aa r X i v : . [ m a t h . SP ] J a n ANALYTIC EIGENBRANCHESIN THE SEMI-CLASSICAL LIMIT
STEFAN HALLER
Abstract.
We consider a one parameter family of Laplacians on a closedmanifold and study the semi-classical limit of its analytically parametrizedeigenvalues. Our results are analogous to a theorem for scalar Schr¨odingeroperators on Euclidean space by Luc Hillairet and apply to geometric operatorslike Witten’s Laplacian associated with a Morse function.
Let M be a closed smooth manifold and suppose E is a complex vector bundleover M . Fix a smooth volume density on M and a smooth fiber wise Hermitianmetric on E . We will denote the associated L scalar product on the space ofsections, Γ( E ), by hh− , −ii and write k − k for the corresponding L norm.Consider a one parameter family of operators acting on Γ( E ),∆ t = ∆ + tA + t V, t ∈ R , (1)where ∆ is a selfadjoint (bounded from below) Laplacian and A, V ∈ Γ(end( E ))are smooth symmetric sections. This is a selfadjoint holomorphic family of type(A) in the sense of [9, Section VII § analytic eigenbranches of ∆ t . More precisely, there exist eigenbranches λ t and eigensections ψ t , both analytic in t ∈ R , such that∆ t ψ t = λ t ψ t and k ψ t k = 1 . (2)Furthermore, it is possible to choose a sequence of analytic eigenbranches, λ ( k ) t ,and corresponding analytic eigensections, ψ ( k ) t , such that at every time t , the se-quence λ ( k ) t exhausts all of the spectrum of ∆ t , including multiplicities, and ψ ( k ) t forms a complete orthonormal basis of eigensections. The analytic parametriza-tion of the spectrum, λ ( k ) t , is unique up to renumbering. The eigensections, onthe other hand, are by no means canonical, and it seems more natural to considerthe spectral projections instead.In this note we study the semi-classical limit of the analytic eigenbranches,i.e., the behavior of λ t as t → ∞ . We will show that t − λ t converges to a finitelimit µ , see part (b) of the theorem below. Moreover, if the potential V is scalarvalued, i.e., if V = v · id E for a smooth function v , then µ has to be a criticalvalue of v , see part (h) in the theorem below. These observations are analogous Mathematics Subject Classification.
Key words and phrases. semi-classical limit; analytic eigenbranches; Witten Laplacian. to a result of Luc Hillairet [8] who considered the scalar case on M = R n with A = 0.While Hillairet’s proof uses basic properties of semi-classical measures, ourproof is entirely elementary and does not make use of this concept. The ideasentering into the proof, however, appear to be essentially the same. Avoidingsemi-classical measures makes the generalization to the vector valued case con-sidered here straight forward. As in Hillairet’s argument, convergence of t − λ t follows from the fact that this quantity, suitably corrected due of the presenceof A , is bounded and monotone, cf. (14) below. The fact that the Laplacian issemi-bounded enters crucially at this point.The asymptotics of the spectral distribution function in the semi-classical limithas applications in quantum mechanics and geometric topology [4, 5, 6, 7]. Wemerely mention Witten’s influential paper [12] and and proofs of the Cheeger–M¨uller theorem [1, 2, 3]. In these geometric applications, a Morse function f provides a deformation of the deRham differential, d t = e − tf de tf = d + tdf . Thecorresponding Witten Laplacian, ∆ t = ( d t + d ∗ t ) = d t d ∗ t + d ∗ t d t = [ d t , d ∗ t ], is aone parameter family of operators acting on differential forms, i.e., E = Λ ∗ T ∗ M ,which is of the type considered here with scalar valued V = | df | . Hence, in thiscase the absolute minima of V coincide with the critical points of f .Let us return to a general one parameter family of operator considered above,see (1), and an analytic eigenbranch as in (2). Subsequently, we will use thenotation ˙ λ t := ∂∂t λ t , ˙ ψ t := ∂∂t ψ t , and˙∆ t := ∂∂t ∆ t = A + 2 tV. (3)We have the following analogue of Theorem 1 in [8]. Theorem.
For each analytic eigenbranch the following hold true:(a) ˙ λ t = O ( t ) and λ t = O ( t ) , as t → ∞ .(b) t − λ t converges to a finite limit, µ := lim t →∞ t − λ t . (4) (c) t ∂∂t ( t − λ t ) is bounded and lim sup t →∞ t ∂∂t ( t − λ t ) = 0 . (d) t − hh ∆ ψ t , ψ t ii is bounded and lim inf t →∞ t − hh ∆ ψ t , ψ t ii = 0 . (e) The limit, µ , has the following interpretations: µ = lim t →∞ t − λ t = lim sup t →∞ hh V ψ t , ψ t ii = lim sup t →∞ (2 t ) − ˙ λ t = lim sup t →∞ ∂∂t ( t − λ t ) . Furthermore, for each sequence t n → ∞ such that, cf. (d) , lim n →∞ t − n hh ∆ ψ t n , ψ t n ii = 0 , (5) NALYTIC EIGENBRANCHES 3 the following hold true:(f ) For every positive integer s ∈ N , lim n →∞ t − sn k ψ t n k H s ( M ) = 0 , where k − k H s ( M ) denotes any Sobolev s norm on Γ( E ) .(g) We have lim n →∞ k ( V − µ ) ψ t n k = 0 . (6) Hence, the eigensections ψ t n localize near Σ µ := { x ∈ M : det( V ( x ) − µ ) = 0 } .In particular, for every open neighborhood U of Σ µ , we have lim n →∞ k ψ t n k L ( M \ U ) = 0 . (7) (h) If, moreover, the potential V is multiplication by a (scalar) function, then lim n →∞ (cid:13)(cid:13) | dV | ψ t n (cid:13)(cid:13) = 0 . (8) Hence, the eigensections ψ t n localize near the critical points of V . For everyneighborhood ˜ U of the critical set of V we have lim n →∞ k ψ t n k L ( M \ ˜ U ) = 0 . (9) In particular, µ has to be a critical value of V .Proof. From (2) we obtain hh ∆ t ψ t , ψ t ii = λ t . (10)Differentiating the second equation in (2) we get hh ˙ ψ t , ψ t ii + hh ψ t , ˙ ψ t ii = 0 . Differentiating (10) and using the selfadjointness of ∆ t this leads to hh ˙∆ t ψ t , ψ t ii = ˙ λ t . (11)Combining this with (3) we obtain˙ λ t = hh ( A + 2 tV ) ψ t , ψ t ii = hh Aψ t , ψ t ii + 2 t hh V ψ t , ψ t ii . (12)Since A and V are bounded operators, this implies ˙ λ t = O ( t ), whence (a).From (1) and (3) we immediately get2∆ t − t ˙∆ t = 2∆ + tA. Combining this with (10) and (11) and using the boundedness of A , we obtain ∂∂t ( t − λ t ) = − t − (cid:0) λ t − t ˙ λ t (cid:1) = − t − hh (2∆ t − t ˙∆ t ) ψ t , ψ t ii = − t − hh (2∆ + tA ) ψ t , ψ t ii = − t − hh ∆ ψ t , ψ t ii + O ( t − ) . (13)Hence, as ∆ is bounded from below, there exists a constant C such that ∂∂t (cid:0) t − λ t + Ct − (cid:1) ≤ , (14) STEFAN HALLER for sufficiently large t . This shows that the quantity t − λ t + Ct − is monotone,for large t . In view of (a) it is bounded too. Whence t − λ t + Ct − converges, as t → ∞ . This immediately implies (b).Similarly, one can show (c) and (d): Rewriting (13) we get − t ∂∂t ( t − λ t ) = 2 t − hh ∆ ψ t , ψ t ii + O ( t − ) . (15)As t − λ t is bounded we must havelim sup t →∞ t ∂∂t ( t − λ t ) ≥ , (16)for otherwise t − λ t would diverge logarithmically. Moreover,lim inf t →∞ t − hh ∆ ψ t , ψ t ii ≥ , (17)since ∆ is bounded from below. Combining (15), (16) and (17) we obtainlim sup t →∞ t ∂∂t ( t − λ t ) = 0 = lim inf t →∞ t − hh ∆ ψ t , ψ t ii . This completes the proof of (c) and (d), the statements on boundedness followimmediately from (a) and (15).To see (e) note that (1) and (10) give t − λ t = t − hh ∆ ψ t , ψ t ii + t − hh Aψ t , ψ t ii + hh V ψ t , ψ t ii . Using (d) we obtain the second equality in (e). The third equality follows from(12). The last one is immediate using ∂∂t ( t − λ t ) = t − ˙ λ t − t − λ t .To see (f), note first that the case s = 1 is immediate from (5) since there existsa constant C such that k ψ k H ( M ) ≤ C ( hh ∆ ψ, ψ ii + k ψ k ). Using the eigensectionequation ∆ t ψ t = λ t ψ t and (a), one obtains constants C s such that k ψ t k H s +2 ( M ) ≤ C s (1 + t ) k ψ t k H s ( M ) . This permits to establish (f) inductively for all odd integers s ∈ N . The evencase can be reduced to the odd one using k ψ k H ( M ) ≤ C ′ k ψ k H ( M ) k ψ k H ( M ) , anestimate which readily follows from the Cauchy–Schwarz inequality.Using the estimate in (f) for s = 2, the eigensection equation, t − ∆ ψ t + t − Aψ t + ( V − µ ) ψ t = ( t − λ t − µ ) ψ t , implies (6). As V − µ is invertible over the compact set M \ U , there existsa constant c > c ≤ ( V − µ ) ∗ ( V − µ ), over each point in M \ U .Consequently, c k ψ k L ( M \ U ) ≤ k ( V − µ ) ψ k L ( M \ U ) . Combining this with (6), we obtain (7).Let us now turn to the proof of (h). Suppose D is a first order differentialoperator acting on sections of E . Since V is scalar valued, the commutator σ :=[ D, V ] = DV − V D is a differential operator of order zero, namely the principal
NALYTIC EIGENBRANCHES 5 symbol, σ = σ D ( dV ) ∈ Γ(end( E )). As ∆ is a Laplacian, the commutator [ D, ∆]is a differential operator of order at most two. Using[ D, t − ∆ t ] = t − [ D, ∆] + t − [ D, A ] + σ, the Cauchy–Schwarz inequality and (f), we thus obtain k σψ t n k = hh [ D, t − n ∆ t n ] ψ t n , σψ t n ii + o (1) , as n → ∞ . (18)Using ∆ t ψ t = λ t ψ t one readily checks hh [ D, t − ∆ t ] ψ t , σψ t ii = hh Dψ t , [ σ, t − ∆ t ] ψ t ii . (19)Since V is scalar valued we have [ σ, V ] = 0 and t [ σ, t − ∆ t ] = t − [ σ, ∆] + [ σ, A ] , where [ σ, ∆] is a differential operator of order at most one. Proceeding as above,the Cauchy–Schwarz inequality and (f) yieldlim n →∞ hh Dψ t n , [ σ, t − n ∆ t n ] ψ t n ii = 0 . Combining the latter with (18) and (19), we arrive atlim n →∞ k σψ t n k = 0 . (20)Specializing to D = ∇ X where ∇ is some linear connection on E and X is avector field on M , we obtain σ = X · V = dV ( X ). Choosing X = grad( V ) withrespect to some auxiliary Riemannian metric on M , we have σ = | dV | , and (20)becomes (8). On M \ ˜ U we have 0 < ˜ c ≤ | dV | for some constant ˜ c , hence˜ c k ψ k L ( M \ ˜ U ) ≤ (cid:13)(cid:13) | dV | ψ (cid:13)(cid:13) L ( M \ ˜ U ) , and thus (8) implies (9). Combining the latter with (g), we see that µ has to bea critical value of V . This completes the proof of the theorem. (cid:3) Concluding remarks
At the very end of Section 3 in [8], Hillairet points out that for M = R and non-degenerate minima, the limit µ has to be the absolute minimum of V .Indeed, in this situation the spectrum is known to be simple, hence the analyticeigenbranches cannot cross and remain in the same order: λ (1) t < λ (2) t < · · · Thesemi-classical asymptotics of the k -th eigenvalue, however, is governed by thedeepest wells. Conjecture.
If the ‘minima’ of V are non-degenerate in the sense of Shubin [10, Condition C on page 378] and M is connected, then t − λ t converges to theabsolute minimum of V . STEFAN HALLER
From (14) we see that there exists a constant C such that µ − Ct − ≤ t − λ t for sufficiently large t . It is unclear to the author, if a similar estimate from aboveholds true, i.e., if we have t − λ t = µ + O ( t − ) as t → ∞ .As t → ∞ , the following statements are equivalent:(a) hh ∆ ψ t , ψ t ii = O ( t )(b) hh ( V − µ ) ψ t , ψ t ii = O ( t − )(c) ∂∂t ( t − α ( λ t − µt )) = O ( t − α ) for one (and then all) real α = 1.(d) ∂∂t ( t − λ t ) = O ( t − )(e) ˙ λ t − tµ = O (1)Indeed, the equivalence ( a ) ⇔ ( d ) follows from (15), the equivalence ( b ) ⇔ ( e ) fol-lows from (12), and the equivalence ( c ) ⇔ ( d ) ⇔ ( e ) follows from (4). Moreover,if these five (equivalent) statements hold true, then clearly t − λ t = µ + O ( t − ) , as t → ∞ . (21)Suppose (21) holds true. Moreover, assume that µ is the absolute minimumof V and the minima are all non-degenerate in the sense of [10, Condition C onpage 378]. Then, according to [10, Theorem 1.1] or [4, Theorem 11.1], there existsan eigenvalue ω of the harmonic oscillator associated with the minima (deepestwells) such that λ t = µt + ωt + O ( t / ) , as t → ∞ . (22)Better estimates are available if the geometry is flat near the minima, cf. [10,Equation (1.15)]. Under additional assumptions [11] one even obtains a full as-ymptotic expansion in terms of integral powers of t . Furthermore, for everyeigenvalue ω of the harmonic oscillator associated with the minima of V , thereexists an analytic eigenbranch for which (22) holds true with µ the absolute min-imum of V and the number of these eigenbranches coincides with the multiplicityof ω . An intriguing problem remains open: Are there analytic eigenbranches witha different asymptotics which are not governed by the deepest wells? Acknowledgments
The author is indebted to Dan Burghelea for encouraging discussions. Part ofthis work was done while the author enjoyed the hospitality of the Max PlanckInstitute for Mathematics in Bonn. He gratefully acknowledges the support ofthe Austrian Science Fund (FWF): project number P31663-N35.
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