Generic singular continuity for measures
and Delone Hamiltonians
Peter Stollmann
Bedlewo 2004-07-08
1
Overview
Geometric disorder
The corresponding Hamiltonians are defined using Delone sets.
1
Overview
Geometric disorder
The corresponding Hamiltonians are defined using Delone sets.
Back to Wonderland
We present a slight generalization of Simon’s Wonderland theorem
...
1
Overview
Geometric disorder
The corresponding Hamiltonians are defined using Delone sets.
Back to Wonderland
We present a slight generalization of Simon’s Wonderland theorem
...
Analysis in spaces of measures
... based on regularity properties of certain sets of measures. As a
byproduct we get that a generic measure is singular continuous.
1
Overview
Geometric disorder
The corresponding Hamiltonians are defined using Delone sets.
Back to Wonderland
We present a slight generalization of Simon’s Wonderland theorem
...
Analysis in spaces of measures
... based on regularity properties of certain sets of measures. As a
byproduct we get that a generic measure is singular continuous.
Delone sets and Hamiltonians
Here we apply our abstract analysis to Delone Hamiltonians: these
constitute models for geometric disorder. We show that these
operators exhibit purely singular continuous spectrum generically.
2
This is joint work with Daniel Lenz, TU Chemnitz.
www.tu-chemnitz.de/˜stollman
3
Geometric disorder
The ions of a solid are assumed to be distributed in space according
to the points of a Delone set ω ⊂ Rd.
3
Geometric disorder
The ions of a solid are assumed to be distributed in space according
to the points of a Delone set ω ⊂ Rd.
Fix an effective potential v and consider the Hamiltonian
X
H(ω) := −∆ +
v(· − x).
x∈ω
3
Geometric disorder
The ions of a solid are assumed to be distributed in space according
to the points of a Delone set ω ⊂ Rd.
Fix an effective potential v and consider the Hamiltonian
X
H(ω) := −∆ +
v(· − x).
x∈ω
Figure 1: The potential of H(ω) for one ω.
4
Aim: under some mild assumptions there exists a dense Gδ -set of
ω’s for which H(ω) exhibits a purely singular continuous spectral
component.
5
Back to Wonderland
Fix a separable Hilbert space H,
5
Back to Wonderland
Fix a separable Hilbert space H,
consider the space S = S(H) of self-adjoint operators in H with the
strong resolvent topology τsrs.
5
Back to Wonderland
Fix a separable Hilbert space H,
consider the space S = S(H) of self-adjoint operators in H with the
strong resolvent topology τsrs.
Theorem 1. Let (X, ρ) be a complete metric space and H : (X, ρ) →
(S, τsrs) a continuous mapping. Assume that, for an open set
U ⊂ R,
5
Back to Wonderland
Fix a separable Hilbert space H,
consider the space S = S(H) of self-adjoint operators in H with the
strong resolvent topology τsrs.
Theorem 1. Let (X, ρ) be a complete metric space and H : (X, ρ) →
(S, τsrs) a continuous mapping. Assume that, for an open set
U ⊂ R,
(1) the set X1 = {x ∈ X| σpp(H(x)) ∩ U = ∅} is dense in X,
5
Back to Wonderland
Fix a separable Hilbert space H,
consider the space S = S(H) of self-adjoint operators in H with the
strong resolvent topology τsrs.
Theorem 1. Let (X, ρ) be a complete metric space and H : (X, ρ) →
(S, τsrs) a continuous mapping. Assume that, for an open set
U ⊂ R,
(1) the set X1 = {x ∈ X| σpp(H(x)) ∩ U = ∅} is dense in X,
(2) the set X2 = {x ∈ X| σac(H(x)) ∩ U = ∅} is dense in X,
5
Back to Wonderland
Fix a separable Hilbert space H,
consider the space S = S(H) of self-adjoint operators in H with the
strong resolvent topology τsrs.
Theorem 1. Let (X, ρ) be a complete metric space and H : (X, ρ) →
(S, τsrs) a continuous mapping. Assume that, for an open set
U ⊂ R,
(1) the set X1 = {x ∈ X| σpp(H(x)) ∩ U = ∅} is dense in X,
(2) the set X2 = {x ∈ X| σac(H(x)) ∩ U = ∅} is dense in X,
(3) the set X3 = {x ∈ X| U ⊂ σ(H(x))} is dense in X.
5
Back to Wonderland
Fix a separable Hilbert space H,
consider the space S = S(H) of self-adjoint operators in H with the
strong resolvent topology τsrs.
Theorem 1. Let (X, ρ) be a complete metric space and H : (X, ρ) →
(S, τsrs) a continuous mapping. Assume that, for an open set
U ⊂ R,
(1) the set X1 = {x ∈ X| σpp(H(x)) ∩ U = ∅} is dense in X,
(2) the set X2 = {x ∈ X| σac(H(x)) ∩ U = ∅} is dense in X,
(3) the set X3 = {x ∈ X| U ⊂ σ(H(x))} is dense in X.
Then, their intersection
{x ∈ X| U ⊂ σ(H(x)), σac(H(x)) ∩ U = ∅, σpp(H(x)) ∩ U = ∅}
is a dense Gδ -set in X.
6
This generalizes a result from
6
This generalizes a result from
B. Simon, Operators with singular continuous spectrum: I. General
operators. Annals of Math. 141, 131 – 145 (1995)
6
This generalizes a result from
B. Simon, Operators with singular continuous spectrum: I. General
operators. Annals of Math. 141, 131 – 145 (1995)
By Baire’s theorem, it suffices to prove that X1 − X3 are Gδ -sets.
6
This generalizes a result from
B. Simon, Operators with singular continuous spectrum: I. General
operators. Annals of Math. 141, 131 – 145 (1995)
By Baire’s theorem, it suffices to prove that X1 − X3 are Gδ -sets.
By continuity it suffices to prove that the corresponding sets of
measures are Gδ -sets.
6
This generalizes a result from
B. Simon, Operators with singular continuous spectrum: I. General
operators. Annals of Math. 141, 131 – 145 (1995)
By Baire’s theorem, it suffices to prove that X1 − X3 are Gδ -sets.
By continuity it suffices to prove that the corresponding sets of
measures are Gδ -sets.
In fact, for fixed ξ ∈ H, the mapping
6
This generalizes a result from
B. Simon, Operators with singular continuous spectrum: I. General
operators. Annals of Math. 141, 131 – 145 (1995)
By Baire’s theorem, it suffices to prove that X1 − X3 are Gδ -sets.
By continuity it suffices to prove that the corresponding sets of
measures are Gδ -sets.
In fact, for fixed ξ ∈ H, the mapping
H(x)
X → M+(U ) := {measures on U }, x 7→ ρξ
is continuous.
|U
6
This generalizes a result from
B. Simon, Operators with singular continuous spectrum: I. General
operators. Annals of Math. 141, 131 – 145 (1995)
By Baire’s theorem, it suffices to prove that X1 − X3 are Gδ -sets.
By continuity it suffices to prove that the corresponding sets of
measures are Gδ -sets.
In fact, for fixed ξ ∈ H, the mapping
H(x)
X → M+(U ) := {measures on U }, x 7→ ρξ
is continuous.
For every dense {ξn|n ∈ N} ⊂ H:
|U
6
This generalizes a result from
B. Simon, Operators with singular continuous spectrum: I. General
operators. Annals of Math. 141, 131 – 145 (1995)
By Baire’s theorem, it suffices to prove that X1 − X3 are Gδ -sets.
By continuity it suffices to prove that the corresponding sets of
measures are Gδ -sets.
In fact, for fixed ξ ∈ H, the mapping
H(x)
X → M+(U ) := {measures on U }, x 7→ ρξ
is continuous.
For every dense {ξn|n ∈ N} ⊂ H:
\
H(x)
X1 =
{x| ρξn |U ∈ Mc(U )}.
n∈N
|U
7
Analysis in spaces of measures
Fix S, a locally compact, σ-compact, separable metric space.
7
Analysis in spaces of measures
Fix S, a locally compact, σ-compact, separable metric space.
Consider M+(S), the set of positive, regular Borel measures.
7
Analysis in spaces of measures
Fix S, a locally compact, σ-compact, separable metric space.
Consider M+(S), the set of positive, regular Borel measures.
µ ∈ M+(S) diffusive or continuous:⇐⇒ µ({x}) = 0 for every
x ∈ S.
7
Analysis in spaces of measures
Fix S, a locally compact, σ-compact, separable metric space.
Consider M+(S), the set of positive, regular Borel measures.
µ ∈ M+(S) diffusive or continuous:⇐⇒ µ({x}) = 0 for every
x ∈ S.
µ ⊥ ν, mutually singular :⇐⇒ ∃ C ⊂ S such that
µ(C) = 0 = ν(S \ C).
7
Analysis in spaces of measures
Fix S, a locally compact, σ-compact, separable metric space.
Consider M+(S), the set of positive, regular Borel measures.
µ ∈ M+(S) diffusive or continuous:⇐⇒ µ({x}) = 0 for every
x ∈ S.
µ ⊥ ν, mutually singular :⇐⇒ ∃ C ⊂ S such that
µ(C) = 0 = ν(S \ C).
Theorem 2. Let S be as above. Then
7
Analysis in spaces of measures
Fix S, a locally compact, σ-compact, separable metric space.
Consider M+(S), the set of positive, regular Borel measures.
µ ∈ M+(S) diffusive or continuous:⇐⇒ µ({x}) = 0 for every
x ∈ S.
µ ⊥ ν, mutually singular :⇐⇒ ∃ C ⊂ S such that
µ(C) = 0 = ν(S \ C).
Theorem 2. Let S be as above. Then
(1) The set {µ ∈ M+(S)| µ is diffusive} is a Gδ -set in M+(S).
7
Analysis in spaces of measures
Fix S, a locally compact, σ-compact, separable metric space.
Consider M+(S), the set of positive, regular Borel measures.
µ ∈ M+(S) diffusive or continuous:⇐⇒ µ({x}) = 0 for every
x ∈ S.
µ ⊥ ν, mutually singular :⇐⇒ ∃ C ⊂ S such that
µ(C) = 0 = ν(S \ C).
Theorem 2. Let S be as above. Then
(1) The set {µ ∈ M+(S)| µ is diffusive} is a Gδ -set in M+(S).
(2) For any λ ∈ M+(S), the set {µ ∈ M+(S)| µ ⊥ λ} is a Gδ -set
in M+(S).
7
Analysis in spaces of measures
Fix S, a locally compact, σ-compact, separable metric space.
Consider M+(S), the set of positive, regular Borel measures.
µ ∈ M+(S) diffusive or continuous:⇐⇒ µ({x}) = 0 for every
x ∈ S.
µ ⊥ ν, mutually singular :⇐⇒ ∃ C ⊂ S such that
µ(C) = 0 = ν(S \ C).
Theorem 2. Let S be as above. Then
(1) The set {µ ∈ M+(S)| µ is diffusive} is a Gδ -set in M+(S).
(2) For any λ ∈ M+(S), the set {µ ∈ M+(S)| µ ⊥ λ} is a Gδ -set
in M+(S).
(3) For any closed F ⊂ S the set {µ ∈ M+(S)| F ⊂ supp(µ)} is a
Gδ -set in M+(S).
8
Proposition 3. Let K ⊂ M+(S) be compact. Then
8
Proposition 3. Let K ⊂ M+(S) be compact. Then
K• := {µ ∈ M+(S)| ∃ν ∈ K : ν ≤ µ}
is closed.
The proof is easy.
8
Proposition 3. Let K ⊂ M+(S) be compact. Then
K• := {µ ∈ M+(S)| ∃ν ∈ K : ν ≤ µ}
is closed.
The proof is easy.
Proposition 4. Let K ⊂ S be compact and a > 0. Then
8
Proposition 3. Let K ⊂ M+(S) be compact. Then
K• := {µ ∈ M+(S)| ∃ν ∈ K : ν ≤ µ}
is closed.
The proof is easy.
Proposition 4. Let K ⊂ S be compact and a > 0. Then
{a · δx| x ∈ K}
is compact in M+(S).
8
Proposition 3. Let K ⊂ M+(S) be compact. Then
K• := {µ ∈ M+(S)| ∃ν ∈ K : ν ≤ µ}
is closed.
The proof is easy.
Proposition 4. Let K ⊂ S be compact and a > 0. Then
{a · δx| x ∈ K}
is compact in M+(S).
The proof is evident from the fact that S → M+(S), x 7→ δx is
continuous.
9
Proposition 5. Let λ ∈ M+(S), K ⊂ S compact and γ > 0 be
given. Then
9
Proposition 5. Let λ ∈ M+(S), K ⊂ S compact and γ > 0 be
given. Then
K := {f ·λ| f ∈ L2(λ), kf kL2(λ) ≤ 1, 0 ≤ f, supp(f ) ⊂ K,
is compact.
Z
f dλ ≥ γ}
9
Proposition 5. Let λ ∈ M+(S), K ⊂ S compact and γ > 0 be
given. Then
K := {f ·λ| f ∈ L2(λ), kf kL2(λ) ≤ 1, 0 ≤ f, supp(f ) ⊂ K,
Z
f dλ ≥ γ}
is compact.
Proof. The densities considered in K form a closed subset of the unit
ball of L2(K, λ).
9
Proposition 5. Let λ ∈ M+(S), K ⊂ S compact and γ > 0 be
given. Then
K := {f ·λ| f ∈ L2(λ), kf kL2(λ) ≤ 1, 0 ≤ f, supp(f ) ⊂ K,
Z
f dλ ≥ γ}
is compact.
Proof. The densities considered in K form a closed subset of the unit
ball of L2(K, λ).
Since the latter is weakly compact and the mapping L2(K, λ)+ →
M+(S), f 7→ f · λ is w-w∗ -continuous we get the desired compactness.
10
Proof of Theorem 2 (2): (Ms(S) is a Gδ .) By assumption on S, we
find a sequence of compacts Kn % S; let
10
Proof of Theorem 2 (2): (Ms(S) is a Gδ .) By assumption on S, we
find a sequence of compacts Kn % S; let
Z
1
2
K2,n = {f λ| f ∈ L (λ), kf k ≤ 1, 0 ≤ f, supp(f ) ⊂ Kn, f dλ ≥ }
n
which is compact by Proposition 5;
10
Proof of Theorem 2 (2): (Ms(S) is a Gδ .) By assumption on S, we
find a sequence of compacts Kn % S; let
Z
1
2
K2,n = {f λ| f ∈ L (λ), kf k ≤ 1, 0 ≤ f, supp(f ) ⊂ Kn, f dλ ≥ }
n
which is compact by Proposition 5;
•
by Proposition 3 we get that F2,n = K2,n
is closed. Since
10
Proof of Theorem 2 (2): (Ms(S) is a Gδ .) By assumption on S, we
find a sequence of compacts Kn % S; let
Z
1
2
K2,n = {f λ| f ∈ L (λ), kf k ≤ 1, 0 ≤ f, supp(f ) ⊂ Kn, f dλ ≥ }
n
which is compact by Proposition 5;
•
by Proposition 3 we get that F2,n = K2,n
is closed. Since
[
M2 =
F2,n
n∈N
is an Fσ and
10
Proof of Theorem 2 (2): (Ms(S) is a Gδ .) By assumption on S, we
find a sequence of compacts Kn % S; let
Z
1
2
K2,n = {f λ| f ∈ L (λ), kf k ≤ 1, 0 ≤ f, supp(f ) ⊂ Kn, f dλ ≥ }
n
which is compact by Proposition 5;
•
by Proposition 3 we get that F2,n = K2,n
is closed. Since
[
M2 =
F2,n
n∈N
is an Fσ and
Mc2 = {µ ∈ M+(S)| µ ⊥ λ},
part (2) of Theorem 2 is proven.
10
Proof of Theorem 2 (2): (Ms(S) is a Gδ .) By assumption on S, we
find a sequence of compacts Kn % S; let
Z
1
2
K2,n = {f λ| f ∈ L (λ), kf k ≤ 1, 0 ≤ f, supp(f ) ⊂ Kn, f dλ ≥ }
n
which is compact by Proposition 5;
•
by Proposition 3 we get that F2,n = K2,n
is closed. Since
[
M2 =
F2,n
n∈N
is an Fσ and
Mc2 = {µ ∈ M+(S)| µ ⊥ λ},
part (2) of Theorem 2 is proven.
11
Corollary 6. Let U be an open subset of Rd. Then the singular
continuous measures Msc(U ) form a dense Gδ in the space M+(U )
of Borel measures.
11
Corollary 6. Let U be an open subset of Rd. Then the singular
continuous measures Msc(U ) form a dense Gδ in the space M+(U )
of Borel measures.
Once more we find that the silent majority consists of rather strange
individuals.
11
Corollary 6. Let U be an open subset of Rd. Then the singular
continuous measures Msc(U ) form a dense Gδ in the space M+(U )
of Borel measures.
Once more we find that the silent majority consists of rather strange
individuals.
In mathematical terms, we owe this fact to Baire and completeness.
11
Corollary 6. Let U be an open subset of Rd. Then the singular
continuous measures Msc(U ) form a dense Gδ in the space M+(U )
of Borel measures.
Once more we find that the silent majority consists of rather strange
individuals.
In mathematical terms, we owe this fact to Baire and completeness.
In this connection let us mention two classical papers:
11
Corollary 6. Let U be an open subset of Rd. Then the singular
continuous measures Msc(U ) form a dense Gδ in the space M+(U )
of Borel measures.
Once more we find that the silent majority consists of rather strange
individuals.
In mathematical terms, we owe this fact to Baire and completeness.
In this connection let us mention two classical papers:
S. Banach, Über die Bairesche Kategorie gewisser
Funktionenmengen. Studia Math. 3 (1931), 174–179
11
Corollary 6. Let U be an open subset of Rd. Then the singular
continuous measures Msc(U ) form a dense Gδ in the space M+(U )
of Borel measures.
Once more we find that the silent majority consists of rather strange
individuals.
In mathematical terms, we owe this fact to Baire and completeness.
In this connection let us mention two classical papers:
S. Banach, Über die Bairesche Kategorie gewisser
Funktionenmengen. Studia Math. 3 (1931), 174–179
S. Mazurkiewicz, Sur les fonctions non dérivables, Studia Math.
3 (1931), 92–94
12
Delone (Delaunay) sets and Hamiltonians
Definition. A set ω ⊂ Rd is called an (r, R)-set if
12
Delone (Delaunay) sets and Hamiltonians
Definition. A set ω ⊂ Rd is called an (r, R)-set if
• ∀x, y ∈ ω, x 6= y : Ur (x) ∩ Ur (y) = ∅,
12
Delone (Delaunay) sets and Hamiltonians
Definition. A set ω ⊂ Rd is called an (r, R)-set if
• ∀x, y ∈ ω, x 6= y : Ur (x) ∩ Ur (y) = ∅,
S
• x∈ω BR(x) = Rd.
12
Delone (Delaunay) sets and Hamiltonians
Definition. A set ω ⊂ Rd is called an (r, R)-set if
• ∀x, y ∈ ω, x 6= y : Ur (x) ∩ Ur (y) = ∅,
S
• x∈ω BR(x) = Rd.
picture
12
Delone (Delaunay) sets and Hamiltonians
Definition. A set ω ⊂ Rd is called an (r, R)-set if
• ∀x, y ∈ ω, x 6= y : Ur (x) ∩ Ur (y) = ∅,
S
• x∈ω BR(x) = Rd.
picture
By Dr,R(Rd) = Dr,R we denote the set of all (r, R)-sets. We say that
ω ⊂ Rd is a Delone set, if it is an (r, R)-set for some 0 < r ≤ R so
S
d
that D(R ) = D = 0<r≤R Dr,R(Rd) is the set of all Delone sets.
12
Delone (Delaunay) sets and Hamiltonians
Definition. A set ω ⊂ Rd is called an (r, R)-set if
• ∀x, y ∈ ω, x 6= y : Ur (x) ∩ Ur (y) = ∅,
S
• x∈ω BR(x) = Rd.
picture
By Dr,R(Rd) = Dr,R we denote the set of all (r, R)-sets. We say that
ω ⊂ Rd is a Delone set, if it is an (r, R)-set for some 0 < r ≤ R so
S
d
that D(R ) = D = 0<r≤R Dr,R(Rd) is the set of all Delone sets.
Dr,R is a compact metric space in the natural topology.
12
Delone (Delaunay) sets and Hamiltonians
Definition. A set ω ⊂ Rd is called an (r, R)-set if
• ∀x, y ∈ ω, x 6= y : Ur (x) ∩ Ur (y) = ∅,
S
• x∈ω BR(x) = Rd.
picture
By Dr,R(Rd) = Dr,R we denote the set of all (r, R)-sets. We say that
ω ⊂ Rd is a Delone set, if it is an (r, R)-set for some 0 < r ≤ R so
S
d
that D(R ) = D = 0<r≤R Dr,R(Rd) is the set of all Delone sets.
Dr,R is a compact metric space in the natural topology.
13
Consider
H(ω) := −∆ +
X
v(· − x) in Rd,
x∈ω
where ω ∈ D, v ≥ 0 is bounded, measurable and compactly
supported. If ρ ∈ Dr,R is crystallographic, i.e.
13
Consider
H(ω) := −∆ +
X
v(· − x) in Rd,
x∈ω
where ω ∈ D, v ≥ 0 is bounded, measurable and compactly
supported. If ρ ∈ Dr,R is crystallographic, i.e.
P er(ρ) := {t ∈ Rd : ρ = t + ρ}
13
Consider
H(ω) := −∆ +
X
v(· − x) in Rd,
x∈ω
where ω ∈ D, v ≥ 0 is bounded, measurable and compactly
supported. If ρ ∈ Dr,R is crystallographic, i.e.
P er(ρ) := {t ∈ Rd : ρ = t + ρ}
is a lattice, then H(ρ) is periodic.
13
Consider
H(ω) := −∆ +
X
v(· − x) in Rd,
x∈ω
where ω ∈ D, v ≥ 0 is bounded, measurable and compactly
supported. If ρ ∈ Dr,R is crystallographic, i.e.
P er(ρ) := {t ∈ Rd : ρ = t + ρ}
is a lattice, then H(ρ) is periodic.
Theorem 7. Let r, R > 0 with 2r < R and v 6= 0. Then there exists
an open ∅ =
6 U ⊂ R and a dense Gδ -set Ωsc ⊂ Dr,R such that
for every ω ∈ Ωsc the spectrum of H(ω) contains U and is purely
singular continuous in U .
Proof. Dr,R =: X, use the “Wonderland” theorem.
13
Consider
H(ω) := −∆ +
X
v(· − x) in Rd,
x∈ω
where ω ∈ D, v ≥ 0 is bounded, measurable and compactly
supported. If ρ ∈ Dr,R is crystallographic, i.e.
P er(ρ) := {t ∈ Rd : ρ = t + ρ}
is a lattice, then H(ρ) is periodic.
Theorem 7. Let r, R > 0 with 2r < R and v 6= 0. Then there exists
an open ∅ =
6 U ⊂ R and a dense Gδ -set Ωsc ⊂ Dr,R such that
for every ω ∈ Ωsc the spectrum of H(ω) contains U and is purely
singular continuous in U .
Proof. Dr,R =: X, use the “Wonderland” theorem.
14
Since 2r < R, there exist crystallographic γ, γ̃ such that U :=
σ(H(γ))◦ \ σ(H(γ̃)) 6= ∅. picture
14
Since 2r < R, there exist crystallographic γ, γ̃ such that U :=
σ(H(γ))◦ \ σ(H(γ̃)) 6= ∅. picture
Have to prove: ∀ω ∈ Dr,R there exists (ωn)n s.t. H(ωn) is purely
singular in U and ωn → ω.
14
Since 2r < R, there exist crystallographic γ, γ̃ such that U :=
σ(H(γ))◦ \ σ(H(γ̃)) 6= ∅. picture
Have to prove: ∀ω ∈ Dr,R there exists (ωn)n s.t. H(ωn) is purely
singular in U and ωn → ω.
To this end consider ωn ∈ Dr,R s.t. ωn ∩ [−n, n]d = ω ∩ [−n, n]d and
ωn ∩ ([−n − R, n + R]d)c = γ̃ ∩ ([−n − R, n + R]d)c, picture
14
Since 2r < R, there exist crystallographic γ, γ̃ such that U :=
σ(H(γ))◦ \ σ(H(γ̃)) 6= ∅. picture
Have to prove: ∀ω ∈ Dr,R there exists (ωn)n s.t. H(ωn) is purely
singular in U and ωn → ω.
To this end consider ωn ∈ Dr,R s.t. ωn ∩ [−n, n]d = ω ∩ [−n, n]d and
ωn ∩ ([−n − R, n + R]d)c = γ̃ ∩ ([−n − R, n + R]d)c, picture
Since H(ωn) and H(γ̃) only differ by a compactly supported potential,
σac(H(ωn)) ∩ U = σac(H(γ̃)) ∩ U = ∅.
14
Since 2r < R, there exist crystallographic γ, γ̃ such that U :=
σ(H(γ))◦ \ σ(H(γ̃)) 6= ∅. picture
Have to prove: ∀ω ∈ Dr,R there exists (ωn)n s.t. H(ωn) is purely
singular in U and ωn → ω.
To this end consider ωn ∈ Dr,R s.t. ωn ∩ [−n, n]d = ω ∩ [−n, n]d and
ωn ∩ ([−n − R, n + R]d)c = γ̃ ∩ ([−n − R, n + R]d)c, picture
Since H(ωn) and H(γ̃) only differ by a compactly supported potential,
σac(H(ωn)) ∩ U = σac(H(γ̃)) ∩ U = ∅.
Similarly, the other sets are shown to be dense and the Wonderland
theorem implies the assertion.
15
Conclusion
• Mc(S) and Ms(S) are Gδ -sets, for polish S.
• This implies the Wonderland theorem and the fact that generic
measures are singular continuous in “nice spaces”.
• A particular example is given by “geometric disorder” (= Delone
Hamiltonians).
16
Strong resolvent topology
Reminder:
An → A in τsrs
16
Strong resolvent topology
Reminder:
An → A in τsrs
:⇐⇒
16
Strong resolvent topology
Reminder:
An → A in τsrs
:⇐⇒
(An + i)−1ξ → (A + i)−1ξ
(ξ ∈ H)
16
Strong resolvent topology
Reminder:
An → A in τsrs
:⇐⇒
(An + i)−1ξ → (A + i)−1ξ
⇐⇒
ϕ(An)ξ → ϕ(A)ξ
(ξ ∈ H)
(ξ ∈ H, ϕ ∈ Cc(R))
16
Strong resolvent topology
Reminder:
An → A in τsrs
:⇐⇒
(An + i)−1ξ → (A + i)−1ξ
⇐⇒
ϕ(An)ξ → ϕ(A)ξ
n
ρA
ξ
back
(ξ ∈ H)
(ξ ∈ H, ϕ ∈ Cc(R))
=⇒
A
→ ρA
vaguely,
where
hρ
ξ
ξ , ϕi := (ϕ(A)ξ|ξ).
17
Delone sets
18
Figure 2: A Delone set ω and r.
back
19
γ and γ̃
Figure 3: crystallographic γ (green) and γ̃ (green+blue) back
20
ωn
21
Figure 4: ωn back