Dynamical upper bounds in quantum mechanics. Laurent Marin. Helsinki, November 2008

Transcription

1 Dynamical upper bounds in quantum mechanics Laurent Marin Helsinki, November 2008 ynamical upper bounds in quantum mechanics (Helsinki, November 2008) Laurent Marin 1 / 30

2 Let H be a discrete self-adjoint Schrödinger operator on separable Hilbert space l 2 (Z) into himself. [Hψ](n) = ψ(n + 1) + ψ(n 1) + V (n)ψ(n) V is a function Z R, the so-called potential. ynamical upper bounds in quantum mechanics (Helsinki, November 2008) Laurent Marin 2 / 30

3 In this work, we consider only Sturmian potential : V (n) = ( (n + 1)β nβ )V V is a positive constant called strength of potential or couplage. β is an irrationnal number. Link to physics : Modeling electron/phonon spectrum in a quasicrystal. Only 2 values possible for the potential. Quasiperiodical potential : no period but any sequence repeats infinitly many times ynamical upper bounds in quantum mechanics (Helsinki, November 2008) Laurent Marin 3 / 30

4 Potential construction : By graphic means : line with β slope, iterated rotation by angle β. By word concatenation : The potential coincide with sturmian words, W 1 = V W 0 = 0 W k+1 = W a k+1 k W k 1, k 0. ynamical upper bounds in quantum mechanics (Helsinki, November 2008) Laurent Marin 4 / 30

5 Unitary dynamical evolution : ψ(t) = e ith ψ(0). In this case, ψ is known to spread out with time over the canonical basis of l 2 (Z). Dynamical field consist to quantify how fast it spreads. ynamical upper bounds in quantum mechanics (Helsinki, November 2008) Laurent Marin 5 / 30

6 The probability for the system to be in n over the time T (in average) : a(n, T ) = 2 T 0 e 2t/T e ith ψ(0), δ n 2 dt. We denote the time average outside probabilities P(N, T ) = a(n, T ), n >N ynamical upper bounds in quantum mechanics (Helsinki, November 2008) Laurent Marin 6 / 30

7 For p > 0, one defines moments of order p X p ψ(0) = n n p a(n, T ), And their growth exponents : β ψ(0) (p) = lim inf T log X p ψ(0) p log T log X p β + ψ(0) ψ(0) (p) = lim sup T p log T. ynamical upper bounds in quantum mechanics (Helsinki, November 2008) Laurent Marin 7 / 30

8 For all α [0, + ], S (α) = lim inf T log P(T α 2, T ) log T and S + (α) = lim sup T log P(T α 2, T ) log T The following critical exponents are particular of interest : α ± l = sup{α 0 : S ± (α) = 0}, α ± u = sup{α 0 : S ± (α) < }. ynamical upper bounds in quantum mechanics (Helsinki, November 2008) Laurent Marin 8 / 30

9 α l ± is the (lower and upper) rates of propagation of the essential part of the wavepacket and α u ± as the rates of propagation of the fastest part. In particular, if γ > α u + then P(T γ, T ) goes to 0 faster than any inverse power of T. They verify 0 α ± l α ± u. lim p 0 β± ψ (p) = α± l lim p β± ψ (p) = α± u. ynamical upper bounds in quantum mechanics (Helsinki, November 2008) Laurent Marin 9 / 30

10 (B. Simon) For this model, α ± u 1. Until 2005, it was an open question, if one can improve this so-called balistic bound. In this work, we improve upper bounds for the fast part of the wavepacket under assumption on the irrational number. ynamical upper bounds in quantum mechanics (Helsinki, November 2008) Laurent Marin 10 / 30

11 q 1 = 0, q 0 = 1, q k+1 = a k+1 q k + q k 1 Let β be an irrational number and H β with a sturmian log q potential associated to β. V > 20. If D = lim sup k k k is finite then α u + 2D log ( ). V 8 3 ynamical upper bounds in quantum mechanics (Helsinki, November 2008) Laurent Marin 11 / 30

12 ψ(n + 1) + ψ(n 1) + V (n)ψ(n) = zψ(n) (1) with z C and ψ a non-zero vector. One can reformulate this in terms of transfer matrix. ( ) ( ) ψ(n + 1) ψ(1) = F (n, z) ψ(n) ψ(0) T (n, z)...t (1, z) n 1, F (n, z) = Id n = 0, [T (n, z)] 1...[T (0, z)] 1 n 1. ( ) z V (m) 1 T (m, z) = 1 0 ynamical upper bounds in quantum mechanics (Helsinki, November 2008) Laurent Marin 12 / 30

13 The transfer matrix play a key role in all the theory. Of course, in eigenvalue/vector study, and spectrum study. But also in dynamical field, with this estimate : (Damanik, Tcheremtchansev) P(N, T ) exp( cn) ( K + T 3 K max 1 k N F (k, E + i 2) 1 T ) de, ynamical upper bounds in quantum mechanics (Helsinki, November 2008) Laurent Marin 13 / 30

14 p 1 = 1, p 0 = 0, q 1 = 0, q 0 = 1, p k+1 = a k+1 p k + p k 1 We denote q k+1 = a k+1 q k + q k 1 M k (z) = F (q k, z) M k+1 (z) = M k 1 (z)m k (z) a k+1 ynamical upper bounds in quantum mechanics (Helsinki, November 2008) Laurent Marin 14 / 30

15 Let t k,p be the trace of the matrix M k 1 M p k. The evolution along the p index is given by and consequently, t k,p+1 = t k+1,0 t k,p t k,p 1, t k,p+1 = S p (t k+1,0 )t k,1 S p 1 (t k+1,0 )t k,0 S p denotes the p th Tchebychev polynomial of the second kind The evolution along the k index is related to the p-evolution by t k+1,0 = t k,ak+1, ynamical upper bounds in quantum mechanics (Helsinki, November 2008) Laurent Marin 15 / 30

16 The behavior of the sequence of trace t k,0 is interesting in dynamics but also for the spectrum of the operator H. σ = {E R, s.t.{t k,0 (E)} k bounded} We can be more precise and tell with the next criteria when this sequence is bounded or not. ynamical upper bounds in quantum mechanics (Helsinki, November 2008) Laurent Marin 16 / 30

17 A necessary and sufficient condition that {t k,0 (E)} k be unbounded is that t N+1,0 (E) > 2, t N,0 (E) > 2, t N 1,0 (E) 2 for some N 0. This N is unique. Denote G k = G k 1 + a k G k 2, G 0 = 1, G 1 = 1. We have t k,0 (E) e cg k N k > N. ynamical upper bounds in quantum mechanics (Helsinki, November 2008) Laurent Marin 17 / 30

18 Denote σ k,p = {E R, t k,p (E) 2}. (i) the set σ k,p is made of pq k + q k 1 distinct intervals, (ii) the σ k,0 coincide with the spectrum of periodic operator H k replacing β with p k /q k in definition) (iii) σ σ k+1,0 σ k,0 Moreover, σ the spectrum of H verify σ = k=n (σ k+1,0 σ k,0 ) ynamical upper bounds in quantum mechanics (Helsinki, November 2008) Laurent Marin 18 / 30

19 For a given k, we call type I gap : a band of σ k,1 included in a band of σ k,0 and therefore in a gap of σ k+1,0, type II band : a band of σ k+1,0 included in a band of σ k, 1 and in a gap of σ k,0, type III band : a band of σ k+1,0 included in a band of σ k,0 and in a gap of σ k,1. ynamical upper bounds in quantum mechanics (Helsinki, November 2008) Laurent Marin 19 / 30

20 (Raymond) At a given level k, (i) a type I gap contains an unique type II band of σ k+2,0. (ii) a type II band contains (a k+1 + 1) bands of type I of σ k+1,1. They are alternated with (a k+1 ) type III bands of σ k+2,0 (iii) a type III band contains (a k+1 ) bands of type I of σ k+1,1. They are alternated with (a k+1 1) type III bands of σ k+2,0 ynamical upper bounds in quantum mechanics (Helsinki, November 2008) Laurent Marin 20 / 30

21 (Liu, Wen) Each band length has a lower bound with coefficient of Q n and an upper bound with coefficient of P n. with c 1 = 3 V 8 Q n = with c 2 = 1 V +5. P n = 0 ca n c 1 /a n 0 c 1 /a n c 1 /a n 0 c 1 /a n 0 c a n c 2 (a n + 2) 3 0 c 2 (a n + 2) 3 c 2 (a n + 2) 3 0 c 2 (a n + 2) 3 ynamical upper bounds in quantum mechanics (Helsinki, November 2008) Laurent Marin 21 / 30

22 By now, we define the periodic approximants spectrum not only in R but in C. σ k,0 = {z C : t k,0 (z) 2} If k 3, and V > 20 then there exist constants c,d > 0 such that q k 1 j=1 q B(x (j) k, r k 1 k) σ k,0 B(x (j) k, R k) j=1 where {x (j) k } 1 j q k 1 are the zeros of t k,0, r k = c. min band length and R k = d.max band length. ynamical upper bounds in quantum mechanics (Helsinki, November 2008) Laurent Marin 22 / 30

23 As x (j) k are real, we have σ k,0 {z C : Imz < R k } {z C : Imz < dq γ(v ) k }. for a suitable γ(v ). This implies σ σ k,0 σ k+1,0 {z C : Imz < dq γ(v ) k }. (2) We should have R k < dq γ(v ) k chosen by taking : so a suitable γ can be γ(v ) lim sup k log c 1. k 2 log q k ynamical upper bounds in quantum mechanics (Helsinki, November 2008) Laurent Marin 23 / 30

24 For T > 1, denote by k(t ) the unique integer with and let q γ(v ) k(t ) 1 d T qγ(v ) k(t ) d N(T ) = q k(t )+ k(t ). N(T ) C ν T 1 γ(v ) T ν. (3) ynamical upper bounds in quantum mechanics (Helsinki, November 2008) Laurent Marin 24 / 30

25 ( K P(N(T ), T ) exp( cn(t )) + T 3 K exp( cn(t )) + T 3 e 2cG k(t ) max 1 q n N(T ) From this bound, we see that P(N(T ), T ) goes to 0 faster than any inverse power of T. As N(T ) C ν T 1 γ(v ) T ν. M n(e we have α + u 1 γ(v ) + ν ynamical upper bounds in quantum mechanics (Helsinki, November 2008) Laurent Marin 25 / 30

26 Let β be an irrational number and H β with a sturmian log q potential associated to β. V > 20. If D = lim sup k k k is finite then α u + 2D log ( ). V 8 3 Moreover, for an irrational with continued fraction expansion containing no 1, the dynamical upper bound becomes α u + D log ( ). V 8 3 ynamical upper bounds in quantum mechanics (Helsinki, November 2008) Laurent Marin 26 / 30

27 It is clear that taking V large enough, one can obtain a non trivial bound that is smaller than 1. The condition D finite is true for almost all irrational. Moreover, (Khintchin) For almost all β with respect to Lebesgue measure, D = lim sup k log q k k = D K = π2 12 log 2. ynamical upper bounds in quantum mechanics (Helsinki, November 2008) Laurent Marin 27 / 30

28 There exist an irrational number β with D = + such that for any V > 20. α + u = 1. Proof : Sturmian potentials of operator H β and H βn same first p n+1 + q n+1 values. have the ynamical upper bounds in quantum mechanics (Helsinki, November 2008) Laurent Marin 28 / 30

29 (Last) Let H 1 = + V 1 and H 2 = + V 2 on l 2 (Z), and such that V 1 (k), V 2 (k) < C for all k Z and C a constant. Let T > 0 and ε > 0 be fixed so there exist L(T, ε), δ > 0 such that V 1 (k) V 2 (k) < δ for all k < L, then X 2 H 1,ψ(0) X 2 H 2,ψ(0) < ε. ynamical upper bounds in quantum mechanics (Helsinki, November 2008) Laurent Marin 29 / 30

30 As H βn is a periodic potential operator, one has X 2 β n T > C n T 2 choose T n big enough such that C n > 1 log T n. One can then choose a n+1 such that L(T n, ε) p n+1 + q n+1. Inductively, we have a sequence T n going to infinity and an irrational number β with X 2 β Tn > T 2 n log T n ε > T 2 δ n, δ > 0 α + u β δ 1 (2) > 1 δ, δ > 0. ynamical upper bounds in quantum mechanics (Helsinki, November 2008) Laurent Marin 30 / 30