The Model First result Second result Thank you ! Lyapunov exponent and integrated density of states for slowly oscillating perturbations of periodic Schrödinger operators Asya M ETELKINA FernUniversität in Hagen Saint-Petersburg July 2010 1 / 8
The Model First result Second result Thank you ! The Model Operator H ( θ ) in L 2 ( R + ) associated to θ ∈ [ 0 , π ) : H ( θ ) = − d 2 dx 2 +[ V ( x )+ W ( x α )] (1) on D ( H ( θ )) = { f ∈ H 2 ( R + ) | f ( 0 ) cos θ + f ′ ( 0 ) sin θ = 0 } . Basic assumptions : ( V ) V ∈ L 2 , loc ( R ) and periodic V ( x + 1 ) = V ( x ) , ( W ) W is smooth and periodic function with W ( x + 2 π ) = W ( x ) . ( α ) α ∈ ( 0 , 1 ) . Remark : For α ∈ ( 0 , 1 ) the function W ( x α ) oscillates slowly at infinity : d dx ( W ( x α )) = lim x → ∞ ( x α − 1 W ′ ( x α )) = 0 . lim x → ∞ Additional assumptions : α ∈ ( 1 2 , 1 ) and W is analytic in S Y = {| ℑ z | < Y } . 2 / 8
The Model First result Second result Thank you ! Resolvent matrix Schrödinger equation : H ( θ ) f ( x , E ) = Ef ( x , E ) is equivalent to matrix equation on resolvent matrix T ( x , y , E ) : d dxT ( x , y , E ) = A ( x , E ) T ( x , y , E ) with T ( y , y , E ) = I (2) � � 0 1 A ( x , E ) = and (3) V ( x )+ W ( x α ) − E 0 Consider a quasi-periodic (periodic) matrix equation depending on z and ε : d dxT z , ε ( x , y , E ) = A z , ε ( x , E ) T z , ε ( x , y , E ) with T z , ε ( y , y , E ) = I (4) � � 0 1 A z , ε ( x , E ) = and (5) V ( x )+ W ( ε x + z ) − E 0 3 / 8
The Model First result Second result Thank you ! Approximation of the resolvent matrix 1 1 α . Pose x n = ( 2 π n ) Claim : There exist C and n 0 such that ∀ n > n 0 one can find ( z n , ε n ) such that : | W ( x α ) − W ( ε n x + z n ) | < C 1 sup (6) n x ∈ [ x n , x n + 1 ] Lemma (A. Metelkina) Suppose basic and additional assumptions are satisfied. ∃ ( z n , ε n ) such that for n big enough the resolvent matrix T ( x n + 1 , x n , E ) of (2) can be approched by the resolvent matrix T z n , ε n ( x n + 1 , x n , E ) of (4) 4 / 8
The Model First result Second result Thank you ! Approximations of the resolvent matrix 2 Symmetries in A z , ε . V ( x ) = V ( x + 1 ) ⇒ [ V ( x + 1 )+ W ( ε ( x + 1 )+( z − ε ))] = [ V ( x )+ W ( ε z + x )] Consistancy condition : T z , ε ( x + 1 , E ) = T z − ε , ε ( x , E ) W ( ε x +( z + 2 π )) = W ( ε x + z ) ⇒ T z + 2 π , ε ( x , E ) satisfies (4) if T z , ε ( x , E ) does. Monodromy matrix M ( z , ε ) : T z + 2 π , ε ( x , E ) = T z , ε ( x , E ) M T ( z , ε , E ) (7) Theorem (A. Metelkina) Suppose basic and additional assumptions are satisfied. ∃ ( z n , ε n ) such that for n big enough the resolvent matrix T ( x n + 1 , x n , E ) of (2) can be approched by the transposed monodromy matrix M T ( z n , ε n , E ) associated to T z n , ε n ( x n + 1 , x n , E ) and defined in (7) 5 / 8
The Model First result Second result Thank you ! Definitions of IDS and LE Let N D ( H l , E ) be the number of Dirichlet eigenvalues H in L 2 ( 0 , l ) . Definition (IDS) We call integrated density of states the following limit when it exists : N D ( H l , E ) k ( E ) = lim l l → ∞ T ( x , 0 , E ) be the resolvent matrix, solution of (2). Definition (LE) We call Lyapunov exponent the following limit when it exists : ln � T ( x , 0 , E ) � γ ( E ) = lim x → ∞ x 6 / 8
The Model First result Second result Thank you ! Theorem : IDS and LE For E ∈ C + lets k p ( E ) be a principal branch of Bloch quasimomentum. For E ∈ R denote k p ( E ) its boudary values. Theorem (A. Metelkina) Suppose only basic assumptions are satisfied. For all energies E ∈ R the integrated density of states for H ( θ ) exists and is given by : � 2 π 1 k ( E ) = ℜ k p ( E − W ( x )) dx 2 π 2 0 For almost all E ∈ R the Lyapunov exponent for H ( θ ) exists and is given by : � 2 π γ ( E ) = 1 ℑ k p ( E − W ( x )) dx 2 π 0 7 / 8
The Model First result Second result Thank you ! Thank you for your attention ! 8 / 8
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