SCATTERING ALONG A CURVE IN THE PLAIN J. DITTRICH NUCLEAR PHYSICS - PowerPoint PPT Presentation

SCATTERING ALONG A CURVE IN THE PLAIN J. DITTRICH NUCLEAR PHYSICS INSTITUTE CAS, ˇ REˇ Z, CZECH REPUBLIC Physical motivation Nanochannels in some solids - effectively 1-D motion of electrons Simplified mathematical models in the plane 1/18

δ Γ in R 2 d > 0 d = 0 quantum quantum leaky quantum waveguide graph graph 2/18

J. F. Brasche, P. Exner, Yu. A. Kuperin and P. ˇ Seba, Schr¨ odinger operators with singular interactions, J. Math. Anal. Appl. 184 (1994) 112–139. P. Exner and T. Ichinose, Geometrically induced spectrum in curved leaky wires, J. Phys. A: Math. Gen. 34 (2001) 1439– 1450. P. Exner, S. Kondej: Scattering by local deformations of a straight leaky wire, J. Phys. A: Math. Gen. 38 (2005), 4865– 4874. P. Exner: Leaky quantum graphs: a review, in: Proc. Sympos. Pure Math., 77 (2008), 523–564. [arXiv: 0710.5903 [math-ph]] 3/18

Present talk based on J. Dittrich: Scattering of particles bounded to infinite planar curve , arXive: 1912.03958 [math-ph] We consider formal Hamiltonian H = − ∆ − αδ Γ , α > 0 in L 2 ( R 2 ) precisely defined as the self-adjoint operator associated with the sesquilinear form ∫ q ( f, g ) = ( ∇ f, ∇ g ) − α f (Γ( s )) g (Γ( s )) ds R D ( q ) = H 1 ( R 2 ) 4/18

✔ ✔ ✔ ✔ ✔ ✔ ✔ ✔ ✔ ✔ ✛ ✙ ✔ ✔ ✚ ✘ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ Exner present excluded Kondej (2005) work case 5/18

Assumptions on the curve Γ : R → R 2 , Γ ∈ C 3 , | Γ ′ ( s ) | = 1 for s ∈ R Curvature K = Γ ′ 1 Γ ′′ 2 − Γ ′ 2 Γ ′′ s →∞ K ′ ( s ) = 0 s →∞ K ( s ) = 0 , lim , lim 1 There exists ρ ∈ (0 , 1] such that for every s, t ∈ R ρ | s − t | ≤ | Γ( s ) − Γ( t ) | ≤ | s − t | 6/18

Let us write a − , a + , v − , v + ∈ R 2 Γ( s ) = a ± + v ± s + ν ± ( s ) , | v − | = | v + | = 1 , v + ̸ = − v − | ν ′ | ν ′ − ( t ) | + ( t ) | ϕ − ( s ) = sup , ϕ + ( s ) = sup t ≤ s t ≥ s There exist R > 0 and δ > 3 such that ∫ − R ∫ ∞ (1+ s 2 ) δ | ϕ − ( s ) | 2 ds < ∞ (1+ s 2 ) δ | ϕ + ( s ) | 2 ds < ∞ , −∞ R i.e. ϕ − ∈ L 2 ϕ + ∈ L 2 δ (( −∞ , − R )) δ (( R, ∞ )) , s →±∞ ν ± ( s ) = 0 lim Then we can choose 7/18

Example: n > 5 | ν ± ( s ) | < c | s | − n | ν ′ ± ( s ) | < c | s | − n − 1 , , 2 s →±∞ ν ′′ s →±∞ ν ′′′ lim ± ( s ) = 0 , lim ± ( s ) = 0 8/18

Theorem 1. There exists at most discrete subset E of open interval ( − α 2 4 , 0) such that σ ess ( H ) = [0 , + ∞ ) , ( ) − α 2 4 , 0 ∩ σ pp ( H ) = E , ( ) − α 2 4 , 0 ⊂ σ ac ( H ) , ( ) − α 2 ∩ σ sc ( H ) = ∅ 4 , 0 . Proof. Based on the limiting absorbtion principle, inspired by the proof of Agmon-Kato-Kuroda theorem for the standard Schr¨ odinger operators, technical and long. 9/18

Scattering - Wave Operators Transversally bounded particles, energy in ( − α 2 4 , 0) Asymptotic lines and free Hamiltonians Γ ± ( s ) = a ± + v ± s H ( ± ) = − ∆ − αδ Γ ± out in (-) out in 10/18

Wave operators < e − iH (+) t J (+) Ω (+) t →−∞ e iHt P (+) in = s − lim > e − iH (+) t J (+) Ω (+) t → + ∞ e iHt P (+) out = s − lim > e − iH ( − ) t J ( − ) Ω ( − ) t →−∞ e iHt P ( − ) in = s − lim < e − iH ( − ) t J ( − ) Ω ( − ) t → + ∞ e iHt P ( − ) out = s − lim where J ( ± ) is the spectral projector of H ( ± ) corresponding to the interval ( − α 2 4 , 0) (spectrum is absolutely continuous) 11/18

P ( ± ) is the spectral projector of the momentum − iv ± · ∇ in ≷ the direction of the asymptotic line corresponding to the inter- val (0 , + ∞ ) resp. ( −∞ , 0) These spectral projectors can be explicitly constructed, e.g. for v + = (1 , 0) √ α 2 e − α P (+) > J (+) = F − 1 2 | y | 1 χ (0 ,α/ 2) F 1 ⊗ ϕ 0 ( ϕ 0 , · ) 2 , ϕ 0 ( y ) = D. W. Fox, Spectral measures and separation of variables, J. of Research of the National Bureau of Standards - B. Mathe- matical Sciences 80 (1976) 347–351. 12/18

Definition. Let B be a self-adjoint operator in a separable Hilbert space H and E λ its spectral projectors. We denote as M ( B ) the set of all ϕ ∈ H such that d ( ϕ, E λ ϕ ) = | f ( λ ) | 2 dλ with f ∈ L ∞ ( R ). Theorem 2 (Generalized Kuroda-Birman) . Let A and B be self-adjoint operators in a separable Hilbert space H , K a bounded operator in H , z a complex number in the resolvent sets of both A and B , K ( B − z ) − 1 − ( A − z ) − 1 K = C 1 + C 2 where C 1 ∈ J 1 ( H ) is a trace class operator and C 2 is a bounded operator in H . Let there further exists a dense set 13/18

M ⊂ M ( B ) such that ∫ + ∞ t → + ∞ C 2 e − iBt ϕ = 0 ∥ C 2 e − iBt ϕ ∥ dt < ∞ lim , 0 for every ϕ ∈ M . Then there exists t → + ∞ e iAt Ke − iBt P ac ( B ) Ω out ( A, B ; K ) = s − lim . Proof. Amendments in the textbook proofs of Pearson and Kuroda-Birman theorems. Theorem 3. The wave operators Ω (+) in , Ω (+) out , Ω ( − ) in , Ω ( − ) out exist. Proof for Ω (+) out , others by symmetry, v + = (1 , 0). 1) Use known formula for the resolvent of H , show that it is expressible through integral operators, kernels given by Mac- donald function K 0 or bounded by such expressions. 14/18

2) Remember H (+) = − ∂ 2 H 2 ,α = − ∂ 2 x ⊗ I 2 + I 1 ⊗ H 2 ,α , y − αδ choose κ > 0 large enough 0 (( −∞ , 0) ∪ (0 , α 2) ∪ ( α M 1 = F − 1 1 C ∞ 2 , + ∞ )) , M 2 = M ( H 2 ,α ) , M = { M 1 × M 2 } lin ( ( H (+) − iκ ) − 1 − ( H − iκ ) − 1 ) θ ( x ) P (+) > J (+) C 1 = ( H (+) − iκ ) − 1 − ( H − iκ ) − 1 ) ( θ ( − x ) P (+) > J (+) C 2 = 3) Verify the assumptions of generalized Kuroda-Birman theo- rem with A = H , B= H (+) , K = P (+) > J (+) . (∑ ) C 1 ∈ J 1 ⇐ ⇒ sup | ( ϕ n , C 1 ψ n ) | < ∞ { ϕ n } , { ψ n } n 15/18

C 1 = D 1 + similar terms D 1 an integral operator with kernel of the form ∫ x, y ∈ R 2 A ( x, s ) B ( s, y ) ds , R Then ∫ ∑ ∑ | ( ϕ n , D 1 ψ n ) | ≤ | ( ϕ n , A ( · , s )) || ( B ( s, · ) , ψ n ) | ds R n n ] 1 / 2 [∑ ] 1 / 2 [∑ ∫ | ( ϕ n , A ( · , s )) | 2 | ( ψ n , B ( s, · )) | 2 ≤ ds R n n ] 1 / 2 [∫ ] 1 / 2 [∫ ∫ R 2 | A ( x, s ) | 2 dx R 2 | B ( s, y ) | 2 dy ≤ ds R Convergence of this integral can be proved. 16/18

Conclusions − α 2 ( ) The interval 4 , 0 belongs to the absolutely continuous spectrum of the considered Hamiltonian, with possible at most discrete set E of embedded eigenvalues. The accumulation of the eigenvalues at − α 2 4 and 0 is not ex- cluded by our proof. Wave operators Ω ( ± ) in,out := Ω in,out ( H, H ( ± ) ; P ( ± ) ≷ J ( ± ) ) for the energy range ( − α 2 4 , 0) exist. 17/18

However, we know nothing on the wave operators of the type Ω in,out ( H ( ± ) , H ) . This means that our wave operators map each scattering state of the ”free” Hamiltonians H ( ± ) (incoming or outgoing along the asymptotic half-lines) to the one of H but we did not prove that all scattering states of H are covered and the opposite map- pings exist. In other words, completeness of the wave operators is not proved. The absence of singular continuous spectrum means that if the wave operators would be complete they would be also asymp- totically complete. 18/18