Eigensystem multiscale analysis for MSA Key step Anderson - PowerPoint PPT Presentation

Alexander Elgart Eigensystem Eigensystem multiscale analysis for MSA Key step Anderson localization in energy intervals II EMSA on intervals implies MSA Alexander Elgart Virginia Tech joint work with Abel Klein QMath13: Mathematical Results in Quantum Physics Georgia Tech October 9, 2016

Alexander Eigensystem multiscale analysis Elgart Eigensystem MSA Key step ◮ We consider the usual Anderson model. EMSA on intervals implies MSA ◮ General strategy: Information about eigensystems at a given scale is used to derive information about eigensystems at larger scales. ◮ Need to carry over deterministic and probabilistic information since the system is random. The probabilistic part is close to the one in the standard MSA, will not be discussed here.

Alexander Level spacing and localization Elgart Eigensystem MSA Definition Key step A box Λ L = [ − L , L ] d + x 0 is called L -level spacing for H if all EMSA on intervals eigenvalues of H Λ L are simple, and implies MSA λ , λ ′ ∈ σ ( H Λ L ) , λ � = λ ′ . � ≥ e − L β � λ − λ ′ � � for all Definition Let Λ L be a box, x ∈ Λ L , and m ≥ 0. Then ϕ ∈ ℓ 2 (Λ L ) is said to be ( x , m )-localized if � ϕ � = 1 and | ϕ ( y ) | ≤ e − m � y − x � � y − x � ≥ L τ . for all y ∈ Λ L with

Alexander Interval localization (na¨ ıve) Elgart Eigensystem MSA Key step Definition (na¨ ıve) EMSA on intervals Let I be a bounded interval and let m > 0. A box Λ L will be implies MSA called ( m , I )-localizing for H if 1 Λ L is level spacing for H . 2 There exists an ( m , I )-localized eigensystem for H Λ L , that is, an eigensystem { ( ϕ ν , ν ) } ν ∈ σ ( H Λ L ) for H Λ L such that for all ν ∈ σ ( H Λ L ) there is x ν ∈ Λ L such that ϕ ν is ( x ν , m )-localized. ◮ Level spacing helps to overcome the small denominator problem (resonances), replaces the Wegner estimate.

Alexander Failure of na¨ ıve approach to EMSA Elgart Eigensystem MSA Key step EMSA on Consider ℓ ≪ L and suppose that intervals implies MSA ◮ A box Λ L is L -level spacing for H ; ◮ Any box Λ ℓ ⊂ Λ L is ( m , I )-localizing for H (in a na¨ ıve sense as above). m , ˆ Can we show that Λ L is ( ˆ I )-localizing for H (allowing for small losses in m and I )? The answer is NO.

Alexander Failure of na¨ ıve approach to EMSA Elgart Eigensystem MSA We don’t know anything about the structure of eigenvectors for Key step EMSA on H Λ ℓ outside I . In particular, the quantum tunneling between intervals implies MSA localized states just inside I for one box Λ ℓ and the delocalized states just outside another box Λ ′ ℓ is possible (when we consider H Λ L as perturbation of decoupled boxes of size ℓ ). ◮ This indicates that on the new scale L , localization on I is no longer uniform (as far as localization length is concerned): As we approach the edges of I , the mass m goes to zero. ◮ Deep inside I we expect localization to survive, since the quantum tunneling between energetically separated states is suppressed by locality of H (Combes-Thomas estimate).

Alexander Correct approach to EMSA Elgart (simplified) Eigensystem MSA Key step ◮ We replace the naive definition with EMSA on intervals Definition implies MSA Let E ∈ R , I = ( E − A , E + A ), and m > 0. A box Λ L will be called ( m , I )-localizing for H if 1 Λ L is level spacing for H . 2 There exists an ( m , I )-localized eigensystem for H Λ L , i.e. an eigensystem { ( ϕ ν , ν ) } ν ∈ σ ( H Λ L ) for H Λ L such that for all ν ∈ σ ( H Λ L ) there is x ν ∈ Λ L such that ϕ ν is ( x ν , mh I ( ν ))-localized. ◮ The modulating function h I satisfies h I ( E ) = 1 and h I ( E ± A ) = 0.

Alexander Key step (simplified version) Elgart Eigensystem MSA Key step Consider ℓ ≪ L and suppose that EMSA on intervals implies MSA ◮ A box Λ L is L -level spacing for H ; ◮ Any box Λ ℓ ⊂ Λ L is ( m , I )-localizing for H . m , ˆ Can we show that Λ L is ( ˆ I )-localizing for H for some choice of the modulating function h I , and allowing for small losses in m and I ? The answer now is YES. ◮ Tricky part: Choice of h I and control over the decay rate.

Alexander EMSA on intervals implies MSA Elgart Eigensystem MSA ◮ The general startegy of going from scale ℓ to scale L Key step concerns the expansion of a true eigenfunction of H Λ L in terms EMSA on intervals of eigenfunctions of Hamiltonians H ℓ . implies MSA ◮ Although the process itself is very natural, preparations can take some time to explain. ◮ Instead, we will illustrate some ideas of the proof by showing how the eigensystem MSA for energy intervals implies the exponential localization of the Green function (the key player in the usual MSA). ◮ It will also reveal our top secret way of choosing the modulating function h I mentioned earlier � .

Alexander EMSA on intervals implies MSA Elgart Eigensystem MSA ◮ Let I = ( E − A , E + A ) with E ∈ R and A > 0. Key step EMSA on ◮ Suppose that Λ L is ( m , I )-localizing for H . intervals implies MSA ◮ Let λ ∈ I L with dist { λ , σ ( H Λ L ) } ≥ e − L β . ◮ WTS: For m not too small and not too large, mh I ( λ ) � x − y � whenever � x − y � ≥ L τ ′ . | G Λ L ( λ ; x , y ) | ≤ e − ˆ ◮ Losses in m should be (controllably) small: 1 − CL − γ � � m ≥ m ˆ for some γ > 0 .

Alexander Analyticity and localization Elgart Eigensystem ◮ We can try to split ( H Λ L − λ ) − 1 into MSA Key step ( H Λ L − λ ) − 1 P I +( H Λ L − λ ) − 1 ¯ P I EMSA on intervals ◮ P I is the spectral projection of H Λ L onto I , ¯ P I = 1 − P I . implies MSA ◮ We have no information on ϕ ν for ν / ∈ I , though, and the decay rate of ϕ ν for ν ∈ I is not uniform. Not good! ◮ Gentler approach: Filter out eigenvalues outside I using an analytic function F I ( H Λ L ) instead of P I : ( H Λ L − λ ) − 1 = ( H Λ L − λ ) − 1 F I ( H Λ L )+( H Λ L − λ ) − 1 ¯ F I ( H Λ L ). ◮ Want (a) F I to be exponentially small outside I and (b) ( z − λ ) − 1 ¯ F I ( z ) to be analytic in a strip that contains real axis (then Combes-Thomas estimate will kick in, and the corresponding term will be exponentially small too).

Alexander Analyticity and localization Elgart Eigensystem MSA Key step To summarize: EMSA on intervals � ϕ ν , ( H Λ L − λ ) − 1 F I ( H Λ L ) ϕ ν � = ( ν − λ ) − 1 F I ( ν ); implies MSA 1 2 | ϕ ν ( x ) ϕ ν ( y ) | ≤ e − mh I ( ν ) � x − y � ; 3 If K ( z ) = ( z − λ ) − 1 ¯ F I ( z ) is analytic and bounded in the strip | Imz | < η by � K � ∞ , then (Aizenman-Graf) |� δ x , K ( H Λ ) δ y �| ≤ C � K � ∞ e − ( log ( 1+ η 4 d )) � x − y � .

Alexander Analyticity and localization Elgart Eigensystem Let’s start tying up loose ends: MSA Key step ◮ Combining (1) – (2), we get the uniform exponential decay EMSA on � � δ x , ( H Λ L − λ ) − 1 F I ( H Λ L ) δ y �� intervals for � as long as � � implies MSA � ( ∗ ) F I ( ν ) e − mh I ( ν ) � x − y � ≤ e − mh I ( λ ) � x − y � for all ν ∈ σ ( H Λ ). ◮ (3) yields exponential decay for |� δ x , K ( H Λ ) δ y �| as long as ( ∗∗ ) � K � ∞ ≤ e ( log ( 1+ η 4 d ) / 2 ) � x − y � . ◮ Are there a filter F I and a modulating function h I out there that satisfy both (*) and (**)?

Alexander End game Elgart Eigensystem MSA Key step ◮ The choice EMSA on intervals F I ( z ) = e − t ( ( z − E ) 2 − ( λ − E ) 2 ); t = m � x − y � implies MSA , A 2 and � t − E � h I ( t ) = h A with � 1 − s 2 if s ∈ [0 , 1) h ( s ) = 0 otherwise does the trick! In fact, it turns Eq. (*) into the identity for ν ∈ I .

Alexander Elgart Eigensystem MSA Key step EMSA on intervals implies MSA THANKS!