Approximation of Laplacians on the Sierpinski Gasket: norm-resolvent - PowerPoint PPT Presentation

Approximation of Laplacians on the Sierpinski Gasket: norm-resolvent and spectral convergence Jan Simmer joint work with Olaf Post University of Trier February 28, 2019

5th level iteration of the Sierpsinski Gasket

The Sierpinski Gasket Definition (Sierpinski Gasket) Let p 1 , p 2 and p 3 be the vertices of an equilateral triangle in R 2 and F j : R 2 → R 2 , F j ( x ) = ( x − p j ) / 2 + p j ( j = 1 , 2 , 3) . Then we call the unique non-empty compact K ⊂ R 2 that satisfies K = F 1 ( K ) ∪ F 2 ( K ) ∪ F 3 ( K ) the Sierpinski Gasket . Moreover, we call V 0 := { p 1 , p 2 , p 3 } the boundary of SG . Let W m := { 1 , 2 , 3 } m . Then there is a natural cell structure on SG given by W m ∋ w �→ F w ( K ) := F w 1 ◦ F w 2 ◦ · · · ◦ F w m ( K ) . We call F w ( K ) an m-cell of K .

Approximating sequence of finite graphs for SG Definition We let G 0 := ( V 0 , E 0 ) be the complete graph and for m ∈ N we define a sequence of finite discrete graphs G m = ( V m , E m ) by � � � � � x ∼ m y { x , y } ⊂ V m V m := F w ( V 0 ) , E m := , w ∈ W m where x ∼ m y ⇐ ⇒ x � = y and ∃ w ∈ W m such that x , y ∈ F w ( K ). Note that V m ⊂ V m +1 for each m ∈ N 0 and � V ⋆ := V m ⊂ K dense . m ∈ N 0 Note also that SG is connected and ∀ m ∈ N , w � = w ′ ∈ W m . F w ( K ) ∩ F w ′ ( K ) ⊂ F w ( V 0 ) ∩ F w ′ ( V 0 )

Energy forms on the approximating graphs Definition On each graph G m = ( V m , E m ) we define an energy form by � 5 � m � � � � 2 � f ( x ) − f ( y ) E m ( f ) = 3 x ∼ m y for f : V m → C . The constant (5 / 3) m is chosen such that the minimisation problem � � � � f : V m +1 → C , f ↾ V m = ̺ E m ( ̺ ) = min E m +1 ( f ) has a unique solution for each ̺ : V m → C .

Energy form on SG Let u : V ⋆ → C . As u ↾ V m is any extension of u ↾ V m − 1 and we have E m − 1 ( u ↾ V m − 1 ) ≤ E m ( u ↾ V m ) and hence the following limit exists in [0 , ∞ ]: E ∞ ( u ) := lim m →∞ E m ( u ↾ V m ) . Theorem ([Ki01] Energy form on SG ) There exists an energy form ( E , dom E ) on SG related to the � � sequence ( G m , E m ) m ∈ N 0 given by E = E ∞ with domain � � � � E ( u ) := lim dom E := u ∈ C( K ) m →∞ E m ( u ↾ V m ) < ∞

Energy form on SG Let u : V ⋆ → C . As u ↾ V m is any extension of u ↾ V m − 1 and we have E m − 1 ( u ↾ V m − 1 ) ≤ E m ( u ↾ V m ) and hence the following limit exists in [0 , ∞ ]: E ∞ ( u ) := lim m →∞ E m ( u ↾ V m ) . Theorem ([Ki01] Energy form on SG ) There exists an energy form ( E , dom E ) on SG related to the � � sequence ( G m , E m ) m ∈ N 0 given by E = E ∞ with domain � � � � E ( u ) := lim dom E := u ∈ C( K ) m →∞ E m ( u ↾ V m ) < ∞

Harmonic functions The compatibility of the sequence {E m } m ∈ N 0 implies: Theorem ([Ki01] m -harmonic functions on SG ) For any boundary value ̺ : V m → C there exists a unique function h ∈ dom E such that h ↾ V m = ̺ and � � � � u ∈ dom E , u ↾ V m = ̺ E m ( ̺ ) = E ( h ) = min E ( u ) . The function h is called m-harmonic function with boundary values ̺ . In the special case where ̺ = ✶ { x } for x ∈ V m , we denote the corresponding m -harmonic function by ψ x , m .

Specifying the Hilbert spaces Let µ be the (homogeneous) self-similar (probability) measure on SG , i.e. for all Borel sets A ⊂ K , � � µ ( A ) = 1 µ ( F − 1 1 ( A )) + µ ( F − 1 2 ( A )) + µ ( F − 1 3 ( A )) . 3 Hence every m -cell has measure µ ( K w ) = 1 / 3 m . Then ( E , dom E ) is a densely defined, closed quadratic form in L 2 ( K , µ ) and we denote the corresponding non-negative and self-adjoint operator by ∆.

On G m = ( V m , E m ) we define a probability measure by � � 1 / 3 m +1 x ∈ V 0 µ m ( x ) := ψ x , m d µ = 2 / 3 m +1 x ∈ V m \ V 0 . K Then our Hilbert space structure is H m = ℓ 2 ( V m , µ m ) with norm � � f � 2 µ m ( x ) | f ( x ) | 2 . ℓ 2 ( V m ,µ m ) = x ∈ V m It is easy to see that ∆ m ≥ 0 acts as � 5 � m � � 25 m � � � � 1 = 3 ∆ m f ( y ) = f ( y ) − f ( x ) f ( y ) − f ( x ) . µ m ( y ) 3 x ∼ x ∼ m y m y

Problem: We have energy forms E m in ℓ 2 ( V m , µ m ) and an energy form ( E , dom E ) in L 2 ( K , µ ) and the spaces are all different. How can we give any sense to the following expression? � (∆ m + 1) − 1 − (∆ + 1) − 1 � → 0

Generalised norm resolvent convergence Let ( E m , H 1 m ) resp. ( E , H 1 ) be energy forms in the separable Hilbert spaces H m resp. H . Definition ([P12] Quasi-unitary equivalence) Let δ m ≥ 0. Then E m and E are called δ m -quasi-unitary equivalent if there exist J m : H m → H , J 1 m : dom E m → dom E and J ′ 1 m : dom E → dom E m such that � J m f � H ≤ (1 + δ m ) � f � H and � f − J ⋆ � u − J m J ⋆ m J m f � H m ≤ δ m � f � E m m u � H ≤ δ m � u � E � J m f − J 1 � J ⋆ m u − J ′ 1 m f � H ≤ δ m � f � E m m u � H m ≤ δ m � u � E |E ( J m f , u ) − E m ( f , J ′ 1 m u ) | ≤ δ m � f � E m � u � E where � u � 2 E := � u � 2 H + E ( u ). Theorem If E m and E are δ m -quasi-unitary equivalent then � J m (∆ m + 1) − 1 − (∆ + 1) − 1 J m � ≤ 4 δ m .

Generalised norm resolvent convergence Let ( E m , H 1 m ) resp. ( E , H 1 ) be energy forms in the separable Hilbert spaces H m resp. H . Definition ([P12] Quasi-unitary equivalence) Let δ m ≥ 0. Then E m and E are called δ m -quasi-unitary equivalent if there exist J m : H m → H , J 1 m : dom E m → dom E and J ′ 1 m : dom E → dom E m such that � J m f � H ≤ (1 + δ m ) � f � H and � f − J ⋆ � u − J m J ⋆ m J m f � H m ≤ δ m � f � E m m u � H ≤ δ m � u � E � J m f − J 1 � J ⋆ m u − J ′ 1 m f � H ≤ δ m � f � E m m u � H m ≤ δ m � u � E |E ( J m f , u ) − E m ( f , J ′ 1 m u ) | ≤ δ m � f � E m � u � E where � u � 2 E := � u � 2 H + E ( u ). Theorem If E m and E are δ m -quasi-unitary equivalent then � J m (∆ m + 1) − 1 − (∆ + 1) − 1 J m � ≤ 4 δ m .

Consequences of quasi-unitary equivalence Theorem ([P12]) Assume that E and E m are δ m -quasi-unitarily equivalent and that U is an open subset such that ∂ U is locally Lipschitz and ∂ U ∩ ( σ (∆ m ) ∪ σ (∆)) = ∅ . Then � η (∆) J m − J m η (∆ m ) � ≤ C η δ m for any holomorphic η : U → C , where the constants C η only depend on η and U. For example choose η ( λ ) = e − t λ then the theorem is about the norm convergence of the approximating heat operators on ( G m , µ m ) to the one on the SG .

Consequences of quasi-unitary equivalence If η = ✶ I ( ∂ I ∩ σ (∆) = ∅ ), then the above theorem states the convergence of the spectral projectors and we conclude: Corollary ([P12]) Let λ k (∆ m ) resp. λ k (∆) be the k-th eigenvalue of ∆ m resp. ∆ . Then | λ k (∆ m ) − λ k (∆) | ≤ C k δ m for all m ∈ N such that dim H m ≥ k and where C k only depends on λ k (∆) . Since the spectrum of ∆ is purely discrete we can approximate an eigenfunction also in energy norm: For λ ∈ σ (∆) with normalised eigenfunction Φ there is a sequence (Φ m ) m of normalised function (linear combinations of eigenfunctions with eigenvalues close to ∆) and C λ > 0 (only depending in λ ) such that � J m Φ m − Φ � dom E ≤ C λ δ m .

Consequences of quasi-unitary equivalence If η = ✶ I ( ∂ I ∩ σ (∆) = ∅ ), then the above theorem states the convergence of the spectral projectors and we conclude: Corollary ([P12]) Let λ k (∆ m ) resp. λ k (∆) be the k-th eigenvalue of ∆ m resp. ∆ . Then | λ k (∆ m ) − λ k (∆) | ≤ C k δ m for all m ∈ N such that dim H m ≥ k and where C k only depends on λ k (∆) . Since the spectrum of ∆ is purely discrete we can approximate an eigenfunction also in energy norm: For λ ∈ σ (∆) with normalised eigenfunction Φ there is a sequence (Φ m ) m of normalised function (linear combinations of eigenfunctions with eigenvalues close to ∆) and C λ > 0 (only depending in λ ) such that � J m Φ m − Φ � dom E ≤ C λ δ m .

Main results In our setting on the SG , this means: � 1 H m := ℓ 2 ( V m , µ m ) where µ m ( x ) := ψ x , m d µ and � 5 � m � � � � 2 � f ( x ) − f ( y ) E m ( f ) := 3 x ∼ m y 2 H := L 2 ( K , µ ) with energy form ( E , dom E ) defined by E ( u ) := lim m →∞ E m ( u ↾ V m ) for each u ∈ { u ∈ C( K ) | E ( u ) := lim m →∞ E m ( u ↾ V m ) < ∞ }

Main results Theorem ([PS18a]) E m and E are δ m -quasi-unitarily equivalent with √ √ δ m = (1 + 3) 2 1 √ · 5 m / 2 . 3 Flavour of the proof: We define J := J m : H m → H by � 1 J ⋆ u ( y ) = Jf = f ( x ) ψ x , m then µ m ( y ) � u , ψ y , m � H x ∈ V m m → H 1 and J ′ 1 : H 1 → H 1 and let J 1 : H 1 m J 1 = J ↾ H 1 J ′ 1 u ( y ) = u ( y ) . and m

Main results Theorem ([PS18a]) E m and E are δ m -quasi-unitarily equivalent with √ √ δ m = (1 + 3) 2 1 √ · 5 m / 2 . 3 Flavour of the proof: We define J := J m : H m → H by � 1 J ⋆ u ( y ) = Jf = f ( x ) ψ x , m then µ m ( y ) � u , ψ y , m � H x ∈ V m m → H 1 and J ′ 1 : H 1 → H 1 and let J 1 : H 1 m J 1 = J ↾ H 1 J ′ 1 u ( y ) = u ( y ) . and m