Perturbations of manifolds and spectral convergence Olaf Post - PowerPoint PPT Presentation

Perturbations of manifolds and spectral convergence Olaf Post Mathematik (Fachbereich 4), Universit¨ at Trier, Germany joint work with Colette Ann´ e (Nantes, France) and Andrii Khrabustovskyi (Graz) 2019-03-01 Differential Operators on Graphs and Waveguides – Graz Motivation: perturbed manifolds and convergence of Laplacians 1 Interlude: Generalised norm resolvent convergence 2 Fading Neumann obstacles 3 Fading Dirichlet obstacles 4 Homogenisation — the critical case 5 Olaf Post (Universit¨ at Trier) Perturbed manifolds, spectral convergence Graz, 02–03/2019 1 / 11

Motivation: perturbed manifolds and convergence of Laplacians Motivation: perturbed manifolds and convergence of Laplacians X Riemannian manifold (or subset of R n or waveguide . . . ) “wild perturbations” (name by Rauch-Taylor [RT75]): remove obstacles B ε , e.g. many small balls, X ε := X \ B ε add many small handles, resulting manifold X ε (neither subset nor superset of X !) Question: Convergence of (Neumann/Dirichlet) Laplacian on X ε ? To what limit? Here mostly free limit: Laplacian on X (we call obstacles B ε with ∆ (Neu / Dir) → ∆ X fading) X ε Another question: Now to define norm resolvent convergence if spaces change? Olaf Post (Universit¨ at Trier) Perturbed manifolds, spectral convergence Graz, 02–03/2019 2 / 11

Interlude: Generalised norm resolvent convergence Interlude: What is a good generalisation of norm resolvent convergence? ∆ ε ≥ 0 in Hilbert space H ε for all ε ≥ 0 Definition (Generalised norm resolvent convergence, P:06, P:12) gnrs → ∆ 0 : ⇔ there exist J = J ε : H 0 → H ε bdd. and δ ε → 0 such that ∆ ε − � (id 0 − J ∗ J ) R 0 � ≤ δ ε , � (id ε − JJ ∗ ) R ε � ≤ δ ε , (1) � R ε J − JR 0 � ≤ δ ε . (2) If (1)–(2) hold for some J , we call ∆ ε , ∆ 0 δ ε -quasi-unitary equivalent. Generalisation of standard norm resolvent convergence δ ε = 0 for (1): J unitary; w.l.o.g. H 0 = H ε , J = id. Then (2) is � R ε − R 0 � → 0 Olaf Post (Universit¨ at Trier) Perturbed manifolds, spectral convergence Graz, 02–03/2019 3 / 11

Interlude: Generalised norm resolvent convergence Interlude: What is a good generalisation of norm resolvent convergence? ∆ ε ≥ 0 in Hilbert space H ε for all ε ≥ 0 Definition (Generalised norm resolvent convergence, P:06, P:12) gnrs ∆ ε − → ∆ 0 : ⇔ there exist J = J ε : H 0 → H ε bdd. and δ ε → 0 such that � (id 0 − J ∗ J ) R 0 � ≤ δ ε , � (id ε − JJ ∗ ) R ε � ≤ δ ε , (1) � R ε J − JR 0 � ≤ δ ε . (2) If (1)–(2) hold for some J , we call ∆ ε , ∆ 0 δ ε -quasi-unitary equivalent. Generalisation of unitary equivalence δ ε = 0 for (1)–(2): J unitary and ∆ ε , ∆ 0 unititarily equivalent Olaf Post (Universit¨ at Trier) Perturbed manifolds, spectral convergence Graz, 02–03/2019 3 / 11

Interlude: Generalised norm resolvent convergence Consequences of generalised norm resolvent convergence Definition (Generalised norm resolvent convergence, P:06, P:12) gnrs ∆ ε − → ∆ 0 : ⇔ there exist J : H 0 → H ε bdd. and δ ε → 0 such that (1) � (id 0 − J ∗ J ) R 0 � ≤ δ ε , � (id ε − JJ ∗ ) R ε � ≤ δ ε , (2) � R ε J − JR 0 � ≤ δ ε . Theorem (P:06, P:12) gnrs If ∆ ε − → ∆ 0 , we have e.g. � ϕ (∆ ε ) − J ϕ (∆ 0 ) J ∗ � ≤ C ϕ δ ε (e.g. ϕ t ( λ ) = e − t λ , ϕ = ✶ I ) σ (∆ ε ) → σ (∆ 0 ) on compact intervals [also for discrete and essential spectrum], convergence of eigenfunctions (even in energy norm) In particular: no spectral pollution, no spurious eigenvalues: λ 0 ∈ σ (∆ 0 ) ⇐ ⇒ ∃ ( λ ε ) ε : λ ε ∈ σ (∆ ε ) , λ ε → λ 0 Cannot have generalised norm resolvent convergence for compact spaces approximating a non-compact one (as essential spectra converge) Olaf Post (Universit¨ at Trier) Perturbed manifolds, spectral convergence Graz, 02–03/2019 4 / 11

Fading Neumann obstacles Fading Neumann obstacles General assumption X complete Riemannian manifold with Laplacian ∆ 0 := ∆ X of bounded geometry (injectivity radius ι > 0 , Ricci curvature bounded from below). Consequences: � Small balls look Euclidean! ∃ C ell > 0 ∀ f ∈ H 2 ( X ): � f � H 2 ( X ) ≤ C ell � (∆ X + 1) f � L 2 ( X ) Definition (Neumann fading obstacles) B ε are Neumann fading obstacles iff B ε ⊂ X , δ ε → 0, � f � L 2 ( B ε ) ≤ δ ε � f � H 1 ( X ) , (4) ∃ E ε : H 1 ( X ε ) → H 1 ( X ) , E ε u ↾ X ε = u : � E ε � H 1 → H 1 ≤ C ext (5) (4) means: eigenfunctions non-concentrating on B ε (5): X ε := X \ B ε strongly connected (in homogenisation theory) Olaf Post (Universit¨ at Trier) Perturbed manifolds, spectral convergence Graz, 02–03/2019 5 / 11

Fading Neumann obstacles Neumann fading obstacles II Theorem (Ann´ e-P:18) gnrs If B ε ⊂ X are Neumann fading obstacles then ∆ Neu − → ∆ X . X ε Flavour of proof: Hilbert spaces: H ε = L 2 ( X ε ), H 0 := L 2 ( X ) X ε | du | 2 , dom E ε = H 1 ( X ε ), Energy forms: E ε ( u ) = � ( � Neumann Laplacian on X ε ) X | du | 2 , dom E 0 = H 1 ( X ) � E 0 ( f ) = Identification operators: J : L 2 ( X 0 ) → L 2 ( X ε ), Jf := f ↾ X ε , u := u ⊕ 0. hence J ∗ u = ¯ Then: u = JJ ∗ u and f − J ∗ Jf = f ↾ B ε ⊕ 0, Hence: (4) � f − J ∗ Jf � 2 = � f � 2 ≤ δ 2 ε � f � 2 H 1 ( X ) = δ 2 ε � (∆ X + 1) 1 / 2 f � 2 L 2 ( X ) , L 2 ( B ε ) (non-concentrating on B ε ) hence (1) fulfilled with δ ε Olaf Post (Universit¨ at Trier) Perturbed manifolds, spectral convergence Graz, 02–03/2019 6 / 11

Fading Neumann obstacles Neumann fading obstacles III Example: For each ε > 0 let η ε > 0 and I ε ⊂ X be an η ε -separated set, i.e., x , y ∈ I ε , x � = y = ⇒ d ( x , y ) ≥ 2 η ε then B ε := B η ε ( I ε ) = � x ∈ I ε B η ε ( x ) is a disjoint union of small balls. We assume that ε 0 < ε ≪ η ε ≪ 1 , i.e., η ε → 0 , → 0 as ε → 0 η ε (e.g. η ε = ε α , α ∈ (0 , 1)). Proposition (Ann´ e-P:18) B ε are Neumann fading obstacles with δ ε = O( ε/η ε ) (n ≥ 3 ). Corollary (Ann´ e-P:18) gnrs If B ε are disjoint balls as above then ∆ Neu − → ∆ X . X ε Olaf Post (Universit¨ at Trier) Perturbed manifolds, spectral convergence Graz, 02–03/2019 7 / 11

Fading Dirichlet obstacles Fading Dirichlet obstacles Definition (Dirichlet fading obstacles) B ε are Dirichlet fading obstacles iff B ε ⊂ X , δ ε → 0, ∃ χ ε : X → [0 , 1], B + ε := { x ∈ X | χ ε ( x ) = 1 } : � f � L 2 ( B + ε ) ≤ δ ε � f � H 1 ( X ) , (6) H 2 ( X ) → L 2 ( T ∗ ( B + ε )) , f �→ fd χ ε ↾ B + (7) ε Theorem ([Ann´ e-P:18]) gnrs Assume B ε ⊂ X are Dirichlet fading obstacles then Then ∆ Dir − → ∆ X . X ε Corollary ([Ann´ e-P:18]) x ∈ I ε B η ε ( x ) is ε α -separated, α ∈ (0 , ( m − 2) / m ) , Assume B ε := B η ε ( I ε ) = � gnrs then ∆ Dir − → ∆ X . X ε Olaf Post (Universit¨ at Trier) Perturbed manifolds, spectral convergence Graz, 02–03/2019 8 / 11

Fading Dirichlet obstacles Meaning of the fading conditions in Neu/Dir case Assume X is compact then by the η ε -separation the number of balls N ε = | I ε | is finite. Moreover if η ε = ε α , then vol B η ε ∼ ε α m and N ε ε α m � 1, hence N ε � ε − α m . (number of balls) When do the obstacles fade away . . . ? Neumann: 0 < α < 1: number of balls less than ε − m , or N ε ε m ∼ = ε (1 − α ) m → 0 (volume of obstacles) . Dirichlet ( m ≥ 3): 0 < α < m − 2 : number of balls less than m ε − ( m − 2) , or N ε ε m − 2 → 0 (capacity of obstacles) . Olaf Post (Universit¨ at Trier) Perturbed manifolds, spectral convergence Graz, 02–03/2019 9 / 11

Homogenisation — the critical case Homogenisation — the critical case Consider the case α = ( m − 2) / m : Let X ε = X \ B ε , B ε union of balls (little obstacles) D ε = ε D of radius ε in a grid of length η ε = ε α cap( D ε ) Let q = lim ε → 0 exist. ε α m Theorem (Khrabustovskiy-Post:18) gnrs We have ∆ Dir − → ∆ X + q. X ε Typical results only include strong resolvent convergence. Olaf Post (Universit¨ at Trier) Perturbed manifolds, spectral convergence Graz, 02–03/2019 10 / 11

Homogenisation — the critical case C. Ann´ e and O. Post, Wildly perturbed manifolds: norm resolvent and spectral convergence, arXiv:1802.01124 (2018). A. Khrabustovskyi and O. Post, Operator estimates for the crushed ice problem, Asymptot. Anal. 110 (2018), 137–161. O. Post and J. Simmer, Approximation of fractals by discrete graphs: norm resolvent and spectral convergence, Integral Equations Operator Theory 90 (2018), 90:68. J. Rauch and M. Taylor, Potential and scattering theory on wildly perturbed domains, J. Funct. Anal. 18 (1975), 27–59. Olaf Post (Universit¨ at Trier) Perturbed manifolds, spectral convergence Graz, 02–03/2019 11 / 11