Partial Semiclassical Limits . Stefan Teufel Mathematisches - PowerPoint PPT Presentation

Partial Semiclassical Limits . Stefan Teufel Mathematisches Institut, Universit¨ at T¨ ubingen Workshop on Quantum Control IHP, Paris, 8–11 December 2010 Jointly with Hans-Michael Stiepan

1. Semiclassics and the Egorov Theorem in quantum mechanics Example: The Schr¨ odinger operator H ε = − ε 2 � 2 ∆ x + V ( x ) is the Weyl-quantization of the symbol 2 p 2 + V ( q ) . H ( q, p ) = 1 In general: Consider an ε -pseudodifferential Operator H ε = H ( x, − i ε ∇ x ) . � We take � H to be the ε -Weyl-quantization of a symbol H : R 2 n → R acting on functions ψ ∈ L 2 ( R n ) as � � 1 � 1 R 2 n e i p · ( x − y ) /ε H H ε ψ )( x ) := ( � 2 ( x + y ) , p ψ ( y ) d p d y . (2 πε ) n Partial semiclassical limits December 2010

1. Semiclassics and the Egorov Theorem in quantum mechanics H ε is self-adjoint, it generates a unitary group If � t �→ U ε ( t ) = e − i � H ε t/ε U ε : R → L ( L 2 ( R n )) , and the asymptotic limit ε → 0 is the semiclassical limit . One way to formulate the semiclassical limit is to look at the way other ε -pseudos transform: e i � A ε e − i � H ε t/ε � H ε t/ε = ? Egorov’s Theorem 1: Let H 0 : R 2 n → R 2 n Φ t be the Hamiltonian flow associated to the principal symbol H 0 of � H ε , then ε e i � A ε e − i � H ε t/ε � H ε t/ε = � A ◦ Φ t + O ( ε ) . H 0 Partial semiclassical limits December 2010

1. Semiclassics and the Egorov Theorem in quantum mechanics A less commonly known improved version is Egorov’s Theorem 2: Let H ε : R 2 n → R 2 n Φ t be the Hamiltonian flow associated to the symbol H ε = H 0 + εH 1 , then ε e i � A ε e − i � H ε t/ε � H ε t/ε = � + O ( ε 2 ) . A ◦ Φ t H ε Remarks: • For H and A from suitable symbol classes, the approximation holds in . norm uniformly on bounded time intervals. • Theorem 1 holds also on T ∗ M with M a Riemannian manifold. • Theorem 2 only holds if M has vanishing curvature. Partial semiclassical limits December 2010

2. Adiabatic slow-fast systems Consider the Hilbert space L 2 ( R n ) ⊗ H f ∼ = L 2 ( R n , H f ) , where H f is the Hilbert space of some quantum mechanical degrees of freedom. For an operator-valued symbol H : R 2 n → L ( H f ) H ε acting on functions ψ ∈ L 2 ( R n , H f ) again as we define � � � 1 � 1 R 2 n e i p · ( x − y ) /ε H H ε ψ )( x ) := ( � 2 ( x + y ) , p ψ ( y ) d p d y . (2 πε ) n Example: The molecular Hamiltonian − ε 2 2 ∆ x − 1 H ε L 2 ( R n x × R m y ) = L 2 ( R n x , L 2 ( R m 2 ∆ y + V ( x, y ) = � on y )) is the Weyl-quantization of the operator-valued symbol 2 p 2 − 1 H ( q, p ) = 1 2 ∆ y + V ( q, y ) . Partial semiclassical limits December 2010

2. Adiabatic slow-fast systems Since H ( q, p ) is operator-valued, it does not generate a Hamiltonian flow on T ∗ M . Can one still prove an Egorov Theorem? H ( q, p ) is self-adjoint for each ( q, p ) ∈ R 2 n . Its eigenvalues E ( q, p ) are real- valued and thus define Hamiltonian functions on phase space. Example: The molecular Hamiltonian − ε 2 2 ∆ x − 1 H ε L 2 ( R n x × R m y ) = L 2 ( R n x , L 2 ( R m 2 ∆ y + V ( x, y ) = � on y )) is the Weyl-quantization of the operator-valued symbol 2 p 2 − 1 H ( q, p ) = 1 2 ∆ y + V ( q, y ) . Partial semiclassical limits December 2010

2. Adiabatic slow-fast systems Adiabatic perturbation theory: (Littlejohn-Flynn, Emmrich-Weinstein, Brummelhuis-Nourrigat, Martinez-Nenciu-Sordoni, Panati-Spohn-T.) If H 0 ( q, p ) has an eigenvalue E ( q, p ) with spectral projection P ( q, p ) that is separated by a gap from the remainder of the spectrum of H 0 ( q, p ), then there exists a unique symbol P ε ( q, p ) P ε ( q, p ) = P ( q, p ) + O ( ε ) with P ε is an orthogonal projection, i.e. such that � P ε ) 2 = � P ε ) ∗ = � P ε , P ε ( � ( � and H ε up to small errors, that commutes with the Hamiltonian � H ε , � P ε ] = O ( ε ∞ ) . [ � Partial semiclassical limits December 2010

2. Adiabatic slow-fast systems Adiabatic perturbation theory: If H 0 ( q, p ) has an eigenvalue E ( q, p ) with spectral projection P ( q, p ) that is separated by a gap from the remainder of the spectrum of H 0 ( q, p ), then there exists a unique symbol P ε ( q, p ) P ε ( q, p ) = P ( q, p ) + O ( ε ) with P ε is an orthogonal projection, i.e. such that � P ε ) 2 = � P ε ) ∗ = � P ε , P ε ( � ( � and H ε up to small errors, that commutes with the Hamiltonian � H ε , � P ε ] = O ( ε ∞ ) . [ � P ε is almost invariant under the group e − i � H ε t/ε , Hence Ran � [e − i � H ε t/ε , � P ε ] = O ( ε ∞ | t | ) and P ε = � P ε + O ( ε ) . H ε � E ε � � Partial semiclassical limits December 2010

2. Adiabatic slow-fast systems Egorov’s Theorem 3: Let E : R 2 n → R 2 n Φ t be the Hamiltonian flow associated to the eigenvalue E of H 0 , then ε � e i � P ε e − i � H ε t/ε � H ε t/ε = � P ε � P ε + O ( ε ) P ε � A ε � a ◦ Φ t E A ε with principle symbol of the form � for any observable A 0 ( q, p ) = a ( q, p ) ⊗ 1 H f . Idea of the proof (PST 2003): Construct a unitary mapping U ε : � P ε L 2 ( R n , H f ) → L 2 ( R n , C ) � and apply the standard Egorov Theorem to the effective Hamiltonian U ε ∗ = � E ε + O ( ε ) . U ε � P ε � H ε � P ε � H ε � eff := � Partial semiclassical limits December 2010

2. Adiabatic slow-fast systems Idea of the proof (PST 2003): Construct a unitary mapping U ε : � P ε L 2 ( R n , H f ) → L 2 ( R n , C ) � and apply the standard Egorov Theorem to the effective Hamiltonian U ε ∗ = � E ε + O ( ε ) . U ε � P ε � H ε � P ε � H ε � eff := � Problem: The construction of this unitary requires the choice of a family of normalized eigenvectors ϕ ( q, p ) ∈ P ( q, p ) H f depending smoothly on ( q, p ) ∈ R 2 n . Put differently, the line bundle over R 2 n defined by the eigenspaces P ( q, p ) H f needs to be trivializable! In important applications, like periodic potentials in strong magnetic fields, the corresponding bundle is not trivializable and this fact has important physical consequences, like the integer quantum Hall effect. Partial semiclassical limits December 2010

3. A general Egorov theorem for adiabatic slow-fast systems Egorov’s Theorem 4: (Stiepan-T.) There is a flow Φ t ε : T ∗ M → T ∗ M ( M either R n or T n ) such that ε � e i � P ε e − i � H ε t/ε � H ε t/ε = � P ε � P ε + O ( ε 2 ) P ε � A ε � a ◦ Φ t ε A ε with principle symbol of the form � for any observable A 0 ( q, p ) = a ( q, p ) ⊗ 1 H f . Here Φ t ε is the Hamiltonian flow of � � �� H ε eff := E + ε i P, H 0 − E, P II 2 tr P =: E + εM with respect to the symplectic form � � � � Ω qq Ω pq 0 1 n ω := + ε . Ω qp Ω pp 0 − 1 n Partial semiclassical limits December 2010

3. A general Egorov theorem for adiabatic slow-fast systems Here � Ω qq � i tr( P [ ∇ q P, ( ∇ q P ) T ]) � � Ω pq i tr( P [ ∇ q P, ( ∇ p P ) T ]) = Ω qp Ω pp i tr( P [ ∇ p P, ( ∇ q P ) T ]) i tr( P [ ∇ p P, ( ∇ p P ) T ]) or shorter Ω ij = i tr P [ ∂ i P, ∂ j P ] is the curvature 2-form of the Berry connection. The Hamiltonian equations of motion have the form ∇ p ( E + εM ) + ε (Ω qq ∇ q E + Ω pq ∇ p E ) q ˙ = −∇ q ( E + εM ) + ε (Ω qp ∇ q E + Ω pp ∇ p E ) p ˙ = or alternatively ∇ p ( E + εM ) − ε (Ω qq ˙ p − Ω pq ˙ q ) + O ( ε 2 ) q ˙ = −∇ q ( E + εM ) − ε (Ω qp ˙ p − Ω pp ˙ q ) + O ( ε 2 ) . ˙ = p Partial semiclassical limits December 2010

3. A general Egorov theorem for adiabatic slow-fast systems As a Corollary we obtain the formula � � P ε � � � P ε � A ε ( t ) � a ◦ Φ t ( q, p ) d ω + O ( ε 2 ) ρ � Tr = T ∗ M ρ ( q, p ) � ε where d ω denotes integration with respect to the volume measure induced by the symplectic form ω , � � 1 − i ε tr ( P { P, P } ) I I d ω = d q d p . Partial semiclassical limits December 2010

4. The semiclassical model for magnetic Bloch Hamiltonians Consider the Hamiltonian � � 2 + V Γ ( x ) − φ ( εx ) H = 1 − i ∇ x + A 0 ( x ) + A ( εx ) II L 2 ( R 3 ) on 2 with a Γ-periodic potential V Γ , smooth electromagnetic potentials A and φ and the vector potential of a constant rational magnetic field B 0 , 2 B 0 x ⊥ . A 0 ( x ) = 1 After a suitable Bloch-Floquet transformation this operator takes the form � � 2 + V Γ ( y ) − φ (i ε ∇ τ H ε = 1 − i ∇ y + k + A 0 ( y ) + A (i ε ∇ τ k ) II � k ) 2 acting on L 2 ( M k , L 2 ( T y )) =: L 2 ( M k , H f ) . H ε is the Weyl-quantization of the operator-valued symbol � � � 2 + V Γ ( y ) − φ ( r ) H ( k, r ) = 1 − i ∇ y + k + A 0 ( y ) + A ( r ) II 2 with H : T ∗ M → L ( L 2 ( T y )). Partial semiclassical limits December 2010

4. The semiclassical model for magnetic Bloch Hamiltonians The eigenvalues E n ( k ) of the periodic Hamiltonian � � 2 + V Γ ( y ) H 0 ( k ) = 1 − i ∇ y + k + A 0 ( y ) II 2 are known as the magnetic Bloch bands and the corresponding spectral projections P n ( k ) define the magnetic Bloch bundle over the torus M . This bundle has in general nonvanishing Chern number and is thus not trivializ- able. Hence the standard techniques can not be applied. Partial semiclassical limits December 2010