Matthias Täufer (TU Chemnitz)

Transcription

1 2. Random breather model. Landau operators Wegner estimate for Landau operators with random breather potential Matthias Täufer (TU Chemnitz) Mainz, 5 September 206 (joint work with I. Veselić) 3. Wegner estimate 4. Proof (sort of) /22

2 2. Random breather model. Landau operators Wegner estimate for Landau operators with random breather potential 3. Wegner estimate 4. Proof (sort of) 2/22

3 . Landau operators H B = (i A) 2 on L 2 (R 2 ) where A = B 2 ( x2 x ). Electron moving on thin plate, perpendicular magnetic field Lorenz force makes it go in circles Quantum mechanics only certain frequencies allowed e 3/22

4 . Landau operators H B = (i A) 2 on L 2 (R 2 ) where A = B 2 ( x2 x ). Electron moving on thin plate, perpendicular magnetic field Lorenz force makes it go in circles Quantum mechanics only certain frequencies allowed e 3/22

5 . Landau operators H B = (i A) 2 on L 2 (R 2 ) where A = B 2 ( x2 x ). Electron moving on thin plate, perpendicular magnetic field Lorenz force makes it go in circles Quantum mechanics only certain frequencies allowed e 3/22

6 . Landau operators H B = (i A) 2 on L 2 (R 2 ) where A = B 2 ( x2 x ). Electron moving on thin plate, perpendicular magnetic field Lorenz force makes it go in circles Quantum mechanics only certain frequencies allowed e 3/22

7 . Landau operators H B = (i A) 2 on L 2 (R 2 ) where A = B 2 ( x2 x ). Electron moving on thin plate, perpendicular magnetic field Lorenz force makes it go in circles Quantum mechanics only certain frequencies allowed e 3/22

8 . Landau operators H B = (i A) 2 on L 2 (R 2 ) where A = B 2 ( x2 x ). infinitely degenerate eigenvalues at Landau Levels B(2N ) Integrated density of states: N(E) = lim Λ Number of Eigenvalues of H B Λ Λ below E.

9 . Landau operators H B = (i A) 2 on L 2 (R 2 ) where A = B 2 ( x2 x ). infinitely degenerate eigenvalues at Landau Levels B(2N ) Integrated density of states: N(E) = lim Λ Number of Eigenvalues of H B Λ Λ below E. N(E) of H B E 4/22

10 . Landau operators H B = (i A) 2 on L 2 (R 2 ) where A = B 2 ( x2 x ). infinitely degenerate eigenvalues at Landau Levels B(2N ) Integrated density of states: N(E) = lim Λ Number of Eigenvalues of H B Λ Λ below E. N(E) of H B N(E) of + V (typically) E E 4/22

11 . Landau operators H B = (i A) 2 on L 2 (R 2 ) where A = B 2 ( x2 x ). infinitely degenerate eigenvalues at Landau Levels B(2N ) Integrated density of states: N(E) = lim Λ Number of Eigenvalues of H B Λ Λ below E. N(E) of H B N(E) of + V (typically) E E Jumps in IDS related to quantum hall effect; since 990 SI definition for electric resistance 4/22

12 2. Random breather model. Landau operators Wegner estimate for Landau operators with random breather potential 3. Wegner estimate 4. Proof (sort of) 5/22

13 2. Random breather model Now add random potential H B,ω = H B + V ω, ω Ω probability space. Fact: IDS still exists almost surely if V ω ergodic 6/22

14 2. Random breather model Now add random potential H B,ω = H B + V ω, ω Ω probability space. Fact: IDS still exists almost surely if V ω ergodic Physisicts fact: IDS is smeared out N(E) of H B N(E) of H B,ω E E 6/22

15 2. Random breather model We are interested in the random breather potential V ω (x) = λ ( ) x j u, ω j > 0, i.i.d., bounded. j Z 2 ω j random dilation of of a single-site potential at every j Z 2 7/22

16 2. Random breather model We are interested in the random breather potential V ω (x) = λ ( ) x j u, ω j > 0, i.i.d., bounded. j Z 2 ω j random dilation of of a single-site potential at every j Z 2 λ > 0 disorder parameter 7/22

17 2. Random breather model We are interested in the random breather potential V ω (x) = λ ( ) x j u, ω j > 0, i.i.d., bounded. j Z 2 ω j random dilation of of a single-site potential at every j Z 2 λ > 0 disorder parameter V ω (x) ω j x λ 7/22

18 2. Random breather model ( x2 H B,ω = (i A) 2 +V ω, A = B ), 2 x V ω (x) = λ ( x j u j Z 2 ω j ), ω j i.i.d., bounded..

19 2. Random breather model ( x2 H B,ω = (i A) 2 +V ω, A = B ), 2 x V ω (x) = λ ( x j u j Z 2 ω j ), ω j i.i.d., bounded.. Goal: Wegner estimate: E [Number of eigenvalues of H B,ω ΛL in I] C I θ Λ L.

20 2. Random breather model ( x2 H B,ω = (i A) 2 +V ω, A = B ), 2 x V ω (x) = λ ( x j u j Z 2 ω j ), ω j i.i.d., bounded.. Goal: Wegner estimate: E [Number of eigenvalues of H B,ω ΛL in I] C I θ Λ L. R Spectrum of H ω,l ω ω ω ω Ω 8/22

21 2. Random breather model ( x2 H B,ω = (i A) 2 +V ω, A = B ), 2 x V ω (x) = λ ( x j u j Z 2 ω j ), ω j i.i.d., bounded.. Goal: Wegner estimate: E [Number of eigenvalues of H B,ω ΛL in I] C I θ Λ L. R Spectrum of H ω,l I ω ω ω ω Ω 8/22

22 2. Random breather model H B,ω = (i A) 2 +V ω, A = B ( ) x2, 2 x V ω (x) = λ ( x j u j Z 2 ω j ), ω j i.i.d., bounded.. Problem : non-linear dependence of V ω on ω j (usually solved by powerful unique continuation principles if operator has bounded coefficient functions), [Nakić, T, Tautenhahn, Veselić 6], [T., Veselić 5] 9/22

23 2. Random breather model H B,ω = (i A) 2 +V ω, A = B ( ) x2, 2 x V ω (x) = λ ( x j u j Z 2 ω j ), ω j i.i.d., bounded.. Problem : non-linear dependence of V ω on ω j (usually solved by powerful unique continuation principles if operator has bounded coefficient functions), [Nakić, T, Tautenhahn, Veselić 6], [T., Veselić 5] Problem 2: Operator H B has unbounded coefficient functions (solvable if linear dependence of V ω on random variables), [Combes, Hislop, Klopp 03/07] 9/22

24 2. Random breather model ( x2 H B,ω = (i A) 2 +V ω, A = B ), 2 x V ω (x) = λ ( x j u j Z 2 ω j ), ω j i.i.d., bounded.. non-linear unbounded Vicious circle! 0/22

25 2. Random breather model ( x2 H B,ω = (i A) 2 +V ω, A = B ), 2 x V ω (x) = λ ( x j u j Z 2 ω j ), ω j i.i.d., bounded.. non-linear unbounded Vicious circle! Way out: Combination of a bit of both strategies, i.e. less strong UCP only for the free Landau Hamiltonian [Raikov, Warzel 02], [Combes, Hislop, Klopp, Raikov 04], [Rojas-Molina 2] and some linearization [Combes, Hislop, Klopp, Nakamura 02], 0/22

26 2. Random breather model ( x2 H B,ω = (i A) 2 +V ω, A = B ), 2 x V ω (x) = λ ( x j u j Z 2 ω j ), ω j i.i.d., bounded.. non-linear unbounded Vicious circle! Way out: Combination of a bit of both strategies, i.e. less strong UCP only for the free Landau Hamiltonian [Raikov, Warzel 02], [Combes, Hislop, Klopp, Raikov 04], [Rojas-Molina 2] and some linearization [Combes, Hislop, Klopp, Nakamura 02], BUT: need λ <<! 0/22

27 2. Random breather model. Landau operators Wegner estimate for Landau operators with random breather potential 3. Wegner estimate 4. Proof (sort of) /22

28 3. Wegner estimate Theorem (Wegner estimate for Landau operator with random breather potential, T,Veselić 6) E [Number of eigenvalues of H B,ω ΛL in I] C(B, E 0, θ) I θ L 2. 2/22

29 3. Wegner estimate Theorem (Wegner estimate for Landau operator with random breather potential, T,Veselić 6) Assume that. u L 0 (R2 ) such that there are C u, r > 0 such that for all t [ω, ω + ] we find x 0 = x 0 (t) (, ) 2 with ( x ) t u C u χ B(x0,r)(x) for almost every x R 2 t 2. the ω j are positive, uniformly bounded, i.i.d random variables with a bounded density. E [Number of eigenvalues of H B,ω ΛL in I] C(B, E 0, θ) I θ L 2. 2/22

30 3. Wegner estimate Theorem (Wegner estimate for Landau operator with random breather potential, T,Veselić 6) Assume that. u L 0 (R2 ) such that there are C u, r > 0 such that for all t [ω, ω + ] we find x 0 = x 0 (t) (, ) 2 with ( x ) t u C u χ B(x0,r)(x) for almost every x R 2 t 2. the ω j are positive, uniformly bounded, i.i.d random variables with a bounded density. Let B > 0, E 0 R, θ (0, ). Then there is λ 0 > 0 such that for all 0 < λ < λ 0, all intervals I (, E 0 ] and all sufficiently large L we have E [Number of eigenvalues of H B,ω ΛL in I] C(B, E 0, θ) I θ L 2. 2/22

31 3. Wegner estimate Recall Assumption from the theorem:... Assume that. u L 0 (R2 ) such that there are C u, r > 0 such that for all t [ω, ω + ] we find x 0 = x 0 (t) (, ) 2 with ( x ) t u C u χ B(x0,r)(x) for almost every x R 2 t... Looks similar to [Combes, Hislop, Klopp, Nakamura 02], but there they have slightly different assumption Very restrictive! Singularities! 3/22

32 3. Wegner estimate Recall Assumption from the theorem:... Assume that. u L 0 (R2 ) such that there are C u, r > 0 such that for all t [ω, ω + ] we find x 0 = x 0 (t) (, ) 2 with ( x ) t u C u χ B(x0,r)(x) for almost every x R 2 t... Looks similar to [Combes, Hislop, Klopp, Nakamura 02], but there they have slightly different assumption Very restrictive! Singularities! Examples: The hat potential u(x) = χ x ( x ), the smooth bump potential u(x) = χ x exp( /( x 2 )). 3/22

33 3. Wegner estimate Corollary The integrated density of states is locally Hölder continuous with respect to every exponent θ (0, ). Physisicts fact verified: N(E) of H B N(E) of H B,ω E E 4/22

34 2. Random breather model. Landau operators Wegner estimate for Landau operators with random breather potential 3. Wegner estimate 4. Proof (sort of) 5/22

35 4. Proof (sort of) Write No. of Eigenvalues as trace [Number of eigenvalues of H B,ω ΛL in I] = Tr [χ I (H B,ω )] 6/22

36 4. Proof (sort of) Write No. of Eigenvalues as trace [Number of eigenvalues of H B,ω ΛL in I] = Tr [χ I (H B,ω )] Decomposition with respect to χ J (H B ) à la [Combes, Hislop, Klopp 03/07], I J, J 2B (distance between Landau levels) = Tr [χ I (H B,ω )χ J (H B )] + Tr [χ I (H B,ω ) (Id χ J (H B ))] I J R 6/22

37 4. Proof (sort of) Write No. of Eigenvalues as trace [Number of eigenvalues of H B,ω ΛL in I] = Tr [χ I (H B,ω )] Decomposition with respect to χ J (H B ) à la [Combes, Hislop, Klopp 03/07], I J, J 2B (distance between Landau levels) = Tr [χ I (H B,ω )χ J (H B )] + Tr [χ I (H B,ω ) (Id χ J (H B ))] Second trace: use that I and J c are far apart I J Tr [χ I (H B,ω )χ J (H B )] + λv ω 2 R dist(i, J c ) 2 Tr [χ I(H B,ω )] 6/22

38 4. Proof (sort of) Tr [χ I (H B,ω )] Tr [χ I (H B,ω )χ J (H B )] + λv ω 2 dist(i, J c ) 2 Tr [χ I(H B,ω )] 7/22

39 4. Proof (sort of) Tr [χ I (H B,ω )] Tr [χ I (H B,ω )χ J (H B )] + λv ω 2 dist(i, J c ) 2 Tr [χ I(H B,ω )] First trace: smuggle in j ω j V ω by estimate on χ J (H B ), needs that J contains at most one Landau level [ ( ) ] C Tr χ I (H B,ω )χ J (H B ) ωj V ω χ J (H B ) + λv ω 2 dist(i, J c ) 2... j 7/22

40 4. Proof (sort of) Tr [χ I (H B,ω )] Tr [χ I (H B,ω )χ J (H B )] + λv ω 2 dist(i, J c ) 2 Tr [χ I(H B,ω )] First trace: smuggle in j ω j V ω by estimate on χ J (H B ), needs that J contains at most one Landau level [ ( ) ] C Tr χ I (H B,ω )χ J (H B ) ωj V ω χ J (H B ) + λv ω 2 dist(i, J c ) 2... Rearrange and hide χ I (H B,ω ) on the left hand side to find [ ( )] Tr [χ I (H B,ω )] C Tr χ I (H B,ω ) ωj V ω. j j 7/22

41 4. Proof (sort of) Tr [χ I (H B,ω )] Tr [χ I (H B,ω )χ J (H B )] + λv ω 2 dist(i, J c ) 2 Tr [χ I(H B,ω )] First trace: smuggle in j ω j V ω by estimate on χ J (H B ), needs that J contains at most one Landau level [ ( ) ] C Tr χ I (H B,ω )χ J (H B ) ωj V ω χ J (H B ) + λv ω 2 dist(i, J c ) 2... Rearrange and hide χ I (H B,ω ) on the left hand side to find [ ( )] Tr [χ I (H B,ω )] C Tr χ I (H B,ω ) ωj V ω. j j Requires λv ω / dist(i, J c ) small. (Here the assumption λ << enters). 7/22

42 4. Proof (sort of) These steps are summarized by the following lemma Lemma (T, Veselić 6) H lower semibounded, purely discrete spectrum, V bdd. symmetric, I J R intervals. Assume there is W 0 such that Then, for V small we have and C is known explicitely χ J (H)W χ J (H) Cχ J (H). Tr [χ I (H + V )] C Tr(χ I (H + V )(W + W 2 )) 8/22

43 4. Proof (sort of) [ ( )] Tr [χ I (H B,ω )] C Tr χ I (H B,ω ) ωj V ω j 9/22

44 4. Proof (sort of) [ ( )] Tr [χ I (H B,ω )] C Tr χ I (H B,ω ) ωj V ω [ ( )] C Tr f (H B,ω ) ωj V ω j j f χ I R 9/22

45 4. Proof (sort of) [ ( )] Tr [χ I (H B,ω )] C Tr χ I (H B,ω ) ωj V ω [ ( )] C Tr f (H B,ω ) ωj V ω [ ] = C Tr ωj (f (H B,ω )) j j j f χ I R 9/22

46 4. Proof (sort of) Take expectation [ ] E Tr [χ I (H B,ω )] CE Tr ωj (f (H B,ω )) j 20/22

47 4. Proof (sort of) Take expectation [ ] E Tr [χ I (H B,ω )] CE Tr ωj (f (H B,ω )) j Rest of the proof is folklore: sum over j gives L 2 (volume) probability estimate gives I θ E Tr [χ I (H B,ω )] C I θ L 2. 20/22

48 Some References Combes, Hislop, Klopp: Hölder continuity of the integrated density of states for some random operators at all energies, IMRN (2003) Combes, Hislop, Klopp: An optimal Wegner estimate and its application to the global continuity of the integrated density of states for random operators, Duke M. Journ. (2007) Combes, Hislop, Nakamura: The Wegner estimate and the integrated density of states for some random operators, Proc. Indian Acad. Sci., 2002 Nakić, T, Tautenhahn, Veselić: Scale-free unique continuation principle, eigenvalue lifting and Wegner estimates for random Schrödinger operators, submitted, 206 Rojas-Molina: Characterization of the Anderson metal-insulator transition for non ergodic operators and appilcation, Ann. Henri Poincaré (202). T., Veselić: Wegner Estimate for Landau-Breather Hamiltonians, J. Math. Phys., 206. T. Veselić: Conditional Wegner estimate for the standard random breather model, J. Stat. Phys., 206 2/22

49 2. Random breather model. Landau operators Thank you for your attention! 3. Wegner estimate 4. Proof (sort of) 22/22