Algebra of waves A. A. Kutsenko Jacobs University, Bremen, Germany - PowerPoint PPT Presentation

Algebra of waves A. A. Kutsenko Jacobs University, Bremen, Germany February 25, 2019 1

Some aspects of mathematical theory of waves Algebras of Spectral Inverse integral theory problems operators Traces and Waves Representation determinants of theory of integral algebras operators Integral continued fractions 2

Some basic facts about waves What are waves, their periods, amplitudes, etc.? 3

Waves are functions In general, waves are oscillations u ( x , t ) = e i ω t − i k · x u 0 ( x ) , where ω is a frequency, k is a wave-number, and u 0 ( x ) is 1-periodic function along the direction of the wave propagation (along the vector k ). Wave length (space-period) L , time-period T , and amplitude U are L = 2 π/ | k | , T = 2 π/ω, U = max | u 0 | . 1) If u 0 ( x ) is periodic along the wave propagation and bounded, non-decreasing along other directions then u ( x , t ) is a volume wave. 2) If u 0 ( x ) is periodic along the wave propagation and decreasing along other directions then u ( x , t ) is a guided wave. 4

Dispersion diagrams (spectrum) Usually, for a given wave u ( x , t ) = e i ω t − i k · x u 0 ( x ) , the parameters ω , k , and u 0 ( x ) are related by certain equations ω = ω ( k ) , u 0 = u 0 [ k , ω ] . To find them, we should substitute u into the wave equation ¨ u = A u , where A is some periodic operator, e.g. A = ρ − 1 ∇ · µ ∇ or discrete A u n = ρ − 1 � n ∼ n ′ µ n ′ ( u n ′ − u n ), etc.. After substitution − ω 2 u 0 = A k u 0 . Hence, ω 2 = ω 2 ( k ) are ”eigenvalues”, and u 0 = u 0 [ k , ω ] are corresponding ”eigenvectors” of −A k . 5

Volume, guided waves and dispersion diagrams ω 5 π 0 k 1 6

About local waves Usually, we can not observe guided (local) waves in uniform and purely periodic structures. To observe them we should consider periodic structures with embedded defects of lower dimension. 7

Example of periodic lattices with defects https://phys.org http://physicsworld.com 8

Periodic lattices We can define N -periodic lattice with M -point unit cell as follows Γ = [1 , ..., M ] × Z N . 9

Periodic operators Any (bounded) operator A : ℓ 2 (Γ) → ℓ 2 (Γ) which commutes with all shift op- erators S m u ( j , n ) = u ( j , n + m ) , u ∈ ℓ 2 (Γ) is called a periodic operator. 10

Fourier-Floquet-Bloch transform The corresponding transformation based on Fourier series N , M := L 2 ([0 , 1] N → C M ) , F : ℓ 2 (Γ) → L 2 � e 2 π i k · n u ( j , n ) ( F u ) j ( k ) = n ∈ Z N allows us to rewrite our periodic operator A as an operator of multiplication by a matrix-valued function A A := FAF − 1 : L 2 ˆ ˆ N , M → L 2 N , M , A u = Au . 11

Periodic operators after F-F-B transform A periodic operator A unitarily equivalent to the following oper- ator ˆ A : L 2 N , M → L 2 N , M , ˆ A u ( k ) = A 0 ( k ) u ( k ) with some (usually continuous) M × M matrix-valued function A 0 ( k ) depending on the ”quasi- momentum” k ∈ [0 , 1] N . 12

Spectrum of periodic operators For the operator of multiplication by the matrix-valued function ˆ A u ( k ) = A 0 ( k ) u ( k ) the spectrum is just eigenvalues of this matrix for different quasi-momentums sp ( ˆ A ) = { λ : det ( A 0 ( k ) − λ I ) = 0 for some k } = � M � { λ j ( k ) } . j =1 k ∈ [0 , 1] N 13

Periodic operators with linear defects ( N = 2) In this case our periodic operator A : L 2 ˆ N , M → L 2 N , M , takes the form ˆ A u = A 0 u + A 1 � B 1 u � 1 with some (usually continuous) matrix-valued functions A , B and � 1 �·� 1 := · dk 1 . 0 14

Periodic operators with linear and point defects ( N = 2) In this case our periodic operator A : L 2 ˆ N , M → L 2 N , M , takes the form ˆ A u = A 0 u + A 1 � B 1 u � 1 + A 2 � B 2 u � 2 with some (usually continuous) matrix-valued functions A , B and � 1 � 1 �·� 2 := · dk 1 dk 2 . 0 0 15

Periodic operator with defects (general case) In general, a periodic operator with defects is unitarily equivalent to the operator ˆ A : L 2 N , M → L 2 N , M of the form ˆ A u = A 0 u + A 1 � B 1 u � 1 + ... + A N � B N u � N . with continuous matrix-valued functions A , B and � 1 � 1 �·� 1 = · dk 1 , �·� j +1 = �·� j dk j +1 . 0 0 Remark. For simplicity we will write A instead of ˆ A . The spectrum of this operator is � sp ( A ) = { λ : A − λ I is non − invertible } = { λ : A is non − invertible } , where � A has the same form as A but with A 0 − λ I instead of A 0 . 16

Test for invertibility of a periodic operator with defects A = A 0 · + A 1 � B 1 ·� 1 + ... + A N � B N ·� N = ( A 0 · )( I + A − 1 0 A 1 � B 1 ·� 1 + ... + A − 1 0 A N � B N ·� N ) = ( A 0 · )( I + A 10 � B 1 ·� 1 + ... + A N 0 � B N ·� N ) = ( A 0 · ) ( I + A 10 � B 1 ·� 1 )( I − A 10 ( I + � B 1 A 10 � 1 ) − 1 � B 1 ·� 1 ) ( I + A 10 � B 1 ·� 1 + ... + A N 0 � B N ·� N ) � �� � = I = ( A 0 · )( I + A 10 � B 1 ·� 1 )( I + A 21 � B 2 ·� 2 + ... + A N 1 � B N ·� N ) = ( A 0 · )( I + A 10 � B 1 ·� 1 )( I + A 21 � B 2 ·� 2 )( I + A 32 � B 3 ·� 3 + ... + A N 2 � B N ·� N ) = ............................... = ( A 0 · )( I + A 10 � B 1 ·� 1 )( I + A 21 � B 2 ·� 2 ) ... ( I + A N , N − 1 � B N ·� N ) 17

Test for invertibility of a periodic operator with defects Theorem (from J. Math. Anal. Appl., 2015) Step 0. Define π 0 = det E 0 , E 0 = A 0 . If π 0 ( k 0 ) = 0 for some k 0 ∈ [0 , 1] N then A is non-invertible else define A j 0 = A − 1 0 A j , j = 1 , ..., N . Step 1. Define π 1 = det E 1 , E 1 = I + � B 1 A 10 � 1 . 1 ∈ [0 , 1] N − 1 then A is non-invertible else define If π 1 ( k 0 1 ) = 0 for some k 0 A j 1 = A j 0 − A 10 E − 1 1 � B 1 A j 0 � 1 , j = 2 , ..., N . Step 2. Define π 2 = det E 2 , E 2 = I + � B 2 A 21 � 2 . 2 ∈ [0 , 1] N − 2 then A is non-invertible else define If π 2 ( k 0 2 ) = 0 for some k 0 A j 2 = A j 1 − A 21 E − 1 2 � B 2 A j 1 � 2 , j = 3 , ..., N . ********* Step N. Define π N = det E N , E N = I + � B N A N , N − 1 � N . If π N = 0 then A is non-invertible else A is invertible. 18

Summary The following expansion as a product of elementary operators is fulfilled A = A 0 · + A 1 � B 1 ·� 1 + ... + A N � B N ·� N = ( A 0 · )( I + A 10 � B 1 ·� 1 )( I + A 21 � B 2 ·� 2 ) ... ( I + A N , N − 1 � B N ·� N ) , where A ij are derived from A n , B n by using algebraic operations (including taking inverse matrices) and a few number of integrations. The inverse is A − 1 = ( I − A N , N − 1 E − 1 N � B N ·� N ) ... ( I − A 10 E − 1 1 � B 1 ·� 1 )( A − 1 0 · ) , where E j = I + � B j A j , j − 1 � j . The determinant is π π π ( A ) = ( π 1 , ..., π N ) , π j = det E j . 19

Embedded defects 20

Determinants in the case of embedded defects In this case the operator has a form A· = A 0 · + A 1 �·� 1 + ... + A N �·� N , where A n does not depend on k 1 , ..., k n . Define the matrix-valued integral continued fractions � � − 1 � I � − 1 I C 0 = A 0 , C 1 = A 1 + , C 2 = A 2 + � � − 1 A 0 I 1 A 1 + A 0 1 2 and so on C j = A j + � C − 1 j − 1 � − 1 . Then j π j ( A ) = det ( � C − 1 j − 1 � j C j ) . Note that if all A j are self-adjoint then A is self-adjoint and all C j are self-adjoint. J. Math. Phys., 2017 21

Spectrum of periodic operators with defects The spectrum of A has the form N � sp ( A ) = σ n , σ n = { λ : � π n = 0 for some k } , n =0 π n ≡ π n ( A − λ I ) ≡ π n ( λ, k n +1 , ..., k N ) . where � The component σ 0 coincides with the spectrum of purely periodic operator A 0 u without defects. All components σ n , n < N are continuous (intervals), the component σ N is discrete. Also note that σ n does not depend on the defects of dimensions greater than n , i.e. of A n +1 , B n +1 , A n +2 , B n +2 and so on. 22

Determinants of periodic operators with defects For all continuous matrix-valued functions A , B on [0 , 1] N of appropriate sizes introduce H = {A : A = A 0 · + A 1 � B 1 ·� 1 + ... + A N � B N ·� N } ⊂ B ( L 2 N , M ) , G = {A ∈ H : A is invertible } . Theorem (arxiv.org, 2015) The set H is a an operator algebra. The subset G is a group. The mapping π π π ( A ) := ( π 0 ( A ) , ..., π N ( A )) is a group homomorphism between G and C 0 × C 1 × ... × C N , where C n is the commutative group of non-zero continuous functions depending on ( k n +1 , ..., k N ) ∈ [0 , 1] N − n . 23

Traces of periodic operators with defects Define π π π ( I + t A ) − π π π ( I ) τ τ τ ( A ) = lim . t t → 0 Then Theorem (arxiv.org, 2015) The following identities are fulfilled τ τ τ ( A ) = ( Tr A 0 , � Tr B 1 A 1 � 1 , ..., � Tr B N A N � N ) , τ τ τ ( α A + β B ) = ατ τ τ ( A ) + βτ τ τ ( B ) , τ τ τ ( AB ) = τ τ τ ( BA ) , π ( e A ) = e τ τ τ ( A ) , π π π π π π π ( AB ) = π π ( A ) π π ( B ) . 24