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
Ivan Veseli´ c (TU Dortmund)
c, Norbert Peyerimhoff, Olaf Post, Constanza Rojas-Molina, Matthias T¨ aufer, Martin Tautenhahn OTIND conference, Wien 2016
Everyday encounter with unique continuation problem
Everyday encounter with unique continuation problem
Everyday encounter with unique continuation problem Possibly with fatal consequences
Let Ω ⊂ Rd 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 ∃ x0 ∈ Ω ∀ N > 0 lim
ǫ→0 ǫ−N
|f |dx = 0 ⇒ f ≡ 0.
Let Ω ⊂ Rd 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 ∃ x0 ∈ Ω ∀ N > 0 lim
ǫ→0 ǫ−N
|f |dx = 0 ⇒ f ≡ 0. vanishing order (at most) M > 0, if for all f ∈ F, f = 0 for each x0 ∈ Ω0 there exist a constant C > 0 and a radius ǫ0 > 0 such that CǫM ≤
|f |dx for all ǫ ∈ (0, ǫ0). These properties are local, i.e. only information in an arbitrarily small ball around x0 are required.
(Non)uniform vanishing order For k ∈ N let fk : C → C, z → zk : fk holomorphic ⇒ strong unique continuation property For large k, however, fk vanishes arbitrarily fast at z0 = 0. Similarly for harmonic functions gk = ℜfk.
(Non)uniform vanishing order For k ∈ N let fk : C → C, z → zk : fk holomorphic ⇒ strong unique continuation property For large k, however, fk vanishes arbitrarily fast at z0 = 0. Similarly for harmonic functions gk = ℜfk. 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
ǫց0 m(ǫ) · m(1/ǫ) = 0 where m(r) := max |z|=r|f (z)|.
Then f ≡ 0. Interplay of local and global properties!
Hadamard’s three circle Theorem Let r1 < r2 < r3. Let f be a holomorphic function in a neighbourhood of the annulus r1 ≤ |z| ≤ r3 and denote m(r) := max|z|=r|f (z)|. Then log r3 r1
r3 r2
r2 r1
Thus log r → log m(r) is a convex function. R iR
Hadamard’s three circle Theorem Let r1 < r2 < r3. Let f be a holomorphic function in a neighbourhood of the annulus r1 ≤ |z| ≤ r3 and denote m(r) := max|z|=r|f (z)|. Then log r3 r1
r3 r2
r2 r1
Thus log r → log m(r) is a convex function. R iR
Hadamard’s three circle Theorem Let r1 < r2 < r3. Let f be a holomorphic function in a neighbourhood of the annulus r1 ≤ |z| ≤ r3 and denote m(r) := max|z|=r|f (z)|. Then log r3 r1
r3 r2
r2 r1
Thus log r → log m(r) is a convex function. R iR
for spectral projectors of a Schr¨
Setting: geometry infinite discrete set Z = (xj)j∈Zd ⊂ Rd,
for spectral projectors of a Schr¨
Setting: geometry infinite discrete set Z = (xj)j∈Zd ⊂ Rd, radius δ ∈ (0, 1/2) such that B(xj, δ) ⊂ Λ1 + j = [−1/2, 1/2]d + j δ-neighbourhood of Z: S = S(δ) = B(Z, δ) =
j∈Zd B(xj, δ)
for spectral projectors of a Schr¨
Setting: geometry infinite discrete set Z = (xj)j∈Zd ⊂ Rd, radius δ ∈ (0, 1/2) such that B(xj, δ) ⊂ Λ1 + j = [−1/2, 1/2]d + j δ-neighbourhood of Z: S = S(δ) = B(Z, δ) =
j∈Zd B(xj, δ)
For L ∈ N, set ΛL = [−L/2, L/2]d, SL(δ) = ΛL ∩ S(δ), and characteristic function WL = χSL(δ).
Setting: Schr¨
For bounded, measurable potential V : Rd → R define s.a. Schr¨
HL = −∆ + V on ΛL, L ∈ N, with Dirichlet, Neumann, or periodic b.c. and spectral projection χ(−∞,E](HL), for energy E > 0. Theorem [Naki´ c, T¨ aufer, Tautenhahn, & Veseli´ c 15, 16] Let δ, Z, L, SL(δ), V , HL, E as above. Then exists K depending only on dimension d such that χ(−∞,E](HL) WL χ(−∞,E](HL) ≥ δ
K
√ E+V 2/3
∞
where WL = χSL(δ). Bound is equivalent to
|ψ|2 ≥ δ
K
√ E+V 2/3
∞
ΛL
|ψ|2 for any ψ =
αkψk ∈ ran χ(−∞,E](HL)
allow to deduce ucp (after some calculations) whole zoo of Carleman estimates exists many with abstract weight functions (satisfying H¨
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¨
Weight function φ: [0, ∞) → [0, ∞) φ(r) = r exp
r 1 − e−t t dt
r 1/φ(r) ⇒ ∀ r ∈ (0, 1) : r/3 ≤ φ(r) ≤ r Theorem [Bourgain & Kenig 05] There are constants C1(d) and C2(d) ∈ [1, ∞) s. t. for all α ≥ C1 and real valued f ∈ C 2(B(0, 1)) with compact support in B(0, 1) \ {0} we have α3
Weight function φ: [0, ∞) → [0, ∞) φ(r) = r exp
r 1 − e−t t dt
r 1/φ(r) ⇒ ∀ r ∈ (0, 1) : r/3 ≤ φ(r) ≤ r Theorem [Bourgain & Kenig 05] There are constants C1(d) and C2(d) ∈ [1, ∞) s. t. for all α ≥ C1 and real valued f ∈ C 2(B(0, 1)) with compact support in B(0, 1) \ {0} we have α3
Possible to scale the inequality to a ball of radius ρ and extend to Sobolev space H2.
A1
A2 A3 A1 A1 A2 A2 A3 A3 Insert χ × ψ = cut-off × eigenfunction in Carleman inequality (e.g. [Bourgain, Kenig 05]) to get three annuli inequality α3
w2−2α|ψ|2
w2−2α|ψ|2 +
w2−2α|ψ|2. profile of cut-off function χ
Carleman inequality (e.g. [Bourgain, Kenig 05]) implies three annuli inequality α3
w2−2α|ψ|2
w2−2α|ψ|2 +
w2−2α|ψ|2. 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
ψ known e.g. ψ = 0 Hψ = Eψ ψ unknown
ψ known e.g. ψ = 0 Hψ = Eψ ψ unknown
inequality with one unknown; solvable
ψ known e.g. ψ = 0 Hψ = Eψ ψ unknown
inequality with one unknown; solvable
ψ known e.g. ψ = 0 Hψ = Eψ ψ known also here
ψ known Hψ = Eψ ψ unknown
ψ known Hψ = Eψ ψ unknown
inequality with two unknowns; unsolvable
ψ known Hψ = Eψ ψ unknown
No unique continuation!
ψ known Hψ = Eψ ψ unknown
ψ known Hψ = Eψ
ψ unknown Hψ = Eψ ψ known
ψ unknown Hψ = Eψ ψ known
w1 w2
1
Each vertex ∈ unique upside triangle.
2
Identify the lower left vertex of upside triangle with the origin in C its other two vertices with w1 = 1 and w2 = eπi/3 ∈ C.
3
Vertex set of X V (X) = (2Zw1 + 2Zw2) · ∪ (w1 + 2Zw1 + 2Zw2) · ∪ (w2 + 2Zw1 + 2Zw2). is Z2-periodic.
4
The the planar graph X ⊂ C defines a metric graph.
5
Graph tiles plan with triangles and hexagons.
Eigenfunction supported in one hexagon For an arbitrary hexagon H ⊂ X with vertices {u0, u1, . . . , u5} there exists a centre w0 ∈ C of H such that {u0, u1, . . . , u5} = { w0 + ekπi/3 | k = 0, 1, . . . , 5 }. The function FH : V → {−1, 0, 1} FH(v) :=
if v ∈ V \ {u0, . . . , u5}, (−1)k, if v = w0 + ekπi/3, (1) satisfies ∆combFH(v) = 1 deg(v)
(FH(v) − FH(w)) = 3 2 FH(v).
ℓ2 and finitely supported eigenfunctions
1
If f ∈ ℓ2(V ) satisfies ∆combf = µf , then µ = 3/2.
2
If F ∈ ℓ2(V ) satisfies ∆combF = µF and has finite support then µ = 3/2 and F is a linear combination of finitely many eigenfunctions FH.
3
Let Hi (i = 1, . . . , k) be a collection of distinct hexagons. Then the set FH1, . . . , FHk is linearly independent.
4
The space of ℓ2(V )-eigenfunctions to the eigenvalue 3/2 is spanned by compactly supported eigenfunctions. Whole spectrum Denote by σac(∆comb) and σpp(∆comb) the absolutely continuous and point spectrum of ∆comb on the Z2-periodic Kagome graph. Then σac(∆comb) =
2
σpp(∆comb) = 3 2
Integrated density of states/Spectral distribution function Let Ncomb be the IDS of the Z2-periodic operator ∆comb, given by Ncomb(µ) = 1 3 Tr[1QPcomb((−∞, µ])], where Tr is the trace and Pcomb denotes the spectral projection. Then
1
Ncomb vanishes on (−∞, 0],
2
is continuous on R \ {3/2},
3
has a jump of size 1/3 at µ = 3/2,
4
Ncomb is strictly monotone increasing on [0, 3/2], and
5
Ncomb(µ) = 1 for µ ≥ 3/2.
Quantum graph with Kirchhoff (or free) vertex conditions: function is continuous at the vertex and the sum of all derivatives add up to zero. Spectrum of Kirchhoff Laplacian ∆0 on equilateral Kagome metric graph Let σpp and σac denote the point spectrum and absolutely continuous spectrum and σcomp denote the spectrum given by the compactly supported eigenfunctions. Then σcomp(−∆0) = σpp(−∆0) = 2k + 2 3 2 π2
k ∈ N
σac(−∆0) =
2 3 2 π2 ∪
3 2 π2,
3 2 π2 .
IDS N0 of the negative Kirchhoff Laplacian −∆0 on the metric Kagome graph N0(λ) = 1 vol(F) Tr[1FP0((−∞, λ])], where Tr is the trace, F the fundamental domain, and P0 denotes the spectral projection.
1
All discontinuities of N0 : R → [0, ∞) are
1
at λ = (2k + 2
3 )2π2, k ∈ Z, with jumps of size 1 6 , 2
at λ = k2π2, k ∈ N, with jumps of size 1
2 . 2
N0 is strictly monotone increasing on the absolutely continuous spectrum σac(−∆0) and
3
N0 is constant on the complement of σ(−∆0).
General correspondence for combinatorial and quantum graphs eigenvalues of compactly supported eigenfucntions = jumps of IDS See e.g.
1
Klassert, Lenz, & Stollmann 03,
2
3
Gruber, Lenz & Ves. 07,
4
Lenz & Ves. 09,
5
Pogorzelski, Schwarzenberger, & Seifert 13
Advertisement with kind permission of the organisers: 5th Najman conference on Spectral Theory and Differential Equations 10th to 15th September 2017 Opatija, Croatia Previous conferences in the series are describe at: http://web.math.pmf.unizg.hr/najman conference (or simply google ”Najman conference”) There will appear detailed information on the 5th Najman conference shortly. Organisers: Jussi Behrndt, Luka Grubiˇ si´ c, Ivica Naki´ c, Ivan Veseli´ c