Delocalization of Schr odinger eigenfunctions Nalini Anantharaman - PowerPoint PPT Presentation

I. Some history II. Quantum ergodicity III. Graphs Delocalization of Schr¨ odinger eigenfunctions Nalini Anantharaman Universit´ e de Strasbourg August 6, 2018

I. Some history II. Quantum ergodicity III. Graphs

I. Some history II. Quantum ergodicity III. Graphs I. Some history

I. Some history II. Quantum ergodicity III. Graphs I. Some history II. Quantum ergodicity

I. Some history II. Quantum ergodicity III. Graphs I. Some history II. Quantum ergodicity III. Toy model : discrete graphs

I. Some history II. Quantum ergodicity III. Graphs I. Some history

I. Some history II. Quantum ergodicity III. Graphs 1913 : Bohr’s model of the hydrogen atom Increasing n = 3 energy orbits n = 2 n = 1 Emitted photon with energy E = h f Kinetic momentum is “quantized” J “ nh , where n P N .

I. Some history II. Quantum ergodicity III. Graphs 1917 : A paper of Einstein Zum Quantensatz von Sommerfeld und Epstein

I. Some history II. Quantum ergodicity III. Graphs 1917 : A paper of Einstein Zum Quantensatz von Sommerfeld und Epstein

I. Some history II. Quantum ergodicity III. Graphs 1917 : A paper of Einstein Zum Quantensatz von Sommerfeld und Epstein

I. Some history II. Quantum ergodicity III. Graphs 1925 : operators / wave mechanics Heisenberg : physical observables are operators (matrices) obeying certain commutation rules r p p , p q s “ i � I . The “spectrum” is obtained by computing eigenvalues of the energy operator p H .

I. Some history II. Quantum ergodicity III. Graphs 1925 : operators / wave mechanics Heisenberg : physical observables are operators (matrices) obeying certain commutation rules r p p , p q s “ i � I . The “spectrum” is obtained by computing eigenvalues of the energy operator p H . De Broglie (1923) : wave particle duality. Schr¨ odinger (1925) : wave mechanics ´ ¯ ´ � 2 i � d ψ d t “ 2 m ∆ ` V ψ ψ p x , y , z , t q is the wave function.

I. Some history II. Quantum ergodicity III. Graphs 1925 : operators / wave mechanics In Heisenberg’s picture the spectrum is computed by diagonalizing the operator p H . ´ ¯ ´ � 2 2 m ∆ ` V In Schr¨ odinger’s picture, we must diagonalize .

I. Some history II. Quantum ergodicity III. Graphs 1925 : operators / wave mechanics In Heisenberg’s picture the spectrum is computed by diagonalizing the operator p H . ´ ¯ ´ � 2 2 m ∆ ` V In Schr¨ odinger’s picture, we must diagonalize . The two theories are mathematically equivalent : Schr¨ odinger’s picture corresponds to a representation of the Heisenberg algebra on the Hilbert space L 2 p R 3 q . But not physically equivalent !

I. Some history II. Quantum ergodicity III. Graphs Wigner 1950’ Random Matrix model for heavy nuclei Figure: Left : nearest neighbour spacing histogram for nuclear data ensemble (NDE). Right : Dyon-Mehta statistic ∆ for NDE. Source O. Bohigas

I. Some history II. Quantum ergodicity III. Graphs Spectral statistics for hydrogen atom in strong magnetic field Figure: Source Delande.

I. Some history II. Quantum ergodicity III. Graphs Billiard tables In classical mechanics, billiard flow φ t : p x , ξ q ÞÑ p x ` t ξ, ξ q . ´ ¯ ´ � 2 In quantum mechanics, i � d ψ dt “ 2 m ∆ ` 0 ψ.

I. Some history II. Quantum ergodicity III. Graphs Spectral statistics for several billiard tables Figure: Random matrices and chaotic dynamics

I. Some history II. Quantum ergodicity III. Graphs A list of questions and conjectures For classically ergodic / chaotic systems, show that the spectrum of the quantum system resembles that of large random matrices (Bohigas-Giannoni-Schmit conjecture);

I. Some history II. Quantum ergodicity III. Graphs A list of questions and conjectures For classically ergodic / chaotic systems, show that the spectrum of the quantum system resembles that of large random matrices (Bohigas-Giannoni-Schmit conjecture); study the probability density | ψ p x q| 2 , where ψ p x q is a solution to the Schr¨ odinger equation (Quantum Unique Ergodicity conjecture);

I. Some history II. Quantum ergodicity III. Graphs A list of questions and conjectures For classically ergodic / chaotic systems, show that the spectrum of the quantum system resembles that of large random matrices (Bohigas-Giannoni-Schmit conjecture); study the probability density | ψ p x q| 2 , where ψ p x q is a solution to the Schr¨ odinger equation (Quantum Unique Ergodicity conjecture); show that ψ p x q resembles a gaussian process ( x P B p x 0 , R � q , R " 1) (Berry conjecture).

I. Some history II. Quantum ergodicity III. Graphs A list of questions and conjectures This is meant in the limit � Ñ 0 (small wavelength). ? ´ ¯ ´ � 2 2 mE 2 m ∆ ` V ψ “ E ψ ù ñ } ∇ ψ } „ �

I. Some history II. Quantum ergodicity III. Graphs II. Quantum ergodicity M a billiard table / compact Riemannian manifold, of dimension d . In classical mechanics, billiard flow φ t : p x , ξ q ÞÑ p x ` t ξ, ξ q (or more generally, the geodesic flow = motion with zero acceleration).

I. Some history II. Quantum ergodicity III. Graphs II. Quantum ergodicity M a billiard table / compact Riemannian manifold, of dimension d . In quantum mechanics : ´ ¯ ´ � 2 i � d ψ dt “ 2 m ∆ ` 0 ψ ´ � 2 2 m ∆ ψ “ E ψ, in the limit of small wavelengths.

I. Some history II. Quantum ergodicity III. Graphs Disk Figure: Billiard trajectories and eigenfunctions in a disk. Source A. B¨ acker.

I. Some history II. Quantum ergodicity III. Graphs Sphere Figure: Spherical harmonics

I. Some history II. Quantum ergodicity III. Graphs Square / torus Figure: Eigenfunctions in a square. Source A. B¨ acker.

I. Some history II. Quantum ergodicity III. Graphs Figure: A few eigenfunctions of the Bunimovich billiard (Heller, 89).

I. Some history II. Quantum ergodicity III. Graphs Figure: Source A. B¨ acker

I. Some history II. Quantum ergodicity III. Graphs Eigenfunctions in a mushroom-shaped billiard. Source A. B¨ acker

I. Some history II. Quantum ergodicity III. Graphs Figure: Propagation of a gaussian wave packet in a cardioid. Source A. B¨ acker.

I. Some history II. Quantum ergodicity III. Graphs Figure: Propagation of a gaussian wave packet in a cardioid. Source A. B¨ acker.

I. Some history II. Quantum ergodicity III. Graphs Eigenfunctions in the high frequency limit M a billiard table / compact Riemannian manifold, of dimension d . ´ � 2 ∆ ψ k “ ´ λ k ψ k or 2 m ∆ ψ “ E ψ, } ψ k } L 2 p M q “ 1 , in the limit λ k Ý Ñ `8 . We study the weak limits of the probability measures on M , ˇ ˇ ˇ 2 dVol p x q . ˇ ψ k p x q

I. Some history II. Quantum ergodicity III. Graphs Let p ψ k q k P N be an orthonormal basis of L 2 p M q , with ´ ∆ ψ k “ λ k ψ k , λ k ď λ k ` 1 . QE Theorem (simplified): Shnirelman 74, Zelditch 85, Colin de Verdi` ere 85 Assume that the action of the geodesic flow is ergodic for the Liouville measure. Let a P C 0 p M q . Then ż ż ˇ ˇ ÿ ˇ ˇ 1 ˇ ˇ 2 d Vol p x q ´ ˇ ˇ ψ k p x q a p x q a p x q d Vol p x q ˇ Ý λ Ñ8 0 . Ñ ˇ N p λ q M M λ k ď λ

I. Some history II. Quantum ergodicity III. Graphs Let p ψ k q k P N be an orthonormal basis of L 2 p M q , with ´ ∆ ψ k “ λ k ψ k , λ k ď λ k ` 1 . QE Theorem (simplified): Shnirelman 74, Zelditch 85, Colin de Verdi` ere 85 Assume that the action of the geodesic flow is ergodic for the Liouville measure. Let a P C 0 p M q . Then ż ż ÿ ˇ ˇ 1 ˇ 2 d Vol p x q ´ ˇ ψ k p x q a p x q a p x q d Vol p x q Ý λ Ñ8 0 . Ñ N p λ q M M λ k ď λ

I. Some history II. Quantum ergodicity III. Graphs Let p ψ k q k P N be an orthonormal basis of L 2 p M q , with ´ ∆ ψ k “ λ k ψ k , λ k ď λ k ` 1 . QE Theorem (simplified): Shnirelman 74, Zelditch 85, Colin de Verdi` ere 85 Assume that the action of the geodesic flow is ergodic for the Liouville measure. Let a P C 0 p M q . Then ż ż ˇ ˇ ÿ ˇ ˇ 1 ˇ ˇ 2 d Vol p x q ´ ˇ ˇ ψ k p x q a p x q a p x q d Vol p x q ˇ Ý λ Ñ8 0 . Ñ ˇ N p λ q M M λ k ď λ

I. Some history II. Quantum ergodicity III. Graphs Equivalently, there exists a subset S Ă N of density 1, such that ż ż ˇ ˇ ˇ 2 dVol p x q ´ ˇ ψ k p x q a p x q ´ ´ ´ ´ Ñ a p x q d Vol p x q . k Ý Ñ`8 M M k P S

I. Some history II. Quantum ergodicity III. Graphs Equivalently, there exists a subset S Ă N of density 1, such that ż ż ˇ ˇ ˇ 2 dVol p x q ´ ˇ ψ k p x q a p x q ´ ´ ´ ´ Ñ a p x q d Vol p x q . k Ý Ñ`8 M M k P S Equivalently, ˇ ˇ ˇ 2 Vol p x q ´ ˇ ψ k p x q ´ ´ ´ ´ Ñ dVol p x q k Ý Ñ`8 k P S in the weak topology.

I. Some history II. Quantum ergodicity III. Graphs The full statement uses analysis on phase space, i.e. � ( T ˚ M “ p x , ξ q , x P M , ξ P T ˚ x M . For a “ a p x , ξ q a “reasonable” function on phase space, we can define an operator on L 2 p M q , ´ ¯ D x “ 1 a p x , D x q i B x .