′ j(1) 1 .Oε(e− η h ) ... . . . ... ... b′ j(2) 2 .Oε(e− η h ) b ′′ m0 m0 = b ′ j(m0) m0 ... ... ...    . Furthermore, the matrix B′′(h) satisfies the structure equations (3.2) and (3.3) with the injec- tive map j : {1, . . . ,m0} → {1, . . . ,m1} replaced by the canonical injection i : {1, . . . ,m0} → {1, . . . ,m1}, i(k) = k. Step 1: For j ∈ {1, . . . , m1} \ {m0}, replace the j-th row Lj by Lj − b′′j m0 b′′m0 m0 Lm0 = Lj − Oε(e− η h ).Lm0 . CUBO 11, 4 (2009) Small singular values of an extracted matrix of a Witten complex. 55 Step 2: Then, for k ∈ {1, . . . , m0 − 1}, replace the k-th column Ck by Ck − b′′m0 k b′′m0 m0 Cm0 = Ck − Oε(e− η h ).Cm0 . Due to the previous operations, only the m0-th row of the new matrix is changed by these operations. Each operation of the two last steps preserves the structure of Assumption 2.2, or more pre- cisely the structure of Remark 3.5 where we have replaced the injective map j by the canonical injection i. Moreover, these operations correspond to left multiplications or right multiplications by quasi-unitary matrices. The new matrix only contains zeros on the m0-th row and m0-th column except for the (m0,m0)- coefficient which is b′ j(m0) m0 = 〈 ψ (1) j(m0) | B(h)ψ(0)m0 〉 . When m0 ≥ 2, iterate the Gaussian elimination, Step 1 with the reference row m0 − ν and Step 2 with the reference column m0 −ν, by taking successively ν = 1, . . . ,m0 − 2. At the end, we obtain a diagonal matrix D(h) ∈ Mm0,m1 (C) such that: ∀k ∈ {1, . . . ,m0} , (D(h))k,k = 〈 ψ (1) j(k) | B(h)ψ(0)k 〉 (1 + Oε(e−η/h)). Moreover, by Lemma 3.1, there exist two quasi-unitary matrices U(h) ∈ Mm0 (C) and V (h) ∈ Mm1 (C) satisfying D(h) = V (h)B ′ (h)U(h) = V (h)C ∗ 1 B(h)C0U(h) . Using again Lemma 3.1, V ′(h) = V (h)C∗ 1 and U′(h) = C0U(h) are quasi-unitary. From D(h) = V ′ (h)B(h)U ′ (h), we conclude using Corollary 3.3 and (3.1). Remark 3.6. a) The square self-adjoint structure A0(h) = B∗(h)B(h) is essential here to be able to conclude. Even a small distortion, A0(h) = B∗(h)CB(h) with C = Id +r with r = Oε(e− η h ), in dimension 2, destroys the above arguments, due to ill-conditioning problem. In the decomposition A0(h) = B ∗ (h)B(h) + B ∗ (h)rB(h) = B ∗ (h)B(h) + B ∗ (h)Oε(e− η h )B(h) , the remainder term B∗(h)Oε(e− η h )B(h) cannot be put in general in the form B∗(h)B(h)Oε(e− η h ): B ∗ (h)rB(h) = B ∗ (h)B(h) ( B(h) −1 rB(h) ) with ∥∥B(h)−1rB(h) ∥∥ ≤ ∥∥B(h)−1 ∥∥‖B(h)‖ ‖r‖ . For example, take η = 1 and B(h) = ( e − 4 h 0 0 e − 2 h ) and C(h) = ( 1 e − 1 h 0 1 ) . In this example the remainder factor equals B(h) −1 rB(h) = ( 0 e + 1 h 0 0 ) 56 D. Le Peutrec CUBO 11, 4 (2009) with a norm of order e 1 h = e 2 h × e− ηh . b) A first attempt at the extension of this analysis to the non self-adjoint case related with Kramers- Fokker-Planck type operators, studied in [12] [13], led to the simple distortion A0(h) = B∗(h)CB(h) with C = Id +Oε(e− η h ). The previous remark shows that it cannot work without including some additional information about the intimate link between these non self-adjoint operators coming from kinetic theory and Witten Laplacians ( [1] [8] [11] [15]). Acknowledgement: The author would like to thank F. Hérau, T. Jecko and F. Nier for profitable discussions, and the anonymous referee for suggesting various improvements. Received: August 2008. Revised: September 2008. References [1] Bismut, J. M. and Lebeau, G., The hypoelliptic Laplacian and Ray-Singer metrics, to appear in the Princeton University Press, 2008. [2] Bovier, A. Eckhoff, M. Gayrard, V. and Klein, M., Metastability in reversible diffusion processes I: Sharp asymptotics for capacities and exit times, JEMS Vol.6 no. 4, pp. 399-424, 2004. [3] Bovier, A. Gayrard, V. and Klein, M., Metastability in reversible diffusion processes II: Precise asymptotics for small eigenvalues, JEMS Vol.7 no. 4, pp. 66-99, 2004. [4] Freidlin, M. I. and Wentzell, A. D., Random perturbations of dynamical systems, Springer-Verlag, New York, 1984. [5] Gohgerg, I. and Krejn, M., Introduction à la théorie des opérateurs linéaires non auto- adjoints dans un espace hilbertien, Monographies Universitaires de Mathématiques, Vol.39, Dunod, 1971. [6] Helffer, B., Introduction to the semi-classical Analysis for the Schrödinger operator and applications. Springer Verlag. Lecture Notes in Mathematics 1336, 1988. [7] Helffer, B. Klein, M. and Nier, F., Quantitative analysis of metastability in reversible diffusion processes via a Witten complex approach, Mémoire 105 Société Mathématique de France, 2006. [8] Helffer, B. and Nier, F., Quantitative analysis of metastability in reversible diffusion processes via a Witten complex approach: the case with boundary, Matematica Contemporanea Vol.26 pp. 41-85, 2004. CUBO 11, 4 (2009) Small singular values of an extracted matrix of a Witten complex. 57 [9] Helffer, B. and Sjöstrand, J.,, Puits multiples en limite semi-classique II -Interaction moléculaire-Symétries-Perturbations. Ann. Inst. H. Poincaré Phys. Théor. 42 (2), p. 127-212 (1985). [10] Helffer, B. and Sjöstrand, J., Multiple wells in the semi-classical analysis IV. Etude du complexe de Witten, Comm. Partial Differential Equations Vol.10 no. 3, pp. 245–340, 1985. [11] Hérau, F. and Nier, F., Isotropic hypoellipticity and trend to the equilibrium for the Fokker-Planck equation with high degree potential, Archive for Rational Mechanics and Analysis 171 (2), p. 151-218, 2004. [12] Hérau, F. Sjöstrand, J. and Stolk, C., Semiclassical analysis for the Kramers-Fokker- Planck equation, Comm. Partial Differential Equations Vol.30 no. 4-6, p. 689-760, 2005. [13] Hérau, F. Hitrik, M. and Sjöstrand, J., Tunnel effect for Kramers-Fokker-Planck type operators, to appear in the Ann. Inst. Henri Poincaré, p.78, 2007. [14] Holley, R. Kusuoka, S. and Stroock, D., Asymptotics of the spectral gap with appli- cations to the theory of simulated annealing, J. Funct. Anal. 83 (2), p.333-347, 1989. [15] Lebeau, G., Geometric Fokker-Planck equations, Portugaliae Mathematica. Nova Série 62, 2005. [16] Nier, F., Quantitative analysis of metastability in reversible diffusion processes via a Witten complex approach, Journées “Equations aux Dérivées Partielles” Forges les eaux, Exp No VIII, Ecole Polytechnique, 2004. [17] Risken, H., The Fokker-Planck equation. Methods of solution and applications, Springer- Verlag, Berlin, second edition, 1989. [18] Simon, B., Trace Ideals and their Applications, Cambridge University Press, Lecture Notes Series vol. 35, 1979. [19] Stroock, D. W. and Varadhan, S. R., Multidimensionnal diffusion processes, Springer- Verlag, Berlin, 1979. Articulo 4