Slow dispersion in Floquet-Dirac Hamiltonians

SLO W DISPERSION IN FLOQUET-DIRA C HAMIL TONIANS ANTHONY BLOCH, AMIR SAGIV, AND STEF AN STEINERBERGER Abstract. W e study dispersive decay for non-autonomous Hamiltonian sys- tems. While the general theory for dispersion in such non-autonomous systems is largely op en, it w as sho wn [33] that there exists a time-perio dically forced one-dimensional Dirac equation with unusually slow dispersive decay rate of t ´ 1 { 5 . It is to b e exp ected that suc h behavior is not generic and requires a very particular forcing term; we provide a more general ansatz and systematic procedure to construct such an equation with a dispersive deca y rate no faster than t ´ 1 { 10 . Our limitations are purely algebraic and it stands to reason that arbitrarily slow decay , t ´ ε for every ε ą 0, should b e ac hiev able. 1. Introduction 1.1. Deca y estimates. Dispersion, the breakdo wn of wa ves ov er time, is a funda- men tal feature of Hamiltonian wa v e dynamics. Disp ersiv e decay estimates quantify the spreading of w av es under unitary evolution and pla y a cen tral role in the anal- ysis of linear and nonlinear disp ersiv e equations. The simplest example is perhaps the one-dimensional Sc hr¨ odinger equation iu t “ u xx whic h admits a deca y estimate } u p t, ¨q} L 8 x ď c | t | 1 { 2 } u p 0 , ¨q} L 1 x , or the one-dimensional Airy equation u t “ u xxx whic h satisfies a similar inequality with a t ´ 1 { 3 rate. These t w o estimates are classical, fairly easy to pro v e and corre- sp ond to our in tuitiv e understanding: a higher degeneracy in the disp ersion relation means that different frequencies tra vel at similar speed, th us yielding slo wer disper- sion. In particular, higher deriv ativ es imply slo wer (polynomial) rate of dispersion for lo w frequencies, which mov e more slowly . The question of disp ersiv e decay b ounds in autonomous and linear Hamiltonian PDEs has been studied extensiv ely , e.g., for time-independent (autonomous) Schr¨ odinger Hamiltonians [28, 29, 30, 39] and, relev an t to this w ork, Dirac Hamiltonians [11, 13, 14, 16, 17, 18, 23, 31, 32]. The dispersive behavior of non-autonomous systems remains largely open, espe- cially when the time-dep enden t term is not lo calized in space. The a v ailable dis- p ersiv e estimates concern Schr¨ odinger equations in dimensions d ě 3 and, crucially , treat the time-dep enden t term as a p erturbation of an autonomous Hamiltonian in some sense [4, 5, 20, 21, 35]. In general non-autonomous settings, many of the tec hniques a v ailable in the autonomous settings no longer apply: one cannot “read” the dynamics off of the instan taneous Hamiltonian and its sp ectral prop erties. AB is supp orted in part b y NSF Grant No. DMS-2103026 and AF OSR Gran ts No. F A9550-23- 1-0215 and No. F A9550-23-1-0400. AS is partially supp orted by NSF Grant No. DMS-2508811. 1 2 1.2. A non-autonomous system. In this work we consider the one-dimensional time-p eriodically forced Dirac equation i B t α p t, x q “ p iσ 3 B x ` ν p t qq α p t, x q , (1.1a) α p 0 , x q “ f P L 2 p R ; C 2 q , (1.1b) where ν p t q is a b ounded T -perio dic 2 ˆ 2 Hermitian matrix-v alued function, and σ 3 is the standard P auli matrix; see (2.1). If the function ν p t q is time-indep enden t, ν p t q “ mσ 1 , then this is the so-called massive Dirac equation [40], for which the follo wing estimate holds } α p t, ¨q} L 8 x ď c t 1 { 2 }x D y 3 { 2 f } L 1 , b y means of F ourier analysis, where D is the Hamiltonian i B x σ 3 ` mσ 1 [15]. But what if ν p t q is non-c onstant in time? This t yp e of equation arises as the effectiv e (homogenized) dynamics of Flo quet materials [25, 19, 37], an emergent and very activ e area of b oth theoretical [2, 3, 22, 25, 36, 38, 37] and exp erimen tal researc h, with applications in condensed matter ph ysics [8], photonics [34], and acoustics [41]. Theorem (informally , Kraisler, Sagiv, W einstein [33]) . Ther e exists a function ν p t q which assumes the values mσ 1 and ´ mσ 1 p erio dic al ly in time so that the (generic) disp ersive de c ay r ate is no faster than t ´ 1 { 3 . Mor e over, ther e exist (nongeneric) choic es of p ar ameters wher e it is no faster than t ´ 1 { 5 . W e note that t ´ 1 { 5 is remark ably slo w decay for an equation of this t yp e with no kno wn analog in autonomous Schr¨ odinger or Dirac equations. W e were motiv ated b y the follo wing question: is a slo w er deca y rate p ossible? The main contribution of this w ork is to demonstrate that time-perio dic forcing allo ws one to systematically engineer high-order degeneracies in the asso ciated Flo quet exp onen ts (disp ersion curv es). These degeneracies directly control the stationary phase structure of the ev olution op erator and therefore the rate of disp ersiv e deca y . Theorem 1.1 (Main Result) . Ther e exists a choic e of ν p t q such that the L 1 Ñ L 8 de c ay r ate is no faster than t ´ 1 { 10 . Mor e pr e cisely: for any r ě 0 , any ine quality of the form } α p t, ¨q} L 8 x ď c t σ }x i B x y r α p 0 , ¨q} L 1 x , may only hold if 0 ă σ ď 1 { 10 . The pro of of this Theorem app ears at the end of Section 2. W e note that no amoun t of smo othing, x i B x y ´ r , can “fix” the low rate of decay . In fact, as w e shall see in Theorem 2.1, for sufficien tly low-energy initial data, this rate app ears in its asymptotic expansion as the result of 10-th order p olynomial degeneracy of the disp ersion relation. Our approac h is completely new and can presumably b e used to obtain similar results for other equations: a zero of high order in the Flo quet exp onent is equiv alent to having a region of extraordinary flatness in the disp ersion relation (see Sec. 2). W e were motiv ated by the idea that if one had a k -dimensional v ector space of ‘generic’ potentials at their disposal, then one should b e able to force the disp ersion relation to v anish to order 2 k somewhere (b y a simple counting heuristic, assuming ev erything b eha v es linearly , which of course need not actually be the case). Since the unitary L 2 p R ; C 2 q flow can b e F ourier-transformed into a parametric family of 3 SU p 2 q matrices, the problem can b e recast in algebraic terms. W e thus pro ceed in t wo steps: (1) By carrying out extensive computations, our ansatz reduces the problem to sho wing that an explicit system of four nonlinear equations in four v ariables has a solution; the most complicated equation is the sum of 295 terms, and there is no indication that the solution admits a simple closed form. (2) W e then pro v e the existence of a solution b y first numerically iden tifying an appr oximate solution (up to an error of 10 ´ 15 ). W e then use the Newton– Kan torovic h theorem to prov e that Newton’s metho d, initialized at this p oin t, con verges to an actual ro ot, establishing the existence of a solution. This construction is carried out for a set of p oten tials ν p t q parametrized b y k “ 4 v ariables, allowing us to eliminate the first nine deriv ativ es, with the tenth deriv ative then being the dominant term. W e do not see an y reason to b elieve that there is anything special about k “ 4. How ev er, the algebraic obstruction persists: as k ě 2 and the num ber of “tuning parameters” in ν p t q increases, the num b er of terms in the asso ciated algebraic equations grows combinatorially. W e therefore conjecture: Conjecture. F or any ε ą 0 , ther e exists a choic e of ν p t q such that the L 1 Ñ L 8 de c ay r ate is no faster than t ´ ε . Mor e pr e cisely: for any r ě 0 , } α p t, ¨q} L 8 x ď c t σ }x i B x y r α p 0 , ¨q} L 1 x r e quir es 0 ă σ ď ε. One could make a stronger conjecture: that ν p t q could actually b e chosen to b e piecewise constant, assuming only the t wo v alues ˘ σ 1 and to b e defined on ď 100 { ε in terv als (which are then p erio dically extended), with only the length of these interv als b e tunable parameters. By the constructive nature of our approach, ho wev er, it is not clear how to show that this approach yields the conjecture for all ε ą 0 (rather than for any giv en ε ), and additional ideas are required. 2. Obt aining dispersive deca y estima tes Let the Pauli matrices b e σ 0 “ I 2 ˆ 2 , σ 1 “ ˆ 0 1 1 0 ˙ , σ 2 “ ˆ 0 ´ i i 0 ˙ , σ 3 “ ˆ 1 0 0 ´ 1 ˙ , (2.1) and set ν p t q “ m p t q σ 1 , where m p t q is a scalar function to b e determined. Since (1.1) is translation inv ariant in space, it can b e F ourier transformed to yield the follo wing ξ P R -parametrized family of 2 ˆ 2 ODEs: i d dt ˆ α p t ; ξ q “ r ξ σ 3 ` m p t q σ 1 s ˆ α p t ; ξ q , ˆ α p 0; ξ q “ ˆ f p ξ q , (2.2) where ˆ f is the F ourier T ransform of the initial data, f , see (1.1). W riting the propagator of (2.2) as ˆ U p t ; ξ q ˆ α 0 p ξ q “ ˆ α p t ; ξ q , the solution of (1.1) can b e written as α p t, x q “ r U p t q f sp x q “ 1 ? 2 π ż R e iξx ˆ U p t ; ξ q f p ξ q dξ . Since m p t q is T -perio dic, to gain insight in to the long-time dynamics w e consider the mono drom y op erator M “ U p T q . Thus for every n ě 0, α p nT , x q “ r M n f sp x q “ 1 ? 2 π ż R e iξx ˆ M n p ξ q ˆ f p ξ q dξ , 4 where ˆ M p ξ q : “ ˆ U p T ; ξ q is the p erio d propagator (mono dromy op erator) associated with (2.2) or, equiv alen tly , the F ourier transform of the op erator M . F or eac h fixed ξ , the instan taneous Hamiltonian H p t ; ξ q ” ξ σ 3 ` m p t q σ 1 is Hermitian, and so for all t , the propagator is a 2 ˆ 2 unitary matrix with determinan t 1, i.e., ˆ U p t ; ξ q P SU p 2 q ; therefore, in particular ˆ M p ξ q P SU p 2 q . In addition, since T r p H p t ; ξ qq “ 0, the p erio d-propagator ˆ M p ξ q has tw o complex-conjugate eigenv alues λ ˘ p ξ q “ exp p˘ iθ p ξ qq and a unitary diagonalizing matrix P p ξ q . Th us the solution at in teger- p eriod times can b e written as the oscillatory integral α p nT , x q “ 1 ? 2 π ż R e iξx P p ξ q ˆ e inθ p ξ q 0 0 e ´ inθ p ξ q ˙ P ˚ p ξ q ˆ f p ξ q dξ . (2.3) This leads to the following strategy for obtaining slo w disp ersion: (1) Construct ν p t q so that θ p j q p 0 q “ 0 for 2 ď j ď k ´ 1 while θ p k q p 0 q ‰ 0; see Section 3. This is the main tec hnical nov elt y of this w ork. (2) Use oscillatory in tegral expansions to show that band-limited wa vepac k ets (i.e. data f with supp ˆ f Ď r´ d, d s ) disp erse at rate t ´ 1 { k . In the remainder of this section w e implement Step (2). T o prov e the main result, w e will pro v e the follo wing asymptotic expansion. Theorem 2.1. Supp ose ν p t q is such that θ p j q p 0 q “ 0 for 2 ď j ď k ´ 1 while θ p k q p 0 q ‰ 0 . Then ther e exists C , d ą 0 and a unitary 2 ˆ 2 matrix P such that for al l initial data f with supp ˆ f Ă p´ d, d q , } α p nT , ¨q} L 8 x „ C n 1 { k ż R p P f qp x q dx , as n Ñ 8 . Since the rates w e derive are very slow, obtaining asymptotic expansions (rather than upp er b ounds) emphasizes that these rates are inherent to the dynamics. Since such expansions are obtained for a broad class of data (for which ş P f ‰ 0), a general dispersive deca y estimate would ha v e a rate slow er or equal to t ´ 1 { k . Since the expansion in Theorem 2.1 is v alid for low- ξ data, no amoun t of smo othing can “fix” that slow rate. Hence, the main result, Theorem 1.1, follows. Pr o of of The or em 2.1. W e first recall the follo wing expansion [26, Eq. 7.7.31]: for ev en k and u P C 8 0 p R q , ż R e iω t k u p t q dt “ C k, 0 ω ´ 1 { k u p 0 q ` O p ω ´ 3 { k q , as ω Ñ `8 , where C k, 0 “ k ´ 1 Γ p k ´ 1 q e π i {p 2 k q . By a standard c hange-of-v ariables argument (see e.g., [33, App endix A]), one obtains the following v ariant: if λ P C 8 satisfies λ 1 p 0 q “ ¨ ¨ ¨ “ λ p k ´ 1 q p 0 q “ 0 and λ p k q p 0 q ‰ 0, then there exists C k ‰ 0 suc h that for u P C 8 0 p R q , ż R e iω λ p t q u p t q dt “ e iω λ p 0 q C k ` ω λ p k q p 0 q ˘ ´ 1 { k u p 0 q ` O p ω ´ 3 { k q , as ω Ñ `8 . (2.4) Analogous expansions are found e.g., in [26] for o dd k , but as we shall see in Sec. 3.2, in this work k is alwa ys ev en. Let b ą 0 b e sufficiently small such that 5 θ p k q p ξ q ‰ 0 for ξ P p´ b, b q , and let f P L 2 p R ; C 2 q satisfy supp ˆ f Ď p´ b, b q . W rite P ˚ p ξ q ˆ f p ξ q “ p ˆ ϕ ` p ξ q , ˆ ϕ ´ p ξ qq J . Then p M n f qp x q “ 1 2 π ż R P p ξ q ˆ e ` inθ p ξ q 0 0 e ´ inθ p ξ q ˙ P ˚ p ξ q ˆ f p ξ q e iξx dξ “ 1 2 π b ż ´ b P p ξ q ˆ e ` inθ p ξ q ˆ ϕ ` p ξ q e ´ inθ p ξ q ˆ ϕ ´ p ξ q ˙ e iξx dξ . Let s 0 : “ ´ θ 1 p 0 q and c hoos e x “ ns 0 . F or the ` phase, define Φ ` p ξ q : “ ξ s 0 ` θ p ξ q so that B ξ Φ ` p 0 q “ s 0 ` θ 1 p 0 q “ 0 . Assuming Theorem 3.1 giv es θ p j q p 0 q “ 0 for 2 ď j ď k ´ 1 and θ p k q p 0 q ‰ 0, we hav e B j ξ Φ ` p 0 q “ θ p j q p 0 q “ 0 p 2 ď j ď k ´ 1 q , B k ξ Φ ` p 0 q “ θ p k q p 0 q ‰ 0 . Applying (2.4) with ω “ n to the oscillatory in tegral defining the first comp onen t (and similarly for the remaining comp onen ts/phases) yields an asymptotic contri- bution of size n ´ 1 { k . In particular, one obtains r M n f sp ns 0 q “ 1 2 π b ż ´ b P p ξ q ˆ e ` inθ p ξ q` iξx ˆ ϕ ` p ξ q e ´ inθ p ξ q` iξx ˆ ϕ ´ p ξ q ˙ dξ “ 1 2 π C k p θ p k q p 0 qq ´ 1 { k P p 0 q ˆ ˆ ϕ ` p 0 q ˆ ϕ ´ p 0 q ˙ n ´ 1 { k ` O p n ´ 3 { k q . Since ˆ h p 0 q “ ş R h p x q dx , this iden tifies the leading co efficien t in terms of the spatial in tegrals of the comp onen ts of f , where P “ P p ξ “ 0 q , completing the pro of. □ Finally , the proof of Theorem 1.1 is a simple consequence of the abov e expansion, and the analysis b elo w in Section 3. pr o of of The or em 1.1. In Section 3, we construct ν p t q such that θ p k q p 0 q “ 0 for all 2 ď k ď 9, but θ p 10 q p 0 q ‰ 0, see Theorem 3.1. Therefore, b y the asymptotic expan- sion, Theorem 2.1, w e ha ve that there exists d ą 0, a unitary matrix P , and initial data, f P L 2 p R ; C 2 q , with (i) supp p ˆ f q Ă p´ d, d q , and (ii) ş R p P f qp x q dx ‰ 0 , 8 , for whic h } α p nT , ¨q} 8 „ n ´ 1 { 10 . Therefore, any upper b ound on } α p t, ¨q} 8 m ust decay no faster than t ´ 1 { 10 , and smo othing of the initial data cannot ameliorate this. □ Remark 2.2. By F ourier T r ansforming the PDE (1.1) into the family of ODEs (2.2) , the pr oblem c an b e c ast in the language of c ontr ol the ory [10] : given the ODE i d dt ˆ α p t ; ξ q “ ξ σ 3 ˆ α , one tries to minimize disp ersion by adding the c ontr ol ν p t q . Sinc e the flow is c onstr aine d to the Lie Gr oup SU p 2 q for e ach ξ , this is a pr oblem in ge ometric c ontr ol the ory [6, 7] . Usual ly, one is inter este d in c ontr ol ling the state, i.e., driving α to a p oint or set at finite time. Her e, we se ek to c ontr ol the ξ -derivatives of the flow (or mor e sp e cific al ly, of the tr ac e of the Flow op er ator). T o the b est of our know le dge, such pr oblems have not b e en pr eviously studie d in c ontr ol the ory. R elate d pr oblems involving p ar ametric al ly for c e d systems and gr owth r ates have b e en studie d in the c ontext of Hil l’s e quations, se e e.g [1] . 6 3. Existence of solutions 3.1. Preliminaries. W e wish to study the Dirac equation (1.1) with piecewise constan t, alternating mass: fix t 1 , . . . , t m ą 0, define τ 0 “ 0, and for 1 ď k ď m set τ k “ ř k j “ 1 t j . Then ν p t q “ m p t q σ 1 is τ m -p eriodic with m p t q “ # 1 , t P r τ 2 k , τ 2 k ` 1 q , 0 ď k ď m { 2 , ´ 1 , t P r τ 2 k ` 1 , τ 2 k ` 2 q , 0 ď k ă m { 2 . W e no w wan t to express the mono drom y op erator (p erio d flow) asso ciated with the F ourier-T ransformed ODEs (2.2), ˆ M p ξ q , in an explicit, algebraic wa y . In what follo ws, we use basic properties of the Sp ecial Unitary Group of order 2, SU p 2 q , and its asso ciated Lie Algebra, su p 2 q . F or a thorough exp osition, see e.g., [24]. Define t wo functions a ˘ : R Ñ su p 2 q via a ˘ p ξ q ” ˘ iσ 1 ` iξ σ 3 . Next, to define the propagator on an y time interv al r τ 2 k , τ 2 k ` 1 s , first recall the standard p olar representation of exponents of elemen ts in su p 2 q : for every S P SU p 2 q there exists s P su p 2 q as s “ ř 3 j “ 1 is j σ j suc h that S “ exp r s s “ cos p| s |q σ 0 ` sin p| s |q | s | s , (3.1) where, by abuse of notation | s | “ |p s 1 , s 2 , s 3 q| , and σ 0 “ I 2 ˆ 2 as usual. Applying the p olar representation (3.1) to a ‹ for ‹ P t` , ´u , then for any t ą 0 consider the asso ciated SU p 2 q -v alued propagator map g ‹ p t, ξ q “ exp ` t a ‹ p ξ q ˘ “ cos p ω t q σ 0 ` i sin p ω t q ω a ˘ p ξ q , ω p ξ q “ a ξ 2 ` 1 . (3.2) Th us, the unit-propagator (mono drom y), ˆ M p ξ q , is a w ord of length m ě 1 defined b y the matrix pro duct ˆ M p ξ q “ g ε m p t m , ξ q ¨ ¨ ¨ g ε 1 p t 1 , ξ q , ε j P t` , ´u , t j ‰ 0 , where the signs alternate, i.e., ε j ε j ` 1 “ ´ for all 1 ď j ă m . Recall that, since ˆ M p ξ q P SU p 2 q , its t w o eigenv alues µ ˘ p ξ q are on the unit circle. F urthermore, since T r p a ˘ p ξ qq “ 0, then it follows from ODE theory [9] that µ ` “ ¯ µ ´ “ exp r iθ p ξ qs , as defined in Section 2. Noting that T r p s q “ 0, for every s P su p 2 q , and using the p olar represen tation in (3.1), w e see that F p ξ q ” 1 2 T r p ˆ M p ξ qq “ cos ` θ p ξ q ˘ . 3.2. The Construction. Henceforward, we will b e interested in words of length 4 ˆ M p ξ q “ g 1 p t 1 , ξ q g ´ 1 p t 2 , ξ q g 1 p t 3 , ξ q g ´ 1 p t 4 , ξ q . Theorem 3.1. Ther e exist t 1 , t 2 , t 3 , t 4 ą 0 such that @ 2 ď k ď 9 θ p k q p 0 q “ 0 . The numerology can be understo od as follows. W e start by trying to construct M in such a w ay that the deriv atives of the trace v anish to very high order. The trace is an ev en function, and so F p 2 k ` 1 q p 0 q “ 0 for all k ě 1 (Lemma 3.2). Therefore, 7 the p o wer series expansion of F around ξ “ 0 only consists of even p o w ers; for t 1 , t 2 , t 3 , t 4 fixed, we ha v e F p ξ q “ a 0 p t 1 , t 2 , t 3 , t 4 q ` a 2 p t 1 , t 2 , t 3 , t 4 q ξ 2 ` a 4 p t 1 , t 2 , t 3 , t 4 q ξ 4 ` a 6 p t 1 , t 2 , t 3 , t 4 q ξ 6 ` a 8 p t 1 , t 2 , t 3 , t 4 q ξ 8 ` O p ξ 10 q , as ξ Ñ 0. This means that v anishing up to 10-th order requires us to find a wa y to solv e the four equations simultaneously . Lemma 3.2. F p ξ q “ 1 2 T r p ˆ M p ξ qq is an even function. Pr o of. Recall that eac h g ε j p t j ; ξ q is giv en b y its polar represen tation (3.2). Since the trace of σ 1 , σ 2 , σ 3 is zero, w e only get contributions to the trace from a pro duct of ev en num ber of terms con taining σ 3 . Thus, eac h such “letter” g is in v ariant under taking p´ ξ , ε j q ÞÑ p ξ , ´ ε j q . But the latter symmetry change only c hanges the sin part of (3.2), which do es not contribute to the trace. Th us F p ξ q “ F p´ ξ q . □ Lemma 3.3. Ther e exist t 1 , t 2 , t 3 , t 4 ą 0 satisfying t 1 ´ t 2 ` t 3 ´ t 4 ‰ 0 such that @ 1 ď k ď 4 a 2 k p t 1 , t 2 , t 3 , t 4 q “ 0 . The key to our ability to write down a 2 k p t 1 , t 2 , t 3 , t 4 q explicitly is the fact that b oth the “letters” g p ξ q and the full “word” ˆ M p ξ q are each elements in SU p 2 q , and therefore admit a p olar representation, see (3.1) and (3.2). Recall the fundamen tal relation for the P auli matrices: σ i σ j “ δ ij σ 0 ` ϵ ij k σ k , @ i, j, k P t 1 , 2 , 3 u , where δ ij is the Kroneck er Delta and ϵ ij k is the Levi-Civita symbol. Therefore, writing ˆ M p ξ q “ 4 ź j “ 1 „ cos p ω t j q σ 0 ` i sin p ω t j q ω a ε j p ξ q ȷ , the terms that contribute to the trace, F p ξ q , are (i) the pro duct of all four cosine terms, and (ii) terms with an even num b er of sine terms and an even n umber of cosine terms. Ha ving collected these terms, we are left with the elementary exercise of expanding ω p ξ q , cos p ω p ξ q t j q , sin p ω p ξ q t j q , 1 { ω p ξ q in T aylor series near ξ “ 0, at which p oint the algebra has help ed us reduce the analytic problem to one of elab orate b ookkeeping, whic h we p erformed using the soft ware Mathematica (though we could do it, in principle, by hand). These equations are very nonlinear in t 1 , . . . , t m , and the num b er of terms in- creases com binatorially with b oth m and k . F rom an algebraic/combinatorial point of view, writing a formula for the trace for general m terms word is interesting. In the case of Theorem 1.1, ho w ev er, w e ha ve 4 terms, and so the complexit y of writing the trace, F , is tractable. W e therefore hav e four equations in four unkno wns, and a bit of optimism suggests that unless the problem is malicious, there should b e a solution. This is indeed the case. Let us first sho w that, indeed, the flatness of F yields the degeneracy in θ : Pr o of of The or em 3.1 using L emma 3.3. T ransferring the flatness of F near zero to that of θ happens as follows: consider F p 0 q “ a 0 p t 1 , t 2 , t 3 , t 4 q “ cos p t 1 ´ t 2 ` t 3 ´ t 4 q . 8 If this v alue is differen t from ˘ 1, then θ p 0 q is not a multiple of π and d dξ cos p θ p ξ qq ˇ ˇ ξ “ 0 ‰ 0 . Lemmas 3.2 and 3.3 imply that there exists ε 0 , C ą 0 such that for all ε P p´ ε 0 , ε 0 q | F p ε q ´ F p 0 q| ď C ε 10 . Therefore, the mean-v alue Theorem implies that for all ε sufficiently small there exists a ξ P p θ p 0 q , θ p ε qq so that cos p θ p ε qq ´ cos p θ p 0 qq “ p´ sin ξ q ¨ p θ p ε q ´ θ p 0 qq whic h forces | θ p ε q ´ θ p 0 q| ď C ε 10 b ecause | sin ξ | is b ounded aw a y from 0. □ W e note that, in [33], m “ 2 and t 1 “ t 2 , whereb y this argument do es not work. Indeed, there the rates are odd (1 { 3 and 1 { 5). It remains to pro v e Lemma 3.3. 3.3. Pro of of Lemma 3.3 - existence of ro ots. The argumen t pro ceeds in the following steps: w e first analyze the structure of the functions a j p t 1 , t 2 , t 3 , t 4 q and discuss some n umerical asp ects. After that, we pro ceed to find approximate solutions and then pro ve the existence of a nearby solution. 1. Algebr aic structur e. An explicit computation sho ws that a 0 p t 1 , t 2 , t 3 , t 4 q “ cos p t 1 ´ t 2 ` t 3 ´ t 4 q . More computation shows a 2 p t 1 , t 2 , t 3 , t 4 q “ ´ t 1 ` t 2 ´ t 3 ` t 4 2 sin p t 1 ´ t 2 ` t 3 ´ t 4 q ` cos p t 1 ´ t 2 ´ t 3 ´ t 4 q ´ cos p t 1 ` t 2 ´ t 3 ´ t 4 q ´ 2 cos p t 1 ´ t 2 ` t 3 ´ t 4 q ` cos p t 1 ` t 2 ` t 3 ´ t 4 q ´ cos p t 1 ´ t 2 ´ t 3 ` t 4 q ` cos p t 1 ` t 2 ´ t 3 ` t 4 q ` cos p t 1 ´ t 2 ` t 3 ` t 4 q . The other three functions, a 4 , a 6 , and a 8 , are similarly structured. More precisely , for some (real-v alued) co efficien ts and summed o ver all p ossible sign p erm utations a 4 “ ÿ ℓ c p 1 q ℓ sin p˘ t 1 ˘ t 2 ˘ t 3 ˘ t 4 q ` c p 2 q ℓ cos p˘ t 1 ˘ t 2 ˘ t 3 ˘ t 4 q ` ÿ ℓ 4 ÿ j “ 1 c p 3 q ℓ,j t j sin p˘ t 1 ˘ t 2 ˘ t 3 ˘ t 4 q ` c p 4 q ℓ,j t j cos p˘ t 1 ˘ t 2 ˘ t 3 ˘ t 4 q . a 4 p t 1 , t 2 , t 3 , t 4 q can be written as the sum of 22 suc h expressions (this may not b e the minimal num ber, the representation is not unique). Moreov er, the largest (in absolute v alue) co efficien t is 2 and they are all explicit rational num bers. The pattern extends to a 6 and a 8 . W e describ e them to the extent that is necessary for the subsequent argument. a 6 is the sum of 130 terms; a representativ e sample of suc h a term is a 6 p t 1 , t 2 , t 3 , t 4 q “ ¨ ¨ ¨ ´ t 2 3 8 cos p t 1 ´ t 2 ` t 3 ` t 4 q ` t 2 t 3 4 cos p t 1 ´ t 2 ` t 3 ` t 4 q ` . . . Sc hematically , a 6 can b e written as a 6 “ 130 ÿ i “ 1 c i p at most cubic monic monomial q sin p˘ t 1 ˘ t 2 ˘ t 3 ˘ t 4 q , 9 where the cubic monomial is a s ingle term of the form 1 , t i , t i t j or t i t j t k and the sine ma y also be a cosine. The co efficien ts are rational n umbers that are completely explicit, they all satisfy | c i | ď 3. Finally , unsurprisingly , a 8 can b e written as the sum of 295 terms of the form a 8 “ 295 ÿ i “ 1 c i p at most quartic monic monomial q sin p˘ t 1 ˘ t 2 ˘ t 3 ˘ t 4 q , where | c i | ď 4. W e note that since all the co efficien ts are of order „ 1, the total n umber of terms is in the hundreds and all expressions are the product of a (monic) monomial of degree ď 4 and a low-frequency trigonometric function, there are no significan t n umerical issues in p oin t wise ev aluating the four functions. Moreov er, all four functions are smooth, they can all be differen tiated in closed form and all the partial deriv ativ es hav e roughly the sam e num b er of terms as the original function. 2. Appr oximate Solutions. W e were motiv ated by the idea that since our ansatz is sufficiently generic, the existence of solutions should follow the basic counting heuristic and we exp ect a finite num ber of solutions (4 nonlinear equations in 4 unkno wns). How ev er, there is no a-priori reason for them to exist or, should they exist, for them to hav e a nice representation in closed form. The remainder of this section concerns the metho d by which we numerically found an approximate solution. Those only interested in the rigorous proof ma y skip directly to step (3), kno wing that in this step we somehow found a p oint p t 1 , t 2 , t 3 , t 4 q for which all equations are satisfied up to a very small n umerical error, see (3.4). T o find an approximate solution, we perform the following procedure. (1) Do the following 1000 times: pic k t 1 , t 2 , t 3 , t 4 uniformly at random from r 0 , 2 π s . If the v alidity conditions min p t 1 , t 2 , t 3 , t 4 q ě 0 . 3 and | t 1 ´ t 2 ` t 3 ´ t 4 | ě 0 . 1 , are satisfied, compute } H p t 1 , t 2 , t 3 , t 4 q} , where H p t 1 , t 2 , t 3 , t 4 q ” p a 2 p t 1 , t 2 , t 3 , t 4 q , a 4 p t 1 , t 2 , t 3 , t 4 q , a 6 p t 1 , t 2 , t 3 , t 4 q , a 8 p t 1 , t 2 , t 3 , t 4 qq , (3.3) and keep the tuple p t 1 , t 2 , t 3 , t 4 q that returns the smallest v alue. (2) T ake the tuple, let γ be a uniformly distributed random v ariable in r´ η, η s 4 for 0 ă η ă 1 and consider replacing the tuple b y p t 1 ` γ 1 , t 2 ` γ 2 , t 3 ` γ 3 , t 4 ` γ 4 q . If the new tuple satisfies the v alidity conditions (see step 1), and the functional } H } decreases, k eep the new tuple. (3) If no update occurs for a while, decrease η and rep eat the previous step. It is important to run step (1) sufficien tly long to b e able to start step (2) with a reasonably go od approximate solution, } H p t 1 , t 2 , t 3 , t 4 q} ď 1 seems to suffice in practice. F urthermore, we note that the role of the v alidity conditions is to preven t the t i all going to 0 as well as F p 0 q ‰ ˘ 1 (the constants 0 . 3 and 0 . 1 are not sp ecial, other choices w ould work as well). The approximate solution which w e will use in the next step is t 1 “ 4 . 088866559569492 t 2 “ 3 . 117488248716022 t 3 “ 2 . 615221023066265 t 4 “ 1 . 762750988714514 (3.4) 10 for which we hav e } H p t 1 , t 2 , t 3 , t 4 q} „ 5 . 2 ¨ 10 ´ 15 . Running many examples fre- quen tly leads to this solution again and again, sometimes with its entries b eing p erm uted. Indeed, the numerical study of these ‘almost-solutions’ lead to the fol- lo wing Prop osition (which, in turn, explains wh y one w ould find p erturbations). Lemma 3.4. If p t 1 , t 2 , t 3 , t 4 q is a solution of @ 1 ď k ď 4 a 2 k p t 1 , t 2 , t 3 , t 4 q “ 0 , then so ar e the cyclic p ermuations of this ve ctor p t 2 , t 3 , t 4 , t 1 q , p t 3 , t 4 , t 1 , t 2 q , p t 4 , t 1 , t 2 , t 3 q , and al l their r eversals, e.g., p t 2 , t 3 , t 4 , t 1 q ÞÑ p t 1 , t 4 , t 3 , t 2 q . Pr o of. Rev ersals are straightforw ard, they amount to running (1.1) (or the ODEs (2.2)). Since the Schr¨ odinger dynamics hav e the phase-conjugate/time rev ersal symmetry , the Flo quet exp onen ts are the same. Regarding the cyclic p erm utations, since ν p t q is p erio dic, it do es not matter for the infinite time dynamics at which p oin t in the cycle we start the dynamics. The solutions will b e different, but the rate of disp ersion is the same. □ It is worth noting that we hav e b een unable to find any other approximate solution except the one listed ab o ve (and its symmetries). This do es not mean that no other appro ximate solutions exist but, dealing with four nonlinear equations in four v ariables, it is not inconceiv able that the list of appro ximate solutions is finite. 3. Existenc e of a solution. It remains to prov e that the approximate solution discussed in the previous section, (3.4), comes from having an actual ro ot nearby . The idea is simple: since } H p t 1 , t 2 , t 3 , t 4 q} „ 5 . 2 ¨ 10 ´ 15 and since H is differentiable with respect to all four parameters (see (3.3)), it should suffice to prov e that the Jacobian D H is lo cally inv ertible to conclude that there exists an actual solution nearb y . This is essen tially the idea behind Implicit/In v erse F unction Theorem; how ev er, that Theorem is frequently stated without quantita- tiv e guaran tees (an exception is the b ook by Hubbard and Hubbard [27]). W e instead make use of a m uc h more commonly stated result, the Newton-Kantoro vic h Theorem, ensuring the con vergence of the Newton method. Theorem 3.5 (Newton-Kan torovic h, adapted from Theorem 2.1 in [12] to our setting) . L et B p x 0 , r q Ă R 4 b e some op en b al l, let G : B p x 0 , r q Ñ R 4 b e a c ontinu- ously differ entiable mapping. F or a starting p oint x 0 P R 4 , let G 1 p x 0 q b e invertible. Assume that (1) for some α ą 0 }p D G qp x 0 q ´ 1 G p x 0 q} ď α (2) ther e exists ω ą 0 such that for al l x, y P B p x 0 , r q }p D G qp x 0 q ´ 1 p D G p x q ´ DG p y qq} ď ω } x ´ y } (3) these two p ar ameters satisfy α ¨ ω ď 1 { 2 (4) and we have ω ´ 1 ` 1 ´ ? 1 ´ 2 αω ˘ ď r . Then Newton iter ation initialize d at B p x 0 , r q is wel l-define d, r emains within B p x 0 , r q , and c onver ges to some r o ot G p x ˚ q “ 0 . In p articular, such a r o ot exists. 11 W e wish to apply this Theorem to H , see (3.3), and the approximate solution x 0 “ p t 1 , t 2 , t 3 , t 4 q P R 4 giv en by (3.4). That H p x 0 q „ 10 ´ 15 allo ws us to work with a fairly small neigh b orhoo d, r ą 0. Moreov er, the function app ears to b e sufficien tly “generic” so as to not conspire against us: it is rather well-posed as evidenced by the following facts. (1) All the partial deriv ativ es are fairly large, we ha ve } ∇ a 2 p x 0 q} „ 3 . 75, } ∇ a 4 p x 0 q} „ 10 . 33, } ∇ a 6 p x 0 q} „ 19 . 14 and } ∇ a 8 p x 0 q} „ 41 . 82. (2) All the partial deriv atives p oin t in very differen t directions, as seen by det ˆ ∇ a 2 p x 0 q } ∇ a 2 p x 0 q} , ∇ a 4 p x 0 q } ∇ a 4 p x 0 q} , ∇ a 6 p x 0 q } ∇ a 6 p x 0 q} , ∇ a 8 p x 0 q } ∇ a 8 p x 0 q} ˙ „ 0 . 413 . Hence, while the ∇ a j do not form an orthogonal basis of R 4 , they are (n umerically) not v ery far from doing so. (3) In particular, the Jacobian of H is relatively small ( ∇ a 8 dominates the op erator norm). Moreo ver, it is nicely inv ertible (and ∇ a 2 dominates the op erator norm of the in v erse). W e ha v e }p D H qp x 0 q} „ 43 . 96 and }p D H qp x 0 q ´ 1 } „ 0 . 35 . T o apply the Newton-Kan toro vich Theorem to H , w e need to find compatible r, α, and ¯ ω . W e start b y noting that }p D H qp x 0 q ´ 1 H p x 0 q} ď }p DH qp x 0 q ´ 1 } ¨ } H p x 0 q} ď 10 ´ 13 “ : α. W e also note that, since }p DH qp x 0 q ´ 1 } ď 1, the second condition is implied by the stronger condition }p D H p x q ´ D H p y qq} ď ω } x ´ y } Lemma 3.6. L et r ď 0 . 5 . Then, for any x, y P B r p x 0 q , the ine quality }p D H qp x q ´ p D H qp y q} ď 10 8 r } x ´ y } . The argument will be incredibly w asteful, muc h better results appear to be true. Numerically , we see that even for a large ball of radius r „ 10 ´ 3 , one still has }p D H qp x q ´ p DH qp y q} ď 100 } x ´ y } . Before pro ving this lemma, let us see how it is enough to apply the Newton-Kan torovic h Theorem: Lemma 3.6 implies that w e can set ω “ 10 8 r . The condition α ¨ ω ď 1 { 2 is satisfied as so on as r ď 10 3 . It remains to chec k whether there exists a c hoice of r that satisfies the inequalit y ω ´ 1 ` 1 ´ ? 1 ´ 2 αω ˘ ď r . This is equiv alen t to 1 ´ ? 1 ´ 2 ¨ 10 ´ 5 r ď 10 8 r 2 whic h is satisfied for r ě 10 ´ 3 . Pr o of of L emma 3.6. W e use the trivial inequality v alid for any X P R 4 ˆ 4 } X } op ď g f f e 4 ÿ i,j “ 1 X 2 ij . This reduces the problem to establishing p oin t wise estimates on B a 2 i B t j p x q ´ B a 2 i B t j p y q , 12 for all 1 ď i, j ď 4. Here, we in v ok e the actual structure of the functions a 2 i . They are the sums of 8, 22, 130 and 295 terms, resp ectiv ely . These terms are all of the form co efficien t ¨ p monic monomial of degree ď 4 q cos p˘ t 1 ˘ t 2 ˘ t 3 ˘ t 4 q where the co efficients are uniformly b ounded in absolute v alue by 4 (and the cosine ma y also b e a sine). Therefore B a 2 i {B t j is the sum of at most 295 ¨ 2 ď 600 terms of exactly the same form except that the coefficients are now bounded in absolute v alue from abov e by ď 16. W e can b ound the difference in deriv ativ es b y using the mean-v alue theorem. The second deriv ative, by the same reasoning, has at most 1200 terms with co efficien ts b ounded b y ď 32. 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T op ological acoustics. Nature Reviews Materials, 7(12):974– 990, 2022. 14 Dep ar tment of Ma thema tics, University of Michigan, Ann Arbor, NJ 48109, USA Email address : abloch@umich.edu Dep ar tment of Ma thema tical Sciences, New Jersey Institute of Technology, Uni- versity Heights, New ark, NJ 07102, USA Email address : amir.sagiv@njit.edu Dep ar tment of Ma thema tics and Dep ar tment of Applied Ma thema tics, University of W ashington, Sea ttle, W A 98195, USA Email address : steinerb@uw.edu