On the correspondence between harmonic analysis and spectral theory - PowerPoint PPT Presentation

On the correspondence between harmonic analysis and spectral theory Zhirayr Avetisyan UCL Maths Imperial, 2017 Zhirayr Avetisyan (UCL Maths) Harmonic analysis vs spectral theory Imperial, 2017 1 / 24

Outline Physics: elementary particle states Old and new examples General approach Zhirayr Avetisyan (UCL Maths) Harmonic analysis vs spectral theory Imperial, 2017 2 / 24

Physics: elementary particle states Zhirayr Avetisyan (UCL Maths) Harmonic analysis vs spectral theory Imperial, 2017 3 / 24

Physics: elementary particle states Quantum system H - complex separable Hilbert space O = { T : H �→ H } - observables, i.e., densely defined normal operators 0 ̸ = C f ⊂ H - a state, i.e., a 1-dimensional Hilbert subspace Zhirayr Avetisyan (UCL Maths) Harmonic analysis vs spectral theory Imperial, 2017 4 / 24

Physics: elementary particle states Dynamics and symmetries Dynamics: Distinguished observable H ∈ O , time variable t ∈ R , d T dt = ı [ H , T ] , ∀ T ∈ O . Symmetries: A group G of unitary operators U ∈ O such that [ U , H ] = 0 . Zhirayr Avetisyan (UCL Maths) Harmonic analysis vs spectral theory Imperial, 2017 5 / 24

Physics: elementary particle states Quantum mechanics G - a real Lie group M - a G -manifold ν - a G -invariant measure (volume form) H = L 2 ( M , ν ) U g f ( x ) = f ( g − 1 x ) , ∀ f ∈ H , ∀ g ∈ G H - a G -invariant Laplacian or Schrödinger operator Zhirayr Avetisyan (UCL Maths) Harmonic analysis vs spectral theory Imperial, 2017 6 / 24

Physics: elementary particle states What is an elementary particle state? Given: S = { T : H → H } - a semigroup of operators Find: (Ω , ˆ ν ) - a measure space ∫ ⊕ F : H → Ω d ˆ ν ( ω ) H ω - a unitary operator (Fourier transform) H ω - irreducible invariant subspace under S for a.e. ω ∈ Ω C f ⊂ H ω - elementary particle states w.r.t. S Zhirayr Avetisyan (UCL Maths) Harmonic analysis vs spectral theory Imperial, 2017 7 / 24

Physics: elementary particle states Dynamical particles = spectral theory { } H n , n ∈ N S = Ω ∋ ω = ( λ, ω λ ) , λ ∈ σ ( H ) , ω λ ∈ Ω λ ∫ ⊕ ∫ ⊕ F : H → d µ ( λ ) H ω σ ( H ) Ω λ µ - spectral measure, H H ω = λ 1 H ω = C f ω - wave function, spectral mode Zhirayr Avetisyan (UCL Maths) Harmonic analysis vs spectral theory Imperial, 2017 8 / 24

Physics: elementary particle states Symmetry particles = harmonic analysis/representation theory { } S = U g , g ∈ G G irrep on H ω = H π , π ∈ ˆ Ω ∋ ω = ( π, ω π ) , ω π = 1 , ..., d π ∫ ⊕ d π ⊕ H ω F : H → d ˆ ν ( π ) ˆ G ω π = 1 ν - ’Plancherel’ measure ˆ H ω - Wigner’s elementary particle states Zhirayr Avetisyan (UCL Maths) Harmonic analysis vs spectral theory Imperial, 2017 9 / 24

Physics: elementary particle states Question : What is an electron, a dynamical or a symmetry particle? Zhirayr Avetisyan (UCL Maths) Harmonic analysis vs spectral theory Imperial, 2017 10 / 24

Physics: elementary particle states Question : What is an electron, a dynamical or a symmetry particle? Answer : It is both. Zhirayr Avetisyan (UCL Maths) Harmonic analysis vs spectral theory Imperial, 2017 11 / 24

Old and new examples Zhirayr Avetisyan (UCL Maths) Harmonic analysis vs spectral theory Imperial, 2017 12 / 24

Old and new examples Example 1: Laplacian on the line. H = ∆ = − ∂ 2 H = L 2 ( R ) . M = R , d ν ( x ) = dx , x , O ∋ X , P, X f ( x ) = xf ( x ) , P = − ı∂ x f ( x ) . ∫ ⊕ ⊕ C e ıλω λ x . Spectral: σ ( H ) = [ 0 , + ∞ ) , H ≃ d λ [ 0 , + ∞ ) ω λ = ± 1 ˆ π ( g ) = e ıπ g . G = R , U g f ( x ) = f ( x − g ) , G = R , H π = C , ∫ ⊕ d π C e ıπ x . Harmonic: H ≃ R Zhirayr Avetisyan (UCL Maths) Harmonic analysis vs spectral theory Imperial, 2017 13 / 24

Old and new examples Example 2: Laplacian on the plane. M = R 2 , d ν ( x ) = dx 1 dx 2 , H = ∆ = − ∂ 2 x 1 − ∂ 2 H = L 2 ( R 2 ) . x 2 , X i f ( x ) = x i f ( x ) , O ∋ X i , P i , P i = − ı∂ x i f ( x ) , i = 1 , 2. ∫ ⊕ ∫ ⊕ S 1 dS ( ω λ ) C e ı ( λω λ , x ) . Spectral: H ≃ d λλ [ 0 , + ∞ ) G = E ( 2 ) = R 2 ⋊ U ( 1 ) , ˆ gx = e ıϕ x + y , G = R + . g = ( y , ϕ ) , π ( g ) f ( ψ ) = e ıπ y 1 f ( ψ − ϕ ) . H π = L 2 ( S 1 ) , ∫ ⊕ π ( g ) e ı ( λω λ , x ) = e ı ( λω ′ λ , x ) . Harmonic: H ≃ d ππ H π , R + Zhirayr Avetisyan (UCL Maths) Harmonic analysis vs spectral theory Imperial, 2017 14 / 24

Old and new examples Example 3: Laplacian on the sphere. M = S 2 , d ν ( ϕ ) = sin ϕ 1 d ϕ 1 d ϕ 2 . 1 1 sin ϕ 1 ∂ ϕ 1 sin ϕ 1 ∂ ϕ 1 − sin 2 ϕ 1 ∂ 2 H = L 2 ( S 2 ) . H = ∆ = − ϕ 2 , l ⊕ ∞ ⊕ C Y m σ ( H ) = { l ( l + 1 ) | l ∈ N 0 } , Spectral: H ≃ l ( ϕ ) . l = 0 m = − l ˆ G = SO ( 3 ) , G = N 0 , ⊕ ∞ H π = C { Y m π ( ϕ ) } π Harmonic: H ≃ H π , m = − π . π = 0 Zhirayr Avetisyan (UCL Maths) Harmonic analysis vs spectral theory Imperial, 2017 15 / 24

Old and new examples Example 4: Solvable Bianchi groups.     1 0 0    e x 3 M  ( x 1 , x 2 , x 3 ) ∈ R 3 g = ( x 1 , x 2 , x 3 ) , x 1 G =  ,  x 2 M ( I ) M ( II ) M ( III ) M ( IV ) M ( V ) ( 0 ) ( 1 ) ( 1 ) ( 1 ) 1 0 1 0 0 0 0 0 0 0 1 0 1 M ( VI q ) , − 1 < q ≤ 1, q ̸ = 0 M ( VII p ) , p ≥ 0 ( 1 ) ( p ) 0 1 0 − q − 1 p Zhirayr Avetisyan (UCL Maths) Harmonic analysis vs spectral theory Imperial, 2017 16 / 24

Old and new examples Example 4: Solvable Bianchi groups (2). Left and right generators             L 1 1 0 0 ∂ x 1 R 1 0 ∂ x 1  e x 3 M ⊤   =       =    L 2 0 1 0 ∂ x 2 R 2 0 ∂ x 2 ( x 1 , x 2 ) M ⊤ L 3 1 ∂ x 3 R 3 0 0 1 ∂ x 3 h − 1 = h ij R i ⊗ R j Left Haar measure √ det h ∗∗ e − x 3 Tr M dx 1 dx 2 dx 3 . d ν h ( x ) = Zhirayr Avetisyan (UCL Maths) Harmonic analysis vs spectral theory Imperial, 2017 17 / 24

Old and new examples Example 4: Laplacian on solvable Bianchi groups. M = R 3 , H = L 2 ( R 3 , ν h ) . d ν ( x ) = d ν h ( x ) , H = h ij R i R j + Tr M h 3 i R i , σ ( H ) = [ 0 , + ∞ ) . ∫ ⊕ ∫ ⊕ ν ( k 1 ) dk 2 e k 2 Tr M ⊕ Spectral: H ≃ d λ R 2 d ˆ C ξ k , h ,λ,ω λ ( x ) . [ 0 , + ∞ ) ω λ = ± 1 G ≃ R 2 / e R M ⊤ , ˆ U g f ( x ) = f ( g − 1 x ) , ∫ ⊕ ⊕ ∞ Harmonic: H ≃ d ˆ ν ( π ) H π , U g ξ k , h ,λ,ω λ ( x ) = ξ k ′ , h ,λ,ω λ ( x ) ˆ G ω π = 1 k ′ = e g 3 M ⊤ k , g = ( g 1 , g 2 , g 3 ) ∈ G . Zhirayr Avetisyan (UCL Maths) Harmonic analysis vs spectral theory Imperial, 2017 18 / 24

General approach Zhirayr Avetisyan (UCL Maths) Harmonic analysis vs spectral theory Imperial, 2017 19 / 24

General approach Fourier transform G - type I unimodular locally compact group, ν - Haar measure, ˆ G - unitary dual, ˆ ν - Plancherel measure ∫ ⊕ F : L 2 ( G , ν ) → ν ( π ) H π ⊗ H ∗ d ˆ π , ˆ G ∫ ∫ ˆ ν ( π ) Tr [ π ∗ ( x )ˆ f ( π ) = d ν ( x ) f ( x ) π ( x ) , f ( x ) = d ˆ f ( π )] , ˆ G G Plancherel theorem: ∫ ν ( π ) Tr [ˆ f ( π ) ∗ ˆ ∥ f ∥ 2 2 = d ˆ f ( π )] . ˆ G Zhirayr Avetisyan (UCL Maths) Harmonic analysis vs spectral theory Imperial, 2017 20 / 24

General approach Fourier multipliers U g f ( x ) = f ( g − 1 x ) , f ∈ L 2 ( G , ν ) , g ∈ G . � � U g f ( π ) = π ( g )ˆ f ∗ h ( π ) = ˆ f ( π )ˆ f ( π ) , g ( π ) , where ∫ d ν ( y ) f ( y ) h ( y − 1 x ) . f ∗ h ( x ) = G H ∈ B ( H ) , [ U g , H ] = 0, ∀ g ∈ G , then H f = ˆ � f ( π ) H π , H π ∈ B ( H π ) , i.e., ∫ ⊕ H ≃ d ˆ ν ( π ) × H π . ˆ G Zhirayr Avetisyan (UCL Maths) Harmonic analysis vs spectral theory Imperial, 2017 21 / 24

General approach Compact groups ˆ G is discrete and d π = dim H π < ∞ , ⊕ L 2 ( G , ν ) ≃ H π ⊗ H ∗ π . ˆ G If { e j } d π j = 1 - eigenfunctions of H π then { } d π ξ π i , j ( x ) = e ∗ i π ∗ ( x ) e j i , j = 1 - eigenfunctions of H, C ( G ) functions. { } ˆ Peter-Weyl theorem: H π ⊗ H ∗ ξ π π = C and i , j { } ˆ π ∈ ˆ ξ π C G dense in C ( G ) . i , j Zhirayr Avetisyan (UCL Maths) Harmonic analysis vs spectral theory Imperial, 2017 22 / 24

General approach Eigenfunction expansions ’... it is perhaps worth posing the general question of what conditions on ∆ are needed for such a theory to exist (some hints in this direction are in Maurin’66)...’ (R. Strichartz, ’Harmonic Analysis as Spectral Theory of Laplacians’, 1989) Gelfand triple: D ⊂ H ⊂ D ′ , D - nuclear, id : D → H continuous. D - core of H, H : D → D continuous. Eigenfunctions ξ λ,ω λ ∈ D ′ . If H hypoelliptic, then ξ λ,ω λ regular. Zhirayr Avetisyan (UCL Maths) Harmonic analysis vs spectral theory Imperial, 2017 23 / 24

General approach Decomposition of continuity T π �→ Tr [ T π π ( x )] , L 1 ( H π ) ≃ E π ⊂ C b ( G ) (Godement’52). {∫ } E = d ˆ ν ( π ) α ( π ) , α ( π ) ∈ E π ⊂ C b ( G ) . ˆ G Generalized Peter-Weyl: E is dense in C b ( G ) . Generalized Bochner: � C b ( G ) is the space of E π -valued finite measures on ˆ G . Zhirayr Avetisyan (UCL Maths) Harmonic analysis vs spectral theory Imperial, 2017 24 / 24