Close singular perturbations of selfadjoint operators Vadym Adamyan - PowerPoint PPT Presentation

Close singular perturbations of selfadjoint operators Vadym Adamyan Department of Theoretical Physics and Astronomy Odessa I.I. Mechnikov National University OTKR-19, Vienna, December 20, 2019

Definition Let H, H 1 be unbounded selfadjoint operators in Hilbert space H ; D , D 1 are the domains and R ( z ) , R 1 ( z ) , Im z � = 0 , are resolvents of H, H 1 , respectively. Singular perturbation H 1 is called singular perturbation of H if D ∩ D 1 is dense in H ; H 1 = H on D ∩ D 1 . Close perturbation A singular perturbation H 1 is close to H if R 1 ( z ) − R ( z ) , Im z � = 0 , is a trace class operator. Motivation: Boundary problems of mathematical physics, Solvable models in quantum mechanics

Definition Let H, H 1 be unbounded selfadjoint operators in Hilbert space H ; D , D 1 are the domains and R ( z ) , R 1 ( z ) , Im z � = 0 , are resolvents of H, H 1 , respectively. Singular perturbation H 1 is called singular perturbation of H if D ∩ D 1 is dense in H ; H 1 = H on D ∩ D 1 . Close perturbation A singular perturbation H 1 is close to H if R 1 ( z ) − R ( z ) , Im z � = 0 , is a trace class operator. Motivation: Boundary problems of mathematical physics, Solvable models in quantum mechanics

Definition Let H, H 1 be unbounded selfadjoint operators in Hilbert space H ; D , D 1 are the domains and R ( z ) , R 1 ( z ) , Im z � = 0 , are resolvents of H, H 1 , respectively. Singular perturbation H 1 is called singular perturbation of H if D ∩ D 1 is dense in H ; H 1 = H on D ∩ D 1 . Close perturbation A singular perturbation H 1 is close to H if R 1 ( z ) − R ( z ) , Im z � = 0 , is a trace class operator. Motivation: Boundary problems of mathematical physics, Solvable models in quantum mechanics

Resolvents of close singular perturbations Let K be another Hilbert space and G ( z ) is a related to the H bounded holomorphic in the open upper and lower half-planes operator function from K to H satisfying the conditions ◮ for any non-real z, z 0 G ( z ) = G ( z 0 ) + ( z − z 0 ) R ( z ) G ( z 0 ) , (1) ◮ at least for one and hence for all non-real z zero is not an eigenvalue of the operator G ( z ) ∗ G ( z ) ◮ the intersection of the subspace N = G ( z 0 ) K ⊂ H and the domain D ( H ) of H is only of the zero-vector. Q ( z ) is a dounded and holomorphic in the open upper and lower half-planes operator function in K such that ◮ Q ( z ) ∗ = Q (¯ z ) , z � = 0 ; ◮ for any non-real z, z 0 z 0 ) ∗ G ( z ) . Q ( z ) − Q ( z 0 ) = ( z − z 0 ) G ( ¯ (2)

Resolvents of close singular perturbations. Krein formula Then for any selfadjoint operator L in K such that the operator function Q ( z ) + L, Im z � = 0 , has bounded inverse there is the singular perturbation H L of H and for the resolvent R L ( z ) of H L the following M.G. Krein formula R L ( z ) = R ( z ) − G ( z ) [ Q ( z ) + L ] − 1 G (¯ z ) ∗ (3) is valid. If, in addition, G ( z ) , Im z � = 0 , is a Hilbert-Schmidt mapping or L is an invertible operator and L − 1 is of trace class, or Q ( z ) , Im z � = 0 , is a trace class operator and L has bounded inverse, then the difference R L ( z ) − R ( z ) , Im z � = 0 , is a trace class operator, that is, H L is a close singular perturbation of H . Sometimes (though not always) to make Krein formula more user-friendly, it is worthwhile to exclude from it the auxiliary Hilbert space K .

Resolvents of close singular perturbations. Krein formula Then for any selfadjoint operator L in K such that the operator function Q ( z ) + L, Im z � = 0 , has bounded inverse there is the singular perturbation H L of H and for the resolvent R L ( z ) of H L the following M.G. Krein formula R L ( z ) = R ( z ) − G ( z ) [ Q ( z ) + L ] − 1 G (¯ z ) ∗ (3) is valid. If, in addition, G ( z ) , Im z � = 0 , is a Hilbert-Schmidt mapping or L is an invertible operator and L − 1 is of trace class, or Q ( z ) , Im z � = 0 , is a trace class operator and L has bounded inverse, then the difference R L ( z ) − R ( z ) , Im z � = 0 , is a trace class operator, that is, H L is a close singular perturbation of H . Sometimes (though not always) to make Krein formula more user-friendly, it is worthwhile to exclude from it the auxiliary Hilbert space K .

MG Krein’s formula Let G ( z ) be the defined as above holomorphic mapping K into H , e 1 , ..., e n , ... is a basis in K , { g n ( z ) = G ( z ) e n } ∞ 1 . By our assumptions ◮ g n ( z ) = g n ( z 0 ) + ( z − z 0 ) R ( z ) g n ( z 0 ) , n = 1 , 2 , ... ; (4) ◮ { g n ( z ) } ∞ 1 are linearly independant. Q ( z ) is a holomorphic in the open upper and lower half-planes infinite matrix function defining at each non-real z a bounded operator in the space l 2 such that ◮ Q ( z ) ∗ = Q (¯ z ) , z � = 0 ; ◮ for any non-real z, z 0 z 0 ))) T Q ( z ) − Q ( z 0 ) = ( z − z 0 ) (( g m ( z ) , g n ( ¯ (5) 1 ≤ m,n ≤ N .

MG Krein’s formula Let G ( z ) be the defined as above holomorphic mapping K into H , e 1 , ..., e n , ... is a basis in K , { g n ( z ) = G ( z ) e n } ∞ 1 . By our assumptions ◮ g n ( z ) = g n ( z 0 ) + ( z − z 0 ) R ( z ) g n ( z 0 ) , n = 1 , 2 , ... ; (4) ◮ { g n ( z ) } ∞ 1 are linearly independant. Q ( z ) is a holomorphic in the open upper and lower half-planes infinite matrix function defining at each non-real z a bounded operator in the space l 2 such that ◮ Q ( z ) ∗ = Q (¯ z ) , z � = 0 ; ◮ for any non-real z, z 0 z 0 ))) T Q ( z ) − Q ( z 0 ) = ( z − z 0 ) (( g m ( z ) , g n ( ¯ (5) 1 ≤ m,n ≤ N .

Theorem. If vectors { g n ( z ) ∞ 1 , Im z � = 0 , form a Risz basis in N , then for any infinite Hermitian matrix ˆ L = ( l mn ) ∞ 1 that defines a selfadjoint operator in l 2 the operator Q ( z ) + ˆ L, Im z � = 0 , has bounded inverse in l 2 and the operator function ∞ �� � − 1 � Q ( z ) + ˆ � R L ( z ) = R ( z ) − ( · , g n (¯ z )) g m ( z ) (6) L mn m,n =1 is the resolvent of some selfadjoint operator H L . If { g n ( z ) ∞ 1 is not a Riesz basis in N , but for the given ˆ L operators Q ( z ) + ˆ L, Im z � = 0 , in l 2 are boundedly invertible, then R L ( z ) is the resolvent of selfadjoint operator H L .Herewith H L is a close � − 1 is a trace � Q ( z ) + ˆ singular perturbation of H at least if L class operator in l 2 or if ∞ | ( g m ( z ) , g n ( z )) | 2 < ∞ , Im z � = 0 . � m,n =1

Theorem. If vectors { g n ( z ) ∞ 1 , Im z � = 0 , form a Risz basis in N , then for any infinite Hermitian matrix ˆ L = ( l mn ) ∞ 1 that defines a selfadjoint operator in l 2 the operator Q ( z ) + ˆ L, Im z � = 0 , has bounded inverse in l 2 and the operator function ∞ �� � − 1 � Q ( z ) + ˆ � R L ( z ) = R ( z ) − ( · , g n (¯ z )) g m ( z ) (6) L mn m,n =1 is the resolvent of some selfadjoint operator H L . If { g n ( z ) ∞ 1 is not a Riesz basis in N , but for the given ˆ L operators Q ( z ) + ˆ L, Im z � = 0 , in l 2 are boundedly invertible, then R L ( z ) is the resolvent of selfadjoint operator H L .Herewith H L is a close � − 1 is a trace � Q ( z ) + ˆ singular perturbation of H at least if L class operator in l 2 or if ∞ | ( g m ( z ) , g n ( z )) | 2 < ∞ , Im z � = 0 . � m,n =1

Theorem. If vectors { g n ( z ) ∞ 1 , Im z � = 0 , form a Risz basis in N , then for any infinite Hermitian matrix ˆ L = ( l mn ) ∞ 1 that defines a selfadjoint operator in l 2 the operator Q ( z ) + ˆ L, Im z � = 0 , has bounded inverse in l 2 and the operator function ∞ �� � − 1 � Q ( z ) + ˆ � R L ( z ) = R ( z ) − ( · , g n (¯ z )) g m ( z ) (6) L mn m,n =1 is the resolvent of some selfadjoint operator H L . If { g n ( z ) ∞ 1 is not a Riesz basis in N , but for the given ˆ L operators Q ( z ) + ˆ L, Im z � = 0 , in l 2 are boundedly invertible, then R L ( z ) is the resolvent of selfadjoint operator H L .Herewith H L is a close � − 1 is a trace � Q ( z ) + ˆ singular perturbation of H at least if L class operator in l 2 or if ∞ | ( g m ( z ) , g n ( z )) | 2 < ∞ , Im z � = 0 . � m,n =1

Wave operators From now on we assume that the spectrum of selfadjoint operator H is absolutely continuous and as long as no mention of another that dim K = ∞ . By virtue of these assumptions the wave operators W ± ( A 1 , A ) defined as strong limits t →±∞ e iH L t e − iHt W ± ( H L , H ) = s − lim (7) exist and are isometric mappings of H onto the absolutely continuous subspace of H L . Let E λ be the spectal function of H . The existence of strong limits in (7) ensures the validity of relations ∞ e − εt e ± iH L t e ∓ iHt W ± ( H L , H ) = s − lim � ε ↓ 0 ε 0 (8) ∞ = s − lim ε ↓ 0 ± iε � R L ( λ ± iε ) dE λ . −∞

Wave operators From now on we assume that the spectrum of selfadjoint operator H is absolutely continuous and as long as no mention of another that dim K = ∞ . By virtue of these assumptions the wave operators W ± ( A 1 , A ) defined as strong limits t →±∞ e iH L t e − iHt W ± ( H L , H ) = s − lim (7) exist and are isometric mappings of H onto the absolutely continuous subspace of H L . Let E λ be the spectal function of H . The existence of strong limits in (7) ensures the validity of relations ∞ e − εt e ± iH L t e ∓ iHt W ± ( H L , H ) = s − lim � ε ↓ 0 ε 0 (8) ∞ = s − lim ε ↓ 0 ± iε � R L ( λ ± iε ) dE λ . −∞