Direct and inverse problems for one-dimensional Dirac operators - PowerPoint PPT Presentation

Akademia Górniczo-Hutnicza im. Stanisława Staszica w Krakowie AGH University of Science and Technology Direct and inverse problems for one-dimensional Dirac operators with nonlocal potentials Kamila Dębowska 1 1 Faculty of Applied Mathematics 2 NASU Institute of Mathematics Department of Functional Analysis 28 February 2019 joint work with L.P. Nizhnik 2 Kamila Dębowska (AGH) Direct and inverse problems for one-dimensional Dirac operators with nonlocal potentials 28 February 2019 1 / 33

Overview Sturm-Liouville operators with nonlocal potentials on the interval 1 First order differential operators with nonlocal potentials on the interval 2 Dirac systems with nonlocal potentials on the interval 3 Literature 4 Kamila Dębowska (AGH) Direct and inverse problems for one-dimensional Dirac operators with nonlocal potentials 28 February 2019 2 / 33

Part I Sturm-Liouville operators with nonlocal potentials on the interval Kamila Dębowska (AGH) Direct and inverse problems for one-dimensional Dirac operators with nonlocal potentials 28 February 2019 3 / 33

Sturm-Liouville operators with nonlocal potentials on the interval Sturm-Liouville eigenvalue problems Problem Consider nonlocal Sturm-Liouville eigenvalue problems of the form ( T ψ ) ( x ) ≡ − d 2 ψ ( x ) + v ( x ) ψ ( 1 ) = λψ ( x ) , 0 ≤ x ≤ 1 , dx 2 with the boundary conditions ψ ( 0 ) = ψ ′ ( 1 ) + � ψ, v � L 2 = 0 , where v ∈ L 2 ( 0 , 1 ) is the nonlocal potential and λ ∈ C is the spectral parameter. Denote �· , ·� L 2 by the usual inner product in L 2 ( 0 , 1 ) . Kamila Dębowska (AGH) Direct and inverse problems for one-dimensional Dirac operators with nonlocal potentials 28 February 2019 4 / 33

Sturm-Liouville operators with nonlocal potentials on the interval Unperturbed operators T ψ = − d 2 ψ ( x ) + v ( x ) ψ ( 1 ) dx 2 D ( T ) = { ψ ∈ W 2 2 ( 0 , 1 ) | ψ ( 0 ) = ψ ′ ( 1 ) + � ψ, v � L 2 = 0 } T 0 ψ = − d 2 ψ ( x ) dx 2 D ( T 0 ) = { ψ ∈ W 2 2 ( 0 , 1 ) | ψ ( 0 ) = ψ ( 1 ) = 0 } T 1 ψ = − d 2 ψ ( x ) dx 2 D ( T 1 ) = { ψ ∈ W 2 2 ( 0 , 1 ) | ψ ( 0 ) = ψ ′ ( 1 ) = 0 } Kamila Dębowska (AGH) Direct and inverse problems for one-dimensional Dirac operators with nonlocal potentials 28 February 2019 5 / 33

Sturm-Liouville operators with nonlocal potentials on the interval Spectrum The operators T 0 and T 1 are self-adjoint and have discrete spectra σ ( T 0 ) = { π 2 n 2 } n ∈ N and σ ( T 1 ) = { π 2 ( n − 1 2 ) 2 } n ∈ N . Lemma The operator T is self-adjoint and has a discrete spectrum { λ n } n ∈ N , where λ 1 ≤ λ 2 ≤ . . . and each eigenvalue is repeated according to its multiplicity. Moreover, T is a rank-one perturbation of the operator T 0 and the spectra of the operators T and T 0 weakly interlace, i.e., λ n ≤ π 2 n 2 ≤ λ n + 1 for every n ∈ N . Kamila Dębowska (AGH) Direct and inverse problems for one-dimensional Dirac operators with nonlocal potentials 28 February 2019 6 / 33

Sturm-Liouville operators with nonlocal potentials on the interval Resolvents of T 0 and T 1 Integral operators � 1 ( T j − z 2 ) − 1 f ( x ) = G j ( x , s ; z ) f ( s ) ds , j = 0 , 1 , 0 Green functions � sin zx sin z ( 1 − s ) 1 for s > x , G 0 ( x , s ; z ) = sin z ( 1 − x ) sin zs for s < x , z sin z 1 � sin zx cos z ( 1 − s ) for s > x , G 1 ( x , s ; z ) = cos z ( 1 − x ) sin zs z cos z for s < x . Kamila Dębowska (AGH) Direct and inverse problems for one-dimensional Dirac operators with nonlocal potentials 28 February 2019 7 / 33

Sturm-Liouville operators with nonlocal potentials on the interval Characteristic function Characteristic function � 1 � 1 � 1 sin zs ( v ( s ) + v ( s )) ds − sin z d ( z ) = cos z + G 0 ( x , s ; z ) v ( s ) v ( x ) dsdx , z z 0 0 0 z ( T − z 2 ) − 1 g ( x ) = ( T 0 − z 2 ) − 1 g ( x ) + sin zd ( z ) ψ ( x ; z ) � g , ψ ( · , z ) � v k be the k-th Fourier coefficient of the function v ( x ) = � ∞ Let ˆ k = 1 ˆ v k sin π kx . Other form ∞ d ( z ) = cos z + sin z a k � z 2 − π 2 k 2 , 2 z k = 1 where v k + ( − 1 ) k 2 π k | 2 − ( 2 π k ) 2 a k = | ˆ Kamila Dębowska (AGH) Direct and inverse problems for one-dimensional Dirac operators with nonlocal potentials 28 February 2019 8 / 33

Sturm-Liouville operators with nonlocal potentials on the interval Direct spectral analysis Theorem Every eigenvalue of the operator T is a squared zero of its characteristic function d and, conversely, every squared zero of d is an eigenvalue of T . The number π 2 n 2 , n ∈ N , is an eigenvalue of T if and only if v n = ( − 1 ) n + 1 2 π n , ˆ and this relation is equivalent to d ( π n ) = 0. All eigenvalues z 2 not in the spectrum of T 0 are simple, and simple are the corresponding zeros z of d (except for the case where z = 0, which is then a zero of even order of d ). If π 2 n 2 for some n ∈ N is an eigenvalue of T , then this eigenvalue is multiple if and only if � 1 � 1 G 1 ( x , s ; π n ) v ( s ) v ( x ) dsdx = 0 , 0 0 in this and only in this case the number π n is a multiple zero of d . Kamila Dębowska (AGH) Direct and inverse problems for one-dimensional Dirac operators with nonlocal potentials 28 February 2019 9 / 33

Sturm-Liouville operators with nonlocal potentials on the interval Direct spectral analysis Theorem The multiplicity of a non-zero eigenvalue z 2 of the operator T equals the order of the corresponding zero z of the characteristic function d , and both do not exceed 2. If z = 0 is an eigenvalue of T , then the order of z = 0 as a zero of d is 2. Asymptotics The eigenvalues λ 1 ≤ λ 2 ≤ . . . satisfy the asymptotic distribution λ n = π ( n − 1 2 ) + µ n � n for some sequence ( µ n ) n ∈ N in ℓ 2 ( N ) . Kamila Dębowska (AGH) Direct and inverse problems for one-dimensional Dirac operators with nonlocal potentials 28 February 2019 10 / 33

Sturm-Liouville operators with nonlocal potentials on the interval Inverse spectral analysis Given the spectrum σ ( T ) of an operator find the nonlocal potential v . Algorithm λ k − z 2 1. Given σ ( T ) , construct the function d via d ( z ) = � 2 ) 2 . k ∈ N π 2 ( k − 1 2. Calculate the values d ( π n ) , n ∈ N . 3. For every n ∈ N , solve the quadratic equations v n + ( − 1 ) n 2 π n ) 2 = ( − 1 ) n ( 2 π n ) 2 d ( π n ) for ˆ (ˆ v n , taking the solution that satisfies the relation ( − 1 ) n + 1 ˆ v n ≤ 2 π n . 4. Put v ( x ) = � n ∈ N ˆ v n sin π nx . Kamila Dębowska (AGH) Direct and inverse problems for one-dimensional Dirac operators with nonlocal potentials 28 February 2019 11 / 33

Sturm-Liouville operators with nonlocal potentials on the interval Example of a solution to an inverse problem Example Let λ 1 = π 2 and λ n = π 2 ( n − 1 2 ) 2 for all n ≥ 2 . Then d ( z ) = z 2 − π 2 cos z , z 2 − π 2 4 so that d ( π k ) = ( − 1 ) k k 2 − 1 4 , ˆ v 1 = 2 π and k 2 − 1 � � k 2 − 1 � v k = ( − 1 ) k + 1 2 π k ˆ 1 − for k ≥ 2 . k 2 − 1 4 Kamila Dębowska (AGH) Direct and inverse problems for one-dimensional Dirac operators with nonlocal potentials 28 February 2019 12 / 33

Part II First order differential operators with nonlocal potentials on the interval Kamila Dębowska (AGH) Direct and inverse problems for one-dimensional Dirac operators with nonlocal potentials 28 February 2019 13 / 33

First order differential operators with nonlocal potentials on the interval Boundary value problem Consider the following nonlocal eigenvalue problems ( L ψ ) ( x ) ≡ i d ψ ( x ) + v ( x ) ψ + = λψ ( x ) , 0 ≤ x ≤ l , (1) dx with the boundary conditions � l ψ − + i ψ ( x ) v ( x ) dx = 0 , (2) 0 ψ + := 1 2 ( ψ ( l ) + ψ ( 0 )) , ψ − := ψ ( l ) − ψ ( 0 ) , v ∈ L 2 ( 0 , l ) . The corresponding operator ( A ψ )( x ) = i d ψ ( x ) + v ( x ) ψ + dx � l � � ψ ∈ W 1 D ( A ) = 2 ( 0 , l ) : ψ − + i ψ ( x ) v ( x ) dx = 0 0 Kamila Dębowska (AGH) Direct and inverse problems for one-dimensional Dirac operators with nonlocal potentials 28 February 2019 14 / 33

First order differential operators with nonlocal potentials on the interval Direct spectral analysis The operator A is self-adjoint. Let A − and A + be the differential operators i d dx on L 2 ( 0 , l ) with the domains D ( A − ) = { ψ ∈ W 1 D ( A + ) = { ψ ∈ W 1 2 ( 0 , l ) : ψ − = 0 } , 2 ( 0 , l ) : ψ + = 0 } , respectively. Both these operators are self-adjoint. Their spectra are discrete, the eigenvalues of A − are λ ( − ) ( n ∈ Z ) and of A + are λ (+) = π ( 2 n − 1 ) = 2 n π n n l l ( x ) = e − i λ ( − ) x and ( n ∈ Z ) with the corresponding eigenfunctions ψ ( − ) n n ( x ) = e − i λ (+) � � ψ (+) ψ (+) x , respectively. The set of eigenfunctions : n ∈ Z is a n n n complete orthogonal system in L 2 ( 0 , l ) , and the potential v ∈ L 2 ( 0 , l ) can be represented by the Fourier series � v n e − i ( 2 n − 1 ) π l x , v ( x ) = n ∈ Z where � l v n = 1 v ( x ) e i ( 2 n − 1 ) π l x dx , n ∈ Z . l 0 Kamila Dębowska (AGH) Direct and inverse problems for one-dimensional Dirac operators with nonlocal potentials 28 February 2019 15 / 33