Unique continuation principle and its absence on continuum space, on - PowerPoint PPT Presentation

1 Unique continuation principle and its absence on continuum space, on lattices, and on quantum graphs Ivan Veseli´ c (TU Dortmund) on joint works joint with Daniel Lenz, Ivica Naki´ c, Norbert Peyerimhoff, Olaf Post, Constanza Rojas-Molina, Matthias T¨ aufer, Martin Tautenhahn OTIND conference, Wien 2016

2 Unique continuation principle Everyday encounter with unique continuation problem

3 Unique continuation principle Everyday encounter with unique continuation problem

4 Unique continuation principle Everyday encounter with unique Possibly with fatal consequences continuation problem

5 Introduce properties of function classes Let Ω ⊂ R d be open, F ⊂ { f : Ω → C measurable } . The class F has the (weak) unique continuation property, if for all f ∈ F ∃ W ⊂ Ω non empty and open, such that f ≡ 0 on W ⇒ f ≡ 0 . E.g. holomorphic functions f : C → C have to this property due to the local uniqueness theorem (analytic continuation). strong unique continuation property, if for all f ∈ F � ǫ → 0 ǫ − N ∃ x 0 ∈ Ω ∀ N > 0 lim | f | d x = 0 ⇒ f ≡ 0 . B ( x 0 ,ǫ )

6 Introduce properties of function classes Let Ω ⊂ R d be open, F ⊂ { f : Ω → C measurable } . The class F has the (weak) unique continuation property, if for all f ∈ F ∃ W ⊂ Ω non empty and open, such that f ≡ 0 on W ⇒ f ≡ 0 . E.g. holomorphic functions f : C → C have to this property due to the local uniqueness theorem (analytic continuation). strong unique continuation property, if for all f ∈ F � ǫ → 0 ǫ − N ∃ x 0 ∈ Ω ∀ N > 0 lim | f | d x = 0 ⇒ f ≡ 0 . B ( x 0 ,ǫ ) vanishing order (at most) M > 0, if for all f ∈ F , f � = 0 for each x 0 ∈ Ω 0 there exist a constant C > 0 and a radius ǫ 0 > 0 such that � C ǫ M ≤ | f | d x for all ǫ ∈ (0 , ǫ 0 ) . B ( x 0 ,ǫ ) These properties are local, i.e. only information in an arbitrarily small ball around x 0 are required.

7 (Non)uniform vanishing order For k ∈ N let f k : C → C , z → z k : f k holomorphic ⇒ strong unique continuation property For large k , however, f k vanishes arbitrarily fast at z 0 = 0. Similarly for harmonic functions g k = ℜ f k .

8 (Non)uniform vanishing order For k ∈ N let f k : C → C , z → z k : f k holomorphic ⇒ strong unique continuation property For large k , however, f k vanishes arbitrarily fast at z 0 = 0. Similarly for harmonic functions g k = ℜ f k . Thus, for F = { f : C → C : f holomorphic } there is no M > 0 such that F has vanishing order (at most) M . Growth vs. vanishing Let f : C → C be holomorphic growing slower at ∞ than vanishing at 0, i.e. lim inf ǫ ց 0 m ( ǫ ) · m (1 /ǫ ) = 0 where m ( r ) := max | z | = r | f ( z ) | . Then f ≡ 0. Interplay of local and global properties!

9 Hadamard’s three circle Theorem Let r 1 < r 2 < r 3 . Let f be a holomorphic function in a neighbourhood of the annulus r 1 ≤ | z | ≤ r 3 and denote m ( r ) := max | z | = r | f ( z ) | . Then � r 3 � r 3 � r 2 � � � log log m ( r 2 ) ≤ log log m ( r 1 ) + log log m ( r 3 ) . r 1 r 2 r 1 Thus log r �→ log m ( r ) is a convex function. i R R

10 Hadamard’s three circle Theorem Let r 1 < r 2 < r 3 . Let f be a holomorphic function in a neighbourhood of the annulus r 1 ≤ | z | ≤ r 3 and denote m ( r ) := max | z | = r | f ( z ) | . Then � r 3 � r 3 � r 2 � � � log log m ( r 2 ) ≤ log log m ( r 1 ) + log log m ( r 3 ) . r 1 r 2 r 1 Thus log r �→ log m ( r ) is a convex function. i R R

11 Hadamard’s three circle Theorem Let r 1 < r 2 < r 3 . Let f be a holomorphic function in a neighbourhood of the annulus r 1 ≤ | z | ≤ r 3 and denote m ( r ) := max | z | = r | f ( z ) | . Then � r 3 � r 3 � r 2 � � � log log m ( r 2 ) ≤ log log m ( r 1 ) + log log m ( r 3 ) . r 1 r 2 r 1 Thus log r �→ log m ( r ) is a convex function. i R R

12 Main Result: Scale free quantitative unique continuation estimate for spectral projectors of a Schr¨ odinger operator on a cube. Setting: geometry infinite discrete set Z = ( x j ) j ∈ Z d ⊂ R d ,

13 Main Result: Scale free quantitative unique continuation estimate for spectral projectors of a Schr¨ odinger operator on a cube. Setting: geometry infinite discrete set Z = ( x j ) j ∈ Z d ⊂ R d , radius δ ∈ (0 , 1 / 2) such that B ( x j , δ ) ⊂ Λ 1 + j = [ − 1 / 2 , 1 / 2] d + j δ -neighbourhood of Z : S = S ( δ ) = B ( Z , δ ) = � j ∈ Z d B ( x j , δ )

14 Main Result: Scale free quantitative unique continuation estimate for spectral projectors of a Schr¨ odinger operator on a cube. Setting: geometry infinite discrete set Z = ( x j ) j ∈ Z d ⊂ R d , radius δ ∈ (0 , 1 / 2) such that B ( x j , δ ) ⊂ Λ 1 + j = [ − 1 / 2 , 1 / 2] d + j δ -neighbourhood of Z : S = S ( δ ) = B ( Z , δ ) = � j ∈ Z d B ( x j , δ ) For L ∈ N , set Λ L = [ − L / 2 , L / 2] d , S L ( δ ) = Λ L ∩ S ( δ ), and characteristic function W L = χ S L ( δ ) .

15 Main Result: Scale free quantitative unique continuation estimate Setting: Schr¨ odinger operator For bounded, measurable potential V : R d → R define s.a. Schr¨ odinger operator H L = − ∆ + V on Λ L , L ∈ N , with Dirichlet, Neumann, or periodic b.c. and spectral projection χ ( −∞ , E ] ( H L ), for energy E > 0. Theorem [Naki´ c, T¨ aufer, Tautenhahn, & Veseli´ c 15, 16] Let δ, Z , L , S L ( δ ) , V , H L , E as above. Then exists K depending only on dimension d such that √ � � E + � V � 2 / 3 K 1+ ∞ χ ( −∞ , E ] ( H L ) W L χ ( −∞ , E ] ( H L ) ≥ δ χ ( −∞ , E ] ( H L ) where W L = χ S L ( δ ) . Bound is equivalent to � √ � � E + � V � 2 / 3 � K 1+ | ψ | 2 ≥ δ | ψ | 2 ∞ S L ( δ ) Λ L for any ψ = � α k ψ k ∈ ran χ ( −∞ , E ] ( H L ) E k ≤ E

16 Concepts and methods for proof of main result.

17 Carleman estimates allow to deduce ucp (after some calculations) whole zoo of Carleman estimates exists many with abstract weight functions (satisfying H¨ ormanders subellipticity condition) we want explicit estimate, thus explicit weight function We start with a formulation of [Bourgain, Kenig 05] since this has given crucial stimulus to the theory of random Schr¨ odinger operators.

18 Carleman estimate as formulated in [Bourgain, Kenig 05] Weight function 1 /φ ( r ) φ : [0 , ∞ ) → [0 , ∞ ) φ ( r ) = � r 1 − e − t � � r exp − d t t 0 w : R d → [0 , ∞ ) , w ( x ) = φ ( | x | ) r ⇒ ∀ r ∈ (0 , 1) : r / 3 ≤ φ ( r ) ≤ r Theorem [Bourgain & Kenig 05] There are constants C 1 ( d ) and C 2 ( d ) ∈ [1 , ∞ ) s. t. for all α ≥ C 1 and real valued f ∈ C 2 ( B (0 , 1)) with compact support in B (0 , 1) \ { 0 } we have � � w 2 − 2 α (∆ f ) 2 d x w − 1 − 2 α f 2 d x ≤ C 2 α 3

19 Carleman estimate as formulated in [Bourgain, Kenig 05] Weight function 1 /φ ( r ) φ : [0 , ∞ ) → [0 , ∞ ) φ ( r ) = � r 1 − e − t � � r exp − d t t 0 w : R d → [0 , ∞ ) , w ( x ) = φ ( | x | ) r ⇒ ∀ r ∈ (0 , 1) : r / 3 ≤ φ ( r ) ≤ r Theorem [Bourgain & Kenig 05] There are constants C 1 ( d ) and C 2 ( d ) ∈ [1 , ∞ ) s. t. for all α ≥ C 1 and real valued f ∈ C 2 ( B (0 , 1)) with compact support in B (0 , 1) \ { 0 } we have � � w 2 − 2 α (∆ f ) 2 d x w − 1 − 2 α f 2 d x ≤ C 2 α 3 Possible to scale the inequality to a ball of radius ρ and extend to Sobolev space H 2 .

20 Consequences/Applications of Carleman estimates: Three annuli inequality Insert χ × ψ = cut-off × eigenfunction A 3 in Carleman inequality A 2 (e.g. [Bourgain, Kenig 05]) to get three annuli inequality A 1 � � w 2 − 2 α | ψ | 2 � α 3 w 2 − 2 α | ψ | 2 A 2 A 1 � w 2 − 2 α | ψ | 2 . + A 3 profile of cut-off function χ A 3 A 2 A 1 A 1 A 2 A 3

21 Compare to Hadamard’s three circle theorem Carleman inequality (e.g. [Bourgain, Kenig 05]) implies three annuli inequality � � � w 2 − 2 α | ψ | 2 � w 2 − 2 α | ψ | 2 + α 3 w 2 − 2 α | ψ | 2 . A 2 A 1 A 3 annuli instead of circles (no worries about regularity and trace map) middle annulus controlled by inner annulus and outer annulus integral over outer annulus plays (typically) the role of a global bound/ bound ’at infinity’/ a priori bound

22 Unique continuation on combinatorial and quantum graphs?

23 Half space with unique continuation for discrete Schr¨ odinger operators

24 Half space with unique continuation for discrete Schr¨ odinger operators ψ known e.g. ψ = 0 ψ unknown H ψ = E ψ

25 Half space with unique continuation for discrete Schr¨ odinger operators ψ known e.g. ψ = 0 ψ unknown H ψ = E ψ inequality with one unknown; solvable

26 Half space with unique continuation for discrete Schr¨ odinger operators ψ known e.g. ψ = 0 ψ unknown H ψ = E ψ inequality with one unknown; solvable

27 Half space with unique continuation for discrete Schr¨ odinger operators ψ known e.g. ψ = 0 ψ known also here H ψ = E ψ

28 Half space without unique continuation for discrete Schr¨ odinger operators ψ known ψ unknown H ψ = E ψ

29 Half space without unique continuation for discrete Schr¨ odinger operators ψ known ψ unknown H ψ = E ψ inequality with two unknowns; unsolvable