On the pointwise convergence of the integral kernels in the Feynman-Trotter formula
aa r X i v : . [ m a t h - ph ] A p r ON THE POINTWISE CONVERGENCE OF THE INTEGRALKERNELS IN THE FEYNMAN-TROTTER FORMULA
FABIO NICOLA AND S. IVAN TRAPASSO
Abstract.
We study path integrals in the Trotter-type form for the Schr¨odingerequation, where the Hamiltonian is the Weyl quantization of a real-valued qua-dratic form perturbed by a potential V in a class encompassing that - consideredby Albeverio and Itˆo in celebrated papers - of Fourier transforms of complex mea-sures. Essentially, V is bounded and has the regularity of a function whose Fouriertransform is in L . Whereas the strong convergence in L in the Trotter formula,as well as several related issues at the operator norm level are well understood,the original Feynman’s idea concerned the subtler and widely open problem of thepointwise convergence of the corresponding probability amplitudes, that are theintegral kernels of the approximation operators. We prove that, for the above classof potentials, such a convergence at the level of the integral kernels in fact occurs,uniformly on compact subsets and for every fixed time, except for certain excep-tional time values for which the kernels are in general just distributions. Actually,theorems are stated for potentials in several function spaces arising in HarmonicAnalysis, with corresponding convergence results. Proofs rely on Banach algebrastechniques for pseudo-differential operators acting on such function spaces. Introduction and main results
The path integral formulation of Quantum Mechanics is by far one of the majorachievements in modern theoretical physics. The first intuition on the issue is attrib-uted to Dirac: in his celebrated 1933 paper [11] he provided several clues indicatingthat the Lagrangian formulation of classical mechanics should have a quantum coun-terpart. While it is debatable whether the entire story was already known to him atthe time of writing, his program has been finalised by Feynman [16], who explicitlyrecognized Dirac’s remarks as the main source of inspiration for his landmark con-tribution to a new formulation of non-relativistic quantum mechanics beyond theSchr¨odinger and Heisenberg pictures.1.1.
The sequential approach to path integrals.
We could argue that the pathintegral formulation comes from a profound understanding of the double-slit exper-iment - in fact, this is precisely the way Feynman introduces the problem in the
Mathematics Subject Classification.
Key words and phrases.
Feynman path integrals, time slicing approximation, modulation spaces,Trotter product formula, metaplectic operators, Schr¨odinger equation. book [17]. While this is an intriguing perspective, we briefly outline this approachfrom a different starting point. Recall that the state of a non-relativistic particlein the Euclidean space R d at time t is represented by the wave function ψ ( t, x ),( t, x ) ∈ R × R d , such that ψ ( t, · ) ∈ L ( R d ). The dynamics under the real-valuedpotential V is regulated by the Cauchy problem for the Schr¨odinger equation :(1) ( i∂ t ψ = ( H + V ( x )) ψψ (0 , x ) = ϕ ( x ) , where H = −△ / propagator U ( t ) = e − itH , t ∈ R : ψ ( t, x ) = U ( t ) ϕ ( x ) . At least on a formal level, one can thus represent U ( t ) as an integral operator: ψ ( t, x ) = Z R d u t ( x, x ) ϕ ( x ) dx , where the kernel u t ( x, x ) intuitively yields the transition amplitude from the posi-tion x at time 0 to the position x at time t . The path integral formulation exactlyconcerns the determination of this kernel: according to Feynman’s prescription, oneshould take into account the many possible interfering alternative paths from x to x that the particle could follow. Each path would contribute to the total probabilityamplitude with a phase factor proportional to the action functional correspondingto the path: S [ γ ] = Z ts L ( γ ( τ ) , ˙ γ ( τ )) dτ, where L is the Lagrangian of the corresponding classical system. In short, a merelyformal representation of the kernel is(2) u t ( x, x ) = Z e iS [ γ ] D γ, namely a sort of integral over the infinite-dimensional space of paths satisfying theaforementioned boundary conditions. In order to shed some light on the heuristicsunderpinning this formula, let us briefly outline the so-called sequential approach to path integrals introduced by Nelson [45], which seems the closest to Feynman’soriginal formulation. First, recall that the free propagator e − itH can be properlyidentified with a Fourier multiplier and the following integral expression holds: e − itH ϕ ( x ) = 1(2 πit ) d/ Z R d exp i | x − x | t ! ϕ ( x ) dx , ϕ ∈ S ( R d ) . We set m = 1 for the mass of the particle and ~ = 1 for the Planck constant. N THE POINTWISE CONVERGENCE IN THE FEYNMAN-TROTTER FORMULA 3
Next, under suitable assumptions for the potential V , the Trotter product formulaholds for the propagator generated by H = H + V : e − it ( H + V ) = lim n →∞ (cid:16) e − i tn H e − i tn V (cid:17) n , where the limit is intended in the strong topology of operators in L ( R d ). Combiningthese two results gives the following representation of the complete propagator e − itH as limit of integral operators (cf. [49, Thm. X.66]):(3) e − it ( H + V ) φ ( x ) = lim n →∞ (cid:18) πi tn (cid:19) − nd Z R nd e iS n ( t ; x ,...,x n − ,x ) ϕ ( x ) dx . . . dx n − , where S n ( t ; x , . . . , x n − , x ) = n X k =1 tn " (cid:18) | x k − x k − | t/n (cid:19) − V ( x k ) , x n = x. In order to grasp the meaning of the phase S n ( t ; x , . . . , x n ), consider the followingargument: given the points x , . . . , x n − , x ∈ R d , let γ be the polygonal path throughthe vertices x k = γ ( kt/n ), k = 0 , . . . , n , x n = x , parametrized as γ ( τ ) = x k + x k +1 − x k t/n (cid:18) τ − k tn (cid:19) , τ ∈ (cid:20) k tn , ( k + 1) tn (cid:21) , k = 0 , . . . , n − . Hence γ prescribes a classical motion with constant velocity along each segment.The action for this path is thus given by S [ γ ] = n X k =1 tn (cid:18) | x k − x k − | t/n (cid:19) − Z t V ( γ ( τ )) dτ. According to Feynman’s interpretation, (3) can be thought as an integral over allpolygonal paths and S n ( x , . . . , x n , t ) is a Riemann-like approximation of the actionfunctional evaluated on them. The limit n → ∞ is now intuitively clear: the set ofpolygonal paths becomes the set of all paths and we recover (2). In fact, it shouldbe noted that the custom in Physics community is to employ the suggestive formula(2) as a placeholder for (3) and the related arguments - see for instance [27, 36].We could not hope to frame the vast literature concerning the problem of puttingthe formula (2) on firm mathematical ground; the interested reader could benefitfrom the monographs [4, 21, 44] as points of departure. We only remark that thethere is in general some relationship between the regularity assumptions on the po-tential and the strength of the convergence of the time-slicing approximation. Whilethe operator theoretic strategy outlined above also allows to treat wild potentials,the convergence in finer operator topologies (for instance, at the level of integralkernels as in Feynman’s original formulation) have been an open problem for a longtime. Nevertheless, there is a variety of schemes to deal with path integrals and FABIO NICOLA AND S. IVAN TRAPASSO pointwise convergence of integral kernels can be achieved by means of other so-phisticated techniques, at least for smooth potentials - see the works of Fujiwara,Ichinose, Kumano-go and coauthors [19, 20, 30, 31, 32, 33, 37, 38, 39, 40, 41, 42, 43].1.2.
Main results.
The present contribution aims at investigating the convergenceat the level of integral kernels for the time-slicing approximation of path inte-grals under low regularity assumptions for the involved potentials. We considerthe Schr¨odinger equation(4) i∂ t ψ = ( H + V ( x )) ψ, where now H = a w is the Weyl quantization of a real quadratic form a ( x, ξ ) on R d and V ∈ L ∞ ( R d ) is complex-valued (so that a linear magnetic potential or aquadratic electric potential are allowed and included in H ). It is well known thatthe propagator U ( t ) = e − itH for the unperturbed problem ( V = 0) is a metaplecticoperator [18]. By a slight abuse of language (essentially, up to a sign factor), we cansuggestively write U ( t ) = µ ( A t ), where t
7→ A t ∈ Sp ( d, R ) is the one-parametersubgroup of symplectic matrices associated with the solution of the classical equa-tions of motion with Hamiltonian H in phase space and µ is the so-called metaplecticrepresentation - see Section 2.3 for the rigorous construction of U ( t ). We expressthe block structure of A t , namely A t = (cid:18) A t B t C t D t (cid:19) , since our results are global in time unless certain exceptional values, namely forany t ∈ R such that det B t = 0 (equivalently, for any t ∈ R such that A t is a freesymplectic matrix - cf. Section 2.3). Consequently, we also introduce the quadraticform(5) Φ t ( x, y ) = 12 xD t B − t x − yB − t x + 12 yB − t A t y, x, y ∈ R d . Recall that (cf. [28]) H is a self-adjoint operator on its domain D ( H ) = { ψ ∈ L ( R d ) : H ψ ∈ L ( R d ) } . Hence, V being bounded, the Trotter product formula holds:(6) e − it ( H + V ) = lim n →∞ E n ( t ) , E n ( t ) = (cid:16) e − i tn H e − i tn V (cid:17) n , where the convergence is again in the strong operator topology in L ( R d ) (see e.g.[12, Cor. 2.7]). We denote by e n,t ( x, y ) the distribution kernel of E n ( t ) and by u t ( x, y ) that of e − it ( H + V ) .In order to state our first result, we need to introduce two spaces of a markedharmonic analysis flavour, defined in terms of the decay of the Fourier transform. N THE POINTWISE CONVERGENCE IN THE FEYNMAN-TROTTER FORMULA 5
Let M ∞ s ( R d ), s ∈ R , denote the subspace of temperate distributions f ∈ S ′ ( R d )such that, for some non-zero Schwartz function g ∈ S ( R d ), k f k M ∞ s = sup x,ξ ∈ R d |F [ f · g ( · − x )] ( ξ ) | (1 + | ξ | ) s < ∞ , where F is the Fourier transform. In addition, consider the space F L s ( R d ) offunctions with weighted integrable Fourier transforms, namely: f ∈ F L s ( R d ) ⇔ k f k F L s = Z R d |F f ( ξ ) | (1 + | ξ | ) s dξ < ∞ . While the space F L s is rather standard, M ∞ s is a special member of a family ofBanach spaces, the so-called modulation spaces , stemming from the branch of har-monic analysis currently known as time-frequency analysis (cf. Section 2.2 for thedetails). Modulation spaces proved to be a fruitful environment for the study ofPDEs, in particular the Schr¨odinger equation (see for instance [6, 7, 8, 54] and thereferences therein), and related problems such as path integrals [46, 47, 48]. Wehave a convergence result for potentials in this space. Theorem 1.1.
Consider H as specified above and V ∈ M ∞ s ( R d ) , with s > d . Forany t ∈ R such that det B t = 0 :(1) the distributions e − πi Φ t e n,t , n ≥ , and e − πi Φ t u t belong to a bounded subsetof M ∞ s ( R d ) ;(2) e n,t → u t in ( F L r ) loc ( R d ) for any < r < s − d , hence uniformly oncompact subsets. We notice that for s > d we have M ∞ s ( R d ) ⊂ ( F L ) loc ( R d ) ∩ L ∞ ( R d ), so thatthe kernels e n,t and u t in the statement are in fact bounded and continuous functions,provided det B t = 0.Also, ∩ s> M ∞ s ( R d ) = C ∞ b ( R d ) is the space of bounded smooth functions withbounded derivatives of any order, which gives the following result. Corollary 1.2.
Let H be as specified above and V ∈ C ∞ b ( R d ) . For any t ∈ R suchthat det B t = 0 :(1) the distributions e − πi Φ t e n,t , n ≥ , and e − πi Φ t u t belong to a bounded subsetof C ∞ b ( R d ) ;(2) e n,t → u t in C ∞ ( R d ) , hence uniformly on compact subsets together with anyderivatives. The same conclusion of Corollary 1.2 is actually known to hold true for shorttimes, as a consequence of Fujiwara’s result [20], but the above result is global intime. The occurrence of a set of exceptional times is to be expected: in these cases,the integral kernel of the propagator degenerates into a distribution. A basic example
FABIO NICOLA AND S. IVAN TRAPASSO of this behaviour is given by the harmonic oscillator, that is H = − π △ + π | x | , V ( x ) = 0, at t = kπ , k ∈ Z . Notice that such exceptional values are exactly thosefor which the upper-right block of the associated Hamiltonian flow A t = (cid:18) (cos t ) I (sin t ) I − (sin t ) I (cos t ) I (cid:19) is non-invertible.We now state our main result, which is subtler than Theorem 1.1 and applies topotentials in a lower regularity space known as the Sj¨ostrand class M ∞ , ( R d ): wesay that f ∈ S ′ ( R d ) belongs to M ∞ , ( R d ) if k f k M ∞ , = Z R d sup x ∈ R d |F [ f · g ( · − x )] ( ξ ) | dξ < ∞ , for some non-zero g ∈ S ( R d ). As a rule of thumb, a function in M ∞ , ( R d ) isbounded on R d and locally enjoys the mild regularity of the Fourier transform of an L function; in fact ( M ∞ , ) loc ( R d ) = ( F L ) loc ( R d ) . Furthermore, we have the following chain of strict inclusions for s > d : C ∞ b ( R d ) ⊂ M ∞ s ( R d ) ⊂ M ∞ , ( R d ) ⊂ ( F L ) loc ( R d ) ∩ L ∞ ( R d ) ⊂ C ( R d ) ∩ L ∞ ( R d ) . Intuitively: we have a scale of low-regularity spaces, the functions in M ∞ s ( R d ) be-coming less regular as s ց d , until the (fractional) differentiability is completelylost in the “maximal” space M ∞ , ( R d ).It seems worth to highlight that results on the convergence of path integrals arealready known for special elements of the Sj¨ostrand class: for instance, a class ofpotentials widely investigated by means of different approaches in the papers ofAlbeverio and coauthors [1, 2, 3] and Itˆo [35] (see also [34]) is F M ( R d ), namelythe space of Fourier transforms of (finite) complex measures on R d . In fact, wehave F M ( R d ) ⊂ M ∞ , ( R d ), cf. Proposition 3.4 below, and the above inclusion isstrict; for instance, f ( x ) = cos | x | , x ∈ R d , clearly belongs to C ∞ b ( R d ), but it iseasy to realize that f / ∈ F M ( R d ) as soon as d >
1, by the known formula for thefundamental solution of the wave equation [13].The following result encompasses these potentials and ultimately yields the desiredpointwise convergence at the level of integral kernels for a wide class of non-smoothpotentials.
Theorem 1.3.
Let H be as specified above and V ∈ M ∞ , ( R d ) . For any t ∈ R such that det B t = 0 : See Section 2 for the choice of the normalization of the Weyl quantization and the classicalflow.
N THE POINTWISE CONVERGENCE IN THE FEYNMAN-TROTTER FORMULA 7 (1) the distributions e − πi Φ t e n,t , n ≥ , and e − πi Φ t u t belong to a bounded subsetof M ∞ , ( R d ) ;(2) e n,t → u t in ( F L ) loc ( R d ) , hence uniformly on compact subsets. Let us conclude this introduction with a few words on the techniques employed forthe proofs. The main idea is to rephrase the problem in terms of pseudodifferentialcalculus and then to exploit the very rich structure enjoyed by the modulationspaces M ∞ s ( R d ) (with s > d ) and M ∞ , ( R d ): in particular, they are Banachalgebras for both pointwise multiplication and twisted product of symbols for theWeyl quantization - see the subsequent Section 2.3 for the details.There is a certain number of questions which seem worthy of further consider-ation. For example, Theorem 1.1 and Corollary 1.2 should hopefully extend toHamiltonians H given by the Weyl quantization of a smooth real-valued functionwith derivatives of order ≥ V could be replaced by a genuine pseudodifferential operator in suitable classes.We preferred to avoid further technicalities here, since the arguments below are al-ready somewhat involved. Finally we observe that the techniques introduced in thepresent paper could hopefully be useful to study similar convergence problems ofthe integral kernels for other approximation formulas arising in semigroup theory;cf. [12].The paper is organized as follows. Sections 2 and 3 are both devoted to preliminaryresults and technical lemmas on function spaces and operators involved. In Section4 we prove Theorem 1.1 and Corollary 1.2. Theorem 1.3 is proved in Section 5.2. Preliminary results
Notation.
We define x = x · x , for x ∈ R d , and xy = x · y is the scalar producton R d . The Schwartz class is denoted by S ( R d ), the space of temperate distributionsby S ′ ( R d ). The brackets h f, g i denote the extension to S ′ ( R d ) × S ( R d ) of the innerproduct h f, g i = R R d f ( x ) g ( x ) dx on L ( R d ).The conjugate exponent p ′ of p ∈ [1 , ∞ ] is defined by 1 /p + 1 /p ′ = 1. The symbol . means that the underlying inequality holds up to a positive constant factor C > x ∈ R d and s ∈ R we set h x i s := (cid:0) | x | (cid:1) s/ . We choose the followingnormalization for the Fourier transform: F f ( ξ ) = Z R d e − πixξ f ( x ) dx, ξ ∈ R d . We define the translation and modulation operators: for any x, ξ ∈ R d and f ∈S ( R d ), ( T x f ) ( y ) := f ( y − x ) , ( M ξ f ) ( y ) := e πiξy f ( y ) . FABIO NICOLA AND S. IVAN TRAPASSO
These operators can be extended by duality on temperate distributions. The com-position π ( x, ξ ) = M ξ T x constitutes a so-called time-frequency shift.Denote by J the canonical symplectic matrix in R d : J = (cid:18) d I d − I d d (cid:19) ∈ Sp( d, R ) , where the symplectic group Sp ( d, R ) is defined as:Sp ( d, R ) = (cid:8) M ∈ GL(2 d, R ) : M ⊤ J M = J (cid:9) and the associated Lie algebra is sp ( d, R ) := { M ∈ R d × d : M J + J M ⊤ = 0 } . Modulation spaces.
The short-time Fourier transform (STFT) of a temper-ate distribution f ∈ S ′ ( R d ) with respect to the window function g ∈ S ( R d ) \ { } isdefined by:(7) V g f ( x, ξ ) := h f, π ( x, ξ ) g i = F ( f · T x g )( ξ ) = Z R d e − πiyξ f ( y ) g ( y − x ) dy. This is a key instrument for time-frequency analysis; the monograph [23] containsa comprehensive treatment of its mathematical properties, especially those men-tioned below. For the sake of conciseness, we only mention that the STFT is deeplyconnected with other well-known phase-space transforms, in particular the Wignertransform W ( f, g )( x, ξ ) = Z R d e − πiyξ f (cid:16) x + y (cid:17) g (cid:16) x − y (cid:17) dy. For this and other aspects of the connection with phase space analysis, we recom-mend [10].Given a non-zero window g ∈ S ( R d ), s ∈ R and 1 ≤ p, q ≤ ∞ , the modulationspace M p,qs ( R d ) consists of all temperate distributions f ∈ S ′ ( R d ) such that V g f ∈ L p,qs ( R d ) (mixed weighted Lebesgue space), that is: k f k M p,qs = k V g f k L p,qs = Z R d (cid:18)Z R d | V g f ( x, ξ ) | p dx (cid:19) q/p h ξ i qs dξ ! /q < ∞ , with trivial adjustments if p or q is ∞ . If p = q , we write M p instead of M p,p .For the unweighted case, corresponding to s = 0, we omit the dependence on s : M p,q ≡ M p,q .It can be proved that M p,qs ( R d ) is a Banach space whose definition does notdepend on the choice of the window g . Just to get acquainted with this family, it isworth to mention that many common function spaces can be equivalently designedas modulation spaces: for instance,(i) M ( R d ) coincides with the Hilbert space L ( R d ); N THE POINTWISE CONVERGENCE IN THE FEYNMAN-TROTTER FORMULA 9 (ii) M s ( R d ) coincides with the usual L -based Sobolev space H s ( R d );(iii) the following continuous embeddings with Lebesgue spaces hold: M p,qr ( R d ) ֒ → L p ( R d ) ֒ → M p,qs ( R d ) , r > d/q ′ and s < − d/q. In particular, M p, ( R d ) ֒ → L p ( R d ) ֒ → M p, ∞ ( R d ) . For these and other properties we address the reader to [14, 15, 23].For a fixed window g ∈ S ( R d ) \ { } , the STFT operator V g is clearly boundedfrom M p,qs ( R d ) to L p,qs ( R d ). The adjoint operator of V g , defined by V ∗ g F = Z R d F ( x, ξ ) π ( x, ξ ) g dxdξ, continuously maps the Banach space L p,qs ( R d ) into M p,qs ( R d ), the integral above tobe intended in a weak sense.The inversion formula for the STFT can be conveniently expressed as follows:for any f ∈ M p,qs ( R d ),(8) f = 1 k g k L V ∗ g V g f, again in a weak sense.The Sj¨ostrand’s class, originally defined in [52], coincides with the choice p = ∞ , q = 1, s = 0. We have that M ∞ , ( R d ) ⊂ C ( R d ) ∩ L ∞ ( R d ) and it is a Banachalgebra under pointwise product. In fact, precise conditions are known on p , q and s in order for M p,qs to be a Banach algebra with respect to pointwise multiplication: Lemma 2.1 ([50, Thm. 3.5]) . Let ≤ p, q ≤ ∞ and s ∈ R . The modulation space M p,qs ( R d ) is a Banach algebra for pointwise multiplication if and only if either s = 0 and q = 1 or s > d/q ′ . Therefore, the Sj¨ostrand’s class M ∞ , ( R d ) and the modulation spaces M ∞ s ( R d )with s > d are Banach algebras for pointwise multiplication. It is worth to pointout that the condition required in Lemma 2.1 are in fact equivalent to assume M p,qs ֒ → L ∞ - cf. [50, Cor. 2.2]. Remark 2.2.
We clarify once for all that the preceding results concern the con-ditions under which the embedding M p,qs · M p,qs ֒ → M p,qs is continuous, hence thereexists a constant C > such that k f g k M p,qs ≤ C k f k M p,qs k g k M p,qs , ∀ f, g ∈ M p,qs . Thus, the algebra property holds up to a constant. It is a well known general factthat one can provide an equivalent norm for which the previous estimate holds with C = 1 and the unit element of the algebra has norm equal to (cf. [51, Thm. 10.2] ). From now on, we assume to work with such equivalent norm whenever concernedwith a Banach algebra.
An important subspace of both M ∞ , ( R d ) and M ∞ s ( R d ) is the space C ∞ b ( R d ) := (cid:8) f ∈ C ∞ ( R d ) : | ∂ α f | ≤ C α ∀ α ∈ N d (cid:9) = \ s ≥ M ∞ s ( R d );see e.g. [26, Lem. 6.1] for this characterization.We briefly mention that the image of modulation spaces under Fourier transformyields another important family of function spaces for the purposes of real harmonicanalysis, which are a very special type of Wiener amalgam spaces : for any 1 ≤ p, q ≤ ∞ , we set W p,q ( R d ) := F M p,q ( R d ) . One can prove that W p,q ( R d ) is a Banach space under the same norm of M p,q ( R d ) butwith flipped order of integration with respect to the time and frequency variables: k f k W p,q := Z R d (cid:18)Z R d | V g f ( x, ξ ) | p ( ξ ) dξ (cid:19) q/p dx ! /q , for g ∈ S ( R d ) \ { } , as usual.2.3. Weyl operators.
The usual definition of the Weyl transform of the symbol σ : R d → C is σ w f ( x ) := Z R d e πi ( x − y ) ξ σ (cid:18) x + y , ξ (cid:19) f ( y ) dydξ. The meaning of this formal integral operator heavily relies on the function space towhich the symbol σ belongs. Instead, we adopt the following definition via dualityfor symbols σ ∈ S ′ ( R d ):(9) σ w : S ( R d ) → S ′ ( R d ) , h σ w f, g i = h σ, W ( g, f ) i , ∀ f, g ∈ S ( R d ) . In particular, M ∞ , ( R d ) and M ∞ s ( R d ) are suitable symbol classes. It is worthto mention that the classical symbol classes investigated within the long traditionof pseudodifferential calculus are usually defined by means of decay/smoothnessconditions (see for instance the general H¨ormander classes S mρ,δ ( R d ) - [29]), whilethe fruitful interplay with time-frequency analysis allows to cover very rough symbolstoo - cf. [24]. Remark 2.3.
Notice that the multiplication by V ( x ) is a special example of Weyloperator with symbol σ V ( x, ξ ) = V ( x ) = ( V ⊗ x, ξ ) , ( x, ξ ) ∈ R d . It is not difficult to prove that the correspondence V σ V is continuous from M ∞ s ( R d ) (resp. M ∞ , ( R d ) ) to M ∞ s ( R d ) (resp. M ∞ , ( R d ) ). In the rest of the paper N THE POINTWISE CONVERGENCE IN THE FEYNMAN-TROTTER FORMULA 11 this identification shall be implicitly assumed; by a slight abuse of notation, we willwrite V also for σ w V for the sake of legibility. The composition of Weyl transforms provides a bilinear form on symbols, the so-called twisted product : σ w ◦ ρ w = ( σ ρ ) w . Although explicit formulas for the twisted product of symbols can be derived (cf.[55]), we will not need them hereafter. Anyway, this is a fundamental notion inorder to establish an algebra structure on symbol spaces. It is a distinctive propertyof M ∞ , ( R d ), as well as M ∞ s ( R d ) with s > d , to enjoy a double Banach algebrastructure: • a commutative one with respect to the pointwise multiplication as detailedabove; • a non-commutative one with respect to the twisted product of symbols ([26,52]).We wish to underline that the latter algebra structure has been deeply investi-gated from a time-frequency analysis perspective. Indeed, it is subtly related toa characterizing property satisfied by pseudodifferential operators with symbols inthose spaces, namely almost diagonalization with respect to time-frequency shifts:we have σ ∈ M ∞ s ( R d ) if and only if |h σ w π ( z ) ϕ, π ( w ) ϕ i| ≤ C h w − z i − s , ∀ ϕ ∈ S ( R d ) \ { } , z, w ∈ R d . Similarly, σ ∈ M ∞ , ( R d ) if and only if there exists H ∈ L ( R d ) such that |h σ w π ( z ) ϕ, π ( w ) ϕ i| ≤ H ( w − z ) , ∀ ϕ ∈ S ( R d ) \ { } , z, w ∈ R d . We address the reader to [5, 6, 9, 25, 26] for further discussions on these aspects.
Remark 2.4.
To unambiguously fix the notation: whenever concerned with a prod-uct of elements a , . . . , a N in a Banach algebra ( A, ⋆ ) , we write N Y k =1 a k := a ⋆ a ⋆ . . . ⋆ a N . This relation is meant to hold even when ( A, ⋆ ) is a non-commutative algebra, pro-vided that the symbol on the LHS exactly designates the ordered product on the RHS. Metaplectic operators.
Given a symplectic matrix
A ∈
Sp( d, R ), we saythat the unitary operator µ ( A ) ∈ U ( L ( R d )) is a metaplectic operator associatedwith A if it does satisfy the following intertwining relation: π ( A z ) = µ ( A ) π ( z ) µ ( A ) − , ∀ z ∈ R d . Strictly speaking, the previous formula defines a whole set of unitary operators upto a constant phase factor: { c A µ ( A ) : c A ∈ C , | c A | = 1 } . The phase factor can beadjusted to either c A = 1 or c A = −
1, namely: µ ( AB ) = ± µ ( A ) µ ( B ) , ∀ A, B ∈ Sp( d, R ) . That is, µ provides a double-valued unitary representation of Sp( d, R ) or, better,a representation of the double covering Mp( d, R ) of Sp( d, R ); we will denote by ρ Mp : Mp( d, R ) → Sp( d, R ) the projection. We refer to [10, 18] for a comprehensivediscussion of these aspects.We confine ourselves to recall that the metaplectic operator corresponding to spe-cial symplectic matrices can be explicitly written as a quadratic Fourier transform.We say that A ∈
Sp( d, R ), with A = (cid:18) A BC D (cid:19) , is a free symplectic matrix whenever det B = 0. We have the following integralformula for µ ( A ). Theorem 2.5 ([10, Sec. 7.2.2]) . Let
A ∈
Sp( d, R ) be a free symplectic matrix. Then, (10) µ ( A ) f ( x ) = c | det B | − / Z R d e πi Φ A ( x,ξ ) f ( y ) dy, f ∈ S ( R d ) , where c ∈ C is a suitable complex factor of modulus and Φ A is the quadratic formgiven by Φ A ( x, y ) = 12 xDB − x − yB − x + 12 yB − Ay.
Incidentally, notice that µ ( J ) = c F − .It is important to recall that a truly distinctive property of the Weyl quantizationis its symplectic covariance [10, Thm. 215], namely: for any A ∈
Sp( d, R ) and σ ∈ S ′ ( R d ), the following relation holds:(11) ( σ ◦ A ) w = µ ( A ) − σ w µ ( A ) . Let now a be a real-valued, time-independent, quadratic homogeneous polynomialon R d , namely: a ( x, ξ ) = 12 xAx + ξBx + 12 ξCξ, We underline that the following quadratic Fourier transform, up to a sign in Φ A , is actuallythe point of departure for the construction of the metaplectic representation in [10]. N THE POINTWISE CONVERGENCE IN THE FEYNMAN-TROTTER FORMULA 13 where
A, C ∈ R d × d are symmetric matrices and B ∈ R d × d . The phase-space flowdetermined by the Hamilton equations π ˙ z = J ∇ z a ( z ) = A , A = (cid:18) B C − A − B ⊤ (cid:19) ∈ sp ( d, R ) , defines a mapping R ∋ t
7→ A t = e ( t/ π ) A ∈ Sp( d, R ) . It follows from the generaltheory of covering manifolds that this path can be lifted in a unique way to amapping R ∋ t M ( t ) ∈ Mp( d, R ) , M (0) = I ; hence ρ Mp ( M ( t )) = A t . Then,with H = a w , the Schr¨odinger equation ( i∂ t ψ = H ψψ (0 , x ) = ϕ ( x ) , ϕ ∈ S ( R d ) , is solved by ψ ( t, x ) = e − itH ϕ ( x ) = µ ( M ( t )) ϕ ( x ) , see [10, Sec. 15.1.3]. By a slight abuse of language we will write µ ( A t ) in place of µ ( M ( t )). We recommend [8, 10, 18] for further details on the matter.2.5. Operators and kernels.
Consider the space L ( X, Y ) of all continuous linearmappings between two Hausdorff topological vector spaces X and Y . It can beendowed with different topologies [53], in which cases we write:(1) L b ( X, Y ), if equipped with the topology of bounded convergence , that isuniform convergence on bounded subsets of X ;(2) L c ( X, Y ), if equipped with the topology of compact convergence , that isuniform convergence on compact subsets of X ;(3) L s ( X, Y ), if equipped with the topology of pointwise convergence , that isuniform convergence on finite subsets of X .Notice that if Y = C , L b ( X, Y ) = X ′ b (the strong dual of X ), while L s ( X, Y ) = X ′ s (the weak dual of X ). We will be mainly concerned with the case X = S ( R d ) and Y = S ′ ( R d ), the latter always endowed with the strong topology unless otherwisespecified (i.e., S ′ ( R d ) = L b ( S ( R d ) , C )). The celebrated Schwartz kernel theoremis usually invoked for proving that any reasonably well-behaved operator is indeedan integral transform, though in a distributional sense. In the following we willneed this identification but at the topological level [22, 53], that is, a linear map A : S ( R d ) → S ′ ( R d ) is continuous if and only if it is generated by a (unique)temperate distribution K ∈ S ′ ( R d ), namely:(12) h Af, g i = h K, g ⊗ f i , ∀ f, g ∈ S ( R d ) , The factor 2 π derives from the normalization of the Fourier transform adopted in the paper. and the correspondence K A above is a topological isomorphism between S ′ ( R d )and the space L b (cid:0) S ( R d ) , S ′ ( R d ) (cid:1) . As mentioned above, S ′ ( R d ) and S ′ ( R d ) areendowed with the strong topology. Proposition 2.6.
Let A n → A in L s (cid:0) S ( R d ) , S ′ ( R d ) (cid:1) . Then we have convergencein S ′ ( R d ) of the corresponding distribution kernels.Proof. Since S ( R d ) is a Fr´echet space and A n , being a sequence, defines a filter withcountable basis on L s ( S ( R d ) , S ′ ( R d )), from [53, Cor. at pag. 348] we have that A n → A also in L c ( S ( R d ) , S ′ ( R d )), which is in turn equivalent to convergence in L b ( S ( R d ) , S ′ ( R d )) since S ( R d ) is a Montel space - cf. [53, Prop. 34.4 and 34.5]. Thedesired conclusion then follows from Schwartz’ kernel theorem. (cid:3) Technical lemmas
The following lemma extends [6, Lem. 2.2 and Prop. 5.2].
Lemma 3.1.
Let X denote either M ∞ s ( R d ) , s ≥ , or M ∞ , ( R d ) .(i) Let σ ∈ X and t
7→ A t ∈ Sp ( d, R ) be a continuous mapping defined onthe compact interval [ − T, T ] ⊂ R , T > . For any t ∈ [ − T, T ] , we have σ ◦ A t ∈ X , with (13) k σ ◦ A t k X ≤ C ( T ) k σ k X . (ii) Let σ ∈ X and A, B, C be real d × d matrices with B invertible, and set Φ( x, y ) = 12 xAx + yBx + 12 yCy. There exists a unique symbol e σ ∈ X such that, for any f ∈ S ( R d ) : (14) σ w Z R d e πi Φ( x,y ) f ( y ) dy = Z R d e πi Φ( x,y ) e σ ( x, y ) f ( y ) dy. Furthermore, the map σ e σ is bounded on X .Proof. The case X = M ∞ , ( R d ) is covered by [6, Lem. 2.2]. We prove here theclaim for X = M ∞ s ( R d ). N THE POINTWISE CONVERGENCE IN THE FEYNMAN-TROTTER FORMULA 15 ( i ) For any non-zero window function Φ ∈ S (cid:0) R d (cid:1) and A ∈
Sp ( d, R ) we have k σ ◦ Ak M ∞ s = sup z,ζ ∈ R d |h σ ◦ A , M ζ T z Φ i| h ζ i s = sup z,ζ ∈ R d (cid:12)(cid:12)(cid:12)D σ, M ( A − ) ⊤ ζ T A z (cid:0) Φ ◦ A − (cid:1)E(cid:12)(cid:12)(cid:12) h ζ i s = sup z,ζ ∈ R d (cid:12)(cid:12)(cid:10) σ, M ζ T z (cid:0) Φ ◦ A − (cid:1)(cid:11)(cid:12)(cid:12) (cid:10) A ⊤ ζ (cid:11) s ≤ (cid:13)(cid:13) A ⊤ (cid:13)(cid:13) s k V Φ ◦A − σ k M ∞ s . kAk s k V Φ ◦A − Φ k L s k σ k M ∞ s , where we used the estimate (cid:10) A ⊤ ζ (cid:11) s ≤ (cid:13)(cid:13) A ⊤ (cid:13)(cid:13) s h ζ i s (here kBk denotes the operatornorm of the matrix B ) and the change-of-window Lemma [23, Lem. 11.3.3] ( k · k L s denoting the weighted L norm with weight h ζ i s ) .We now prove the uniformity with respect to the parameter t , when A = A t .The subset {A t : t ∈ [ − T, T ] } ⊂ Sp ( d, R ) is bounded and thus kA t k ≤ C ( T ).Furthermore, n V Φ ◦A − t Φ : t ∈ [ − T, T ] o is a bounded subset of S ( R d ) (this follows atonce by inspecting the Schwartz seminorms of Φ ◦A − t ), hence k V Φ ◦A − Φ k L s ≤ C ( T ).( ii ) The proof is similar to that of the case X = M ∞ , ( R d ) in [6, Prop. 5.2]. Inparticular, e σ is explicitly derived from σ as follows: e σ = U U U σ , where U , U , U are the mappings U σ ( x, y ) = σ ( x, y + Ax ) , U σ ( x, y ) = σ ( x, B ⊤ y ) , c U σ ( ξ, η ) = e πiξη b σ ( ξ, η ) . U and U are isomorphisms of M ∞ s ( R d ), as a consequence of the previous item.For what concerns U , an inspection of the proof of [23, Cor. 14.5.5] shows that anymodulation space M p,qs ( R d ) is invariant under the action of U . (cid:3) We will also make use of the following easy result.
Lemma 3.2.
Let A be a Banach algebra of complex-valued functions on R d withrespect to pointwise multiplication and assume u ∈ A . For any real t and integer n ≥ we have e − i tn u = 1 + i tn u , where u ∈ A and the following estimate holds: k u k ≤ k u k e | t |k u k . Proof.
We have e − i tn u = ∞ X k =0 (cid:18) − i tn (cid:19) k u k k ! = 1 + i tn u with u = − u ∞ X k =0 (cid:18) − i tn (cid:19) k u k ( k + 1)! . We can estimate the norm of u as follows: k u k ≤ k u k ∞ X k =0 | t | k k u k k ( k + 1)! ! = 1 | t | (cid:0) e | t |k u k − (cid:1) ≤ k u k e | t |k u k . (cid:3) Thanks to the subsequent result, we are able to treat Theorem 1.3 as a perturba-tion of Theorem 1.1.
Lemma 3.3.
For any ǫ > and f ∈ M ∞ , ( R d ) , there exist f ∈ C ∞ b ( R d ) and f ∈ M ∞ , ( R d ) such that f = f + f , k f k M ∞ , ≤ ǫ. Proof.
Fix g ∈ S ( R d ) \ { } with k g k L = 1, and set(15) f ( y ) = V ∗ g ( V g f · A R ) ( y ) = Z A R V g f ( x, ξ ) e πiyξ g ( y − x ) dxdξ, in the sense of distributions, where 1 A R denotes the characteristic function of theset A R = (cid:8) ( x, ξ ) ∈ R d : | ξ | ≤ R (cid:9) , and R > ǫ .The integral in (15) actually converges for every y and defines a bounded function.Indeed, setting S ( ξ ) = sup x ∈ R d | V g f ( x, ξ ) | we have S ∈ L ( R d ) by the assumption f ∈ M ∞ , ( R d ), and for any y ∈ R d , | f ( y ) | = (cid:12)(cid:12)(cid:12)(cid:12)Z A R V g f ( x, ξ ) e πiyξ g ( y − x ) dxdξ (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z A R | V g f ( x, ξ ) | | g ( y − x ) | dxdξ ≤ (cid:18)Z R d | g ( y − x ) | dx (cid:19) (cid:18)Z | ξ |≤ R S ( ξ ) dξ (cid:19) ≤ k g k L k S k L . Similarly one shows that all the derivatives ∂ α f are bounded, using that ξ α S ( ξ ) isintegrable on | ξ | ≤ R . Differentiation under the integral sign is permitted becausefor y in a neighbourhood of any fixed y ∈ R d and every N , | V g f ( x, ξ ) ∂ αy [ e πiyξ g ( y − x )] | ≤ C N (1 + | ξ | ) | α | S ( ξ )(1 + | y − x | ) − N , N THE POINTWISE CONVERGENCE IN THE FEYNMAN-TROTTER FORMULA 17 which is integrable in A R . Hence f ∈ C ∞ b ( R d ).Now, let f = f − f = V ∗ g (cid:0) V g f · A cR (cid:1) where in the second equality we used the inversion formula for the STFT (8). Thecontinuity of V ∗ g : L ∞ , ( R d ) → M ∞ , ( R d ) yields k f k M ∞ , = (cid:13)(cid:13) V ∗ g (cid:0) V g f · A cR (cid:1)(cid:13)(cid:13) M ∞ , . (cid:13)(cid:13) V g f · A cR (cid:13)(cid:13) L ∞ , = Z | ξ | >R S ( ξ ) dξ ≤ ǫ provided that R = R ǫ is large enough. (cid:3) As already claimed in the Introduction, we prove that the Sj¨ostrand class includesthe Fourier transforms of (finite) complex measures.
Proposition 3.4.
Let M ( R d ) denote the space of complex Radon measures on R d .The image of M ( R d ) under the Fourier transform is contained in M ∞ , ( R d ) , thatis: F M ( R d ) ⊂ M ∞ , ( R d ) . Proof.
We regard M ( R d ) ⊂ S ′ ( R d ). Therefore, for any non-zero window g ∈ S ( R d )we can explicitly write the STFT of µ : V g µ ( x, ξ ) = h µ, M ξ T x g i = Z R d e − πixξ g ( y − x ) dµ ( y ) . In view of the relation between the Wiener amalgam space W p,q ( R d ) and M p,q ( R d ),the claimed result is equivalent to prove that M ( R d ) ⊂ W ∞ , ( R d ). Indeed, k µ k W ∞ , = Z R d sup ξ ∈ R d | V g µ ( x, ξ ) | dx ≤ Z R d sup ξ ∈ R d Z R d (cid:12)(cid:12) e − πixξ g ( y − x ) (cid:12)(cid:12) d | µ | ( y ) dx = Z R d Z R d (cid:12)(cid:12) g ( y − x ) (cid:12)(cid:12) d | µ | ( y ) dx = Z R d Z R d (cid:12)(cid:12) g ( y − x ) (cid:12)(cid:12) dxd | µ | ( y )= k g k L | µ | ( R d ) . (cid:3) Proof of Theorem 1.1 and Corollary 1.2
Proof of Theorem 1.1.
Recall that H = a w is the Weyl quantization of thereal quadratic form a ( x, ξ ) on R d and we are assuming V ∈ M ∞ s ( R d ), with s > d (the multiplication by V coincides with σ w V , as discussed in Remark 2.3). The proofwill be carried on for t >
0, since the case t < A − t = A − t is − B ⊤ t (cf. [10, Eq. (2.6)]), hence det B t = 0 if andonly if det B − t = 0.Having in mind the framework outlined in the introductory Section 1.2, we startfrom Trotter formula (6). We employ Lemma 3.2 and the notation e − itH = µ ( A t )from Section 2.4 in order to write E n ( t ) = (cid:16) e − i tn H e − i tn V (cid:17) n = (cid:18) µ (cid:0) A t/n (cid:1) (cid:18) i tn V (cid:19)(cid:19) n for a suitable V = V ,n,t . According to Remark 2.3, we identify 1 + i tn V with theWeyl operator with symbol 1 + i tn σ V , where σ V = V ⊗
1. By the assumption V ∈ M ∞ s ( R d ), Remark 2.3 and Lemma 3.2 we have(16) k σ V k M ∞ s ≤ C ( t )for some constant C ( t ) > n . By applying (11) repeatedly, theordered product of operators in E n ( t ) can be expanded as E n ( t ) = " n Y k =1 (cid:18) I + i tn (cid:16) σ V ◦ A − k tn (cid:17) w (cid:19) µ (cid:0) A t/n (cid:1) n = a w n,t µ ( A t ) , where, for any t and n ≥ k a n,t k M ∞ s = (cid:13)(cid:13)(cid:13)(cid:13) n Y k =1 (cid:18) i tn (cid:16) σ V ◦ A − k tn (cid:17)(cid:19) (cid:13)(cid:13)(cid:13)(cid:13) M ∞ s ≤ n Y k =1 (cid:18) tn (cid:13)(cid:13) σ V ◦ A − k tn (cid:13)(cid:13) M ∞ s (cid:19) , where in the first product symbol we mean the twisted product T = t and (16), we then have(17) k a n,t k M ∞ s ≤ (cid:18) tn C ( t ) (cid:19) n ≤ e C ( t ) t , for some new locally bounded constant C ( t ) > n . N THE POINTWISE CONVERGENCE IN THE FEYNMAN-TROTTER FORMULA 19
Since A t is a free symplectic matrix by assumption, by (10) and (14) we explicitlyhave E n ( t ) ψ ( x ) = a w n,t µ ( A t ) ψ ( x )= c ( t ) | det B t | − / Z R d e πi Φ t ( x,y ) f a n,t ( x, y ) ψ ( y ) dy, where Φ t is given in (5) and c ( t ) is a suitable complex factor of modulus 1.Therefore, we managed to write E n ( t ) as an integral operator with kernel e n,t ( x, y ) = c ( t ) | det B t | − / e πi Φ t ( x,y ) f a n,t ( x, y ) , Now, consider the integral kernel u t of the propagator U ( t ) = e − it ( H + V ) and definefor consistency e a t ∈ S ′ ( R d ) such that u t ( x, y ) = c ( t ) | det B t | − / e πi Φ t ( x,y ) e a t ( x, y ) . Since we know by the usual Trotter formula (6) that for any fixed t k E n ( t ) ψ − U ( t ) ψ k L → , ∀ ψ ∈ L ( R d ) , we have E n ( t ) → U ( t ) in L s ( S ( R d ) , S ′ ( R d )), because S ( R d ) ֒ → L ( R d ) ֒ → S ′ ( R d ).As a consequence of Proposition 2.6, we get e n,t → u t in S ′ ( R d ). This is equivalentto(18) f a n,t → e a t in S ′ ( R d ) . Therefore, for any non-zero Ψ ∈ S ( R d ) we have pointwise convergence of the cor-responding short-time Fourier transforms: for any fixed ( z, ζ ) ∈ R d ,(19) V Ψ f a n,t ( z, ζ ) = h f a n,t , M ζ T z Ψ i → h e a t , M ζ T z Ψ i = V Ψ e a t ( z, ζ ) . By (17) and Lemma 3.1 we see that the sequence f a n,t , for any fixed t , is bounded in M ∞ s ( R d ). Hence, there exists a constant C = C ( t ) independent of n such that(20) | V Ψ f a n,t ( z, ζ ) | ≤ C h ζ i − s , ∀ z, ζ ∈ R d . Combining this estimate with (19) immediately yields e a t ∈ M ∞ s ( R d ) as well, hencethe first claim of Theorem 1.1.For the remaining part, we argue as follows: choose a non-zero window Ψ ∈ C ∞ c ( R d ) and set Θ ∈ C ∞ c ( R d ) with Θ = 1 on suppΨ; for any fixed z ∈ R d and0 < r < s − d , we have (cid:13)(cid:13) F (cid:2) ( e n,t − u t ) T z Ψ (cid:3)(cid:13)(cid:13) L r = | det B t | − / (cid:13)(cid:13) F (cid:2) e πi Φ t ( f a n,t − e a t ) T z Ψ (cid:3)(cid:13)(cid:13) L r = | det B t | − / (cid:13)(cid:13) F (cid:2)(cid:0) T z Θ e πi Φ t (cid:1) ( f a n,t − e a t ) T z Ψ (cid:3)(cid:13)(cid:13) L r = | det B t | − / (cid:13)(cid:13) F (cid:2) T z Θ e πi Φ t (cid:3) ∗ F (cid:2) ( f a n,t − e a t ) T z Ψ (cid:3)(cid:13)(cid:13) L r . | det B t | − / (cid:13)(cid:13) F (cid:2) T z Θ e πi Φ t (cid:3)(cid:13)(cid:13) L r (cid:13)(cid:13) F (cid:2) ( f a n,t − e a t ) T z Ψ (cid:3)(cid:13)(cid:13) L r , where the convolution inequality in the last step is an easy consequence of Peetre’sinequality .Clearly, T z Θ e πi Ψ t ∈ C ∞ c ( R d ), while (cid:13)(cid:13) F (cid:2) ( f a n,t − e a t ) T z Ψ (cid:3)(cid:13)(cid:13) L r → |F (cid:2) ( f a n,t − e a t ) T z Ψ (cid:3) |h ζ i r = | V Ψ ( f a n,t − e a t ) ( z, ζ ) | h ζ i r ≤ C h ζ i r − s ∈ L ( R d ) , because s − r > d , where in the last inequality we used (20) and the fact that e a t ∈ M ∞ s ( R d ).This gives the claimed convergence in ( F L r ) loc ( R d ).To conclude, we have that (cid:13)(cid:13) ( e n,t − u t ) T z Ψ (cid:13)(cid:13) L ∞ ≤ k V Ψ ( e n,t − u t ) ( z, · ) k L → , and in particular this yields uniform convergence on compact subsets: for any com-pact K ⊂ R d , choose Ψ ∈ S ( R d ), Ψ = 1 on K .4.2. Proof of Corollary 1.2.
The proof of Corollary 1.2 is then immediate, since C ∞ b ( R d ) = T s ≥ M ∞ s ( R d ) and C ∞ ( R d ) = T r> ( F L r ) loc ( R d ); we leave the easyproof of the latter equality to the interested reader.5. Proof of Theorem 1.3
We now assume V ∈ M ∞ , ( R d ). Therefore for an arbitrary ǫ >
0, Lemma 3.3allows us to write V = V + V , with V ∈ C ∞ b ( R d ) and V ∈ M ∞ , ( R d ) with k V k M ∞ , ≤ ǫ and clearly k V k M ∞ , ≤ k V k M ∞ , + k V k M ∞ , ≤ k V k M ∞ , + ǫ ≤ k V k M ∞ , , assuming, from now on, ǫ ≤ e − i tn ( V + V ) = 1 + ∞ X k =1 k ! (cid:18) − i tn (cid:19) k ( V + V ) k = 1 + i tn V ′ + i tn V ′ , where we set V ′ = − V ∞ X k =1 k ! (cid:18) − i tn (cid:19) k − V k − , Namely, h x − y i r ≤ C r h x i r h y i r , for x, y ∈ R d , r ≥ N THE POINTWISE CONVERGENCE IN THE FEYNMAN-TROTTER FORMULA 21 V ′ = − ∞ X k =1 k ! (cid:18) − i tn (cid:19) k − (( V + V ) k − V k ) . Now, fix once for all any s > d . The norms of V ′ = V ′ ,n,t and V ′ = V ′ ,n,t can beestimated as follows for any t > k V ′ k M ∞ , ≤ k V k M ∞ , e t k V k M ∞ , ≤ (1 + k V k M ∞ , ) e t ( k V k M ∞ , ) =: C ( t ) , (22) k V ′ k M ∞ s ≤ k V k M ∞ s e t k V k M ∞ s =: C ( t, ǫ ) . Similarly, using the elementary inequality( a + b ) k − a k ≤ kb ( a + b ) k − , a, b ≥ , k ≥ , we obtain(23) k V ′ k M ∞ , ≤ k V k M ∞ , e t ( k V k M ∞ , + k V k M ∞ , ) ≤ ǫe t ( k V k M ∞ , ) =: ǫ C ( t ) . Here C ( t ) and C ( t ) are independent of n and ǫ and C ( t, ǫ ) is independent of n .The approximate propagator E n ( t ) thus becomes E n ( t ) = (cid:16) e − i tn H e − i tn ( V + V ) (cid:17) n = (cid:18) µ (cid:0) A t/n (cid:1) (cid:18) i tn V ′ + i tn V ′ (cid:19)(cid:19) n , and similar arguments to those of the previous section yield E n ( t ) = " n Y k =1 (cid:18) I + i tn (cid:16) σ V ′ ◦ A − k tn (cid:17) w + i tn (cid:16) σ V ′ ◦ A − k tn (cid:17) w (cid:19) µ (cid:0) A t/n (cid:1)(cid:1) n (24) =: (cid:2) a w n,t + b w n,t (cid:3) µ ( A t ) , where we set a n,t = n Y k =1 (cid:18) i tn (cid:16) σ V ′ ◦ A − k tn (cid:17)(cid:19) , and in the latter product we mean the twisted product a w n,t can be estimated as in the proof of Theorem 1.1; in particular, using(22), we get(25) k a n,t k M ∞ s ≤ C ( t, ǫ )cf. (17).In order to estimate the M ∞ , norm of the remainder b n,t , it is useful the followingresult, which can be easily proved by induction on n . Lemma 5.1.
Let A be a Banach algebra. For any u , . . . , u n , v , . . . , v n ∈ A , with k u i k ≤ R and k v i k ≤ S for any i = 1 , . . . , n and some R, S > , and setting w k = u k + v k , we have n Y k =1 ( u k + v k ) = u u . . . u n + z n , where z n = v w . . . w n + u v w . . . w n + . . . + u u . . . u n − v n − w n + u u . . . u n − v n , and therefore k z n k ≤ nS ( R + S ) n − . Setting u k = 1 + i tn (cid:16) σ V ′ ◦ A − k tn (cid:17) , v k = i tn (cid:16) σ V ′ ◦ A − k tn (cid:17) , k = 1 , . . . , n, and applying Lemma 3.1 with T = t , and (21) and (23), we get k u k k M ∞ , = (cid:13)(cid:13) i tn (cid:16) σ V ′ ◦ A − k tn (cid:17) (cid:13)(cid:13) M ∞ , ≤ tn (cid:13)(cid:13) σ V ′ ◦ A − k tn (cid:13)(cid:13) M ∞ , ≤ tn C ( t ) , k v k k M ∞ , = tn (cid:13)(cid:13) σ V ′ ◦ A − k tn (cid:13)(cid:13) M ∞ , ≤ tn C ( t ) ǫ, for some locally bounded constant C ( t ) > n and ǫ . Therefore, byLemma 5.1,(26) k b n,t k M ∞ , ≤ n tn C ( t ) ǫ (cid:16) tn C ( t ) (cid:17) n − ≤ ǫtC ( t ) e tC ( t ) . Following the pathway of the proof of Theorem 1.1, we write E n ( t ) as an integraloperator with kernel e n,t ( x, y ) = c ( t ) | det B t | − / e πi Φ t ( x,y ) (cid:0) f a n,t + f b n,t (cid:1) ( x, y )= c ( t ) | det B t | − / e πi Φ t ( x,y ) k n,t ( x, y ) , that is k n,t = f a n,t + f b n,t , and the Trotter formula (6) combined with Proposition 2.6imply that k n,t → k t in S ′ ( R d ), where the distribution k t is conveniently introducedto rephrase the integral kernel u t of the propagator U ( t ) = e − it ( H + V ) as u t ( x, y ) = c ( t ) | det B t | − / e πi Φ t ( x,y ) k t ( x, y ) . By repeating this argument with V = 0 (hence f b n,t = 0 and k n,t = f a n,t ) we seethat f a n,t converges in S ′ ( R d ) as well, hence f b n,t converges in S ′ ( R d ) by difference.Therefore, for any non-zero Ψ ∈ S ( R d ) the functions V Ψ f a n,t and V Ψ f b n,t convergepointwise in R d .We need a technical lemma at this point. N THE POINTWISE CONVERGENCE IN THE FEYNMAN-TROTTER FORMULA 23
Lemma 5.2.
Let F n and G n be two sequences of complex-valued functions on R d such that F n → F , G n → G pointwise, and assume | F n | ≤ H ∈ L ( R d ) and k G n k L ≤ ǫ for any n ∈ N . Then, lim sup n →∞ k F n + G n − ( F + G ) k L ≤ ǫ. Proof.
First, notice that k G k L ≤ ǫ by Fatou’s lemma. Now, k F n + G n − ( F + G ) k L ≤ k F n − F k L + k G n − G k L , where the first term on the right-hand side goes to zero by dominated convergence,while for the other one we have k G n − G k L ≤ ǫ . The desired conclusion is thenimmediate. (cid:3) For any fixed z ∈ R d , set F n ( ζ ) = V Ψ f a n,t ( z, ζ ) and G n ( ζ ) = V Ψ f b n,t ( z, ζ ).By Lemma 3.1 and (25) we havesup ζ ∈ R d h ζ i s | F n ( ζ ) | . k f a n,t k M ∞ s . k a n,t k M ∞ s ≤ C ( t, ǫ ) . Similarly, by Lemma 3.1 and (26), k G n k L . k f b n,t k M ∞ , . k b n,t k M ∞ , ≤ ǫ C ( t ) . These estimates yield two results: on the one hand, the first claim of Theorem 1.3is proved. On the other hand, the assumptions of Lemma 5.2 are satisfied: we have( F n + G n ) ( ζ ) = V Ψ k n,t ( z, ζ ) and ( F + G ) ( ζ ) = V Ψ k t ( z, ζ ), and therefore we obtainlim sup n →∞ (cid:13)(cid:13) F (cid:2) ( k n,t − k t ) T z Ψ (cid:3)(cid:13)(cid:13) L ≤ ǫ C ( t ) . Since ǫ can be made arbitrarily small and the left-hand side is independent of ǫ , weconclude that lim n →∞ (cid:13)(cid:13) F (cid:2) ( k n,t − k t ) T z Ψ (cid:3)(cid:13)(cid:13) L = 0 , in particular k n,t → k t in ( F L ) loc ( R d ).Finally, with the help of a suitable bump function Θ as in the preceding section,for any fixed z ∈ R d we deduce (cid:13)(cid:13) F (cid:2) ( e n,t − u t ) T z Ψ (cid:3)(cid:13)(cid:13) L ≤ | det B t | − / (cid:13)(cid:13) F (cid:2)(cid:0) T z Θ e πi Φ t (cid:1)(cid:3)(cid:13)(cid:13) L (cid:13)(cid:13) F (cid:2) ( k n,t − k t ) T z Ψ (cid:3)(cid:13)(cid:13) L , and thus (cid:13)(cid:13) F (cid:2) ( e n,t − u t ) T z Ψ (cid:3)(cid:13)(cid:13) L → . This gives e n,t → u t in ( F L ) loc ( R d ) and therefore uniformly on compact subsets of R d . References [1] S. Albeverio, and R. Høegh-Krohn: Oscillatory integrals and the method of stationary phasein infinitely many dimensions, with applications to the classical limit of quantum mechanics.I.
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