Homogeneous wavefront sets

Radially homogeneous wavefront sets

Pseudodiﬀerential calculus

Proof of main theorem

Propagation of singularities for Schrödinger
equations on manifolds with ends
Shota Fukushima (D3)
Graduate School of Mathematical Science, the University of Tokyo

Himeji conference on Partial Diﬀerential Equations
March 2–4, 2022

Shota Fukushima (D3)

Graduate School of Mathematical Science, the University of Tokyo

Propagation of singularities for Schrödinger equations on manifolds with ends

Homogeneous wavefront sets

Radially homogeneous wavefront sets

Pseudodiﬀerential calculus

Proof of main theorem

Contents

1 Homogeneous wavefront sets
2 Radially homogeneous wavefront sets
3 Pseudodiﬀerential calculus
4 Proof of main theorem

Shota Fukushima (D3)

Graduate School of Mathematical Science, the University of Tokyo

Propagation of singularities for Schrödinger equations on manifolds with ends

Homogeneous wavefront sets

Radially homogeneous wavefront sets

Pseudodiﬀerential calculus

Proof of main theorem

Homogeneous wavefront sets (HWF)
Deﬁnition 1
For u ∈ L2 (Rn ), we deﬁne a homogeneous wavefront set
HWF(u) ⊂ T ∗ Rn \ {(0, 0)} as follows: a point(x0 , ξ0 ) ∈ T ∗ Rn \ {(0, 0)} is not in HWF(u) if there exists a
symbol a ∈ Cc∞ (T ∗ Rn ) such thata = 1 near (x0 , ξ0 ),∥aw (ℏx, ℏD)u∥L2 = O(ℏ∞ ).
Here aw (ℏx, ℏD) is deﬁned asZ1ℏx + ℏyw
a (ℏx, ℏD)u(x) :=a, ℏξ eiξ·(x−y) u(y) dydξ.
(2π)n T ∗ Rn
2
Shota Fukushima (D3)

Graduate School of Mathematical Science, the University of Tokyo

Propagation of singularities for Schrödinger equations on manifolds with ends

Homogeneous wavefront sets

Radially homogeneous wavefront sets

Pseudodiﬀerential calculus

Proof of main theorem

Theorem 2 (Nakamura (2005))
Let H be a Hamiltonian of the form
1 X
H := −∂xj (ajk (x)∂xk ) + V (x)
2
n

j,k=1

with
|∂xα (ajk (x) − δjk )| ≤ Cα ⟨x⟩−µ−|α| (∃µ > 0),|∂xα V (x)| ≤ Cα ⟨x⟩ν−|α| (∃ν < 2).
Let (x(t), ξ(t)) be aPnontrapping classical orbit with respect to the
Hamiltonian h0 := nj,k=1 ajk (x)ξj ξk /2 and let
ξ∞ := limt→∞ ξ(t). Then, for any t0 > 0 and u ∈ L2 (Rn ),(x(0), ξ(0)) ∈ WF(u) =⇒ (t0 ξ∞ , ξ∞ ) ∈ HWF(e−it0 H u).
Shota Fukushima (D3)

Graduate School of Mathematical Science, the University of Tokyo

Propagation of singularities for Schrödinger equations on manifolds with ends

Homogeneous wavefront sets

Radially homogeneous wavefront sets

Pseudodiﬀerential calculus

Proof of main theorem

HWF on manifolds?
ℏx”??? Instead of ℏx, we want to consider polar
On manifolds: “ℏcoordinates (r, θ) and replace ℏx to (ℏr, θ).
=⇒ We want to describe the regularity such asaw (ℏr, θ, ℏDr , ℏDθ )u = OL2 (ℏ∞ ).
⇐⇒ radially homogeneous wavefront sets (Ito-Nakamura (Amer.
J. Math., 2009)).
RemarkThe (complement of) HWF is described asaw (ℏr, θ, ℏDr , ℏ2 Dθ )u = OL2 (ℏ∞ ).
For x ̸= 0, (x, ξ) ∈ WFrh (u) =⇒ (x, (ξ · x̂)x̂) ∈ HWF(u)where x̂ := x/|x|.
Shota Fukushima (D3)

Graduate School of Mathematical Science, the University of Tokyo

Propagation of singularities for Schrödinger equations on manifolds with ends

Homogeneous wavefront sets

Radially homogeneous wavefront sets

Pseudodiﬀerential calculus

Proof of main theorem

Manifolds with ends
Let M be an n-dimensional non-compact manifold.
Assumption 1
There exist
an open subset E of M ,an (n − 1)-dimensional compact manifold S anda diﬀeomorphism Ψ : E → R+ × S
such that the set M \ Ψ−1 ((1, ∞)) is a compact subset of M .
Here R+ := (0, ∞).
The set E is called the end of M .
Shota Fukushima (D3)

Graduate School of Mathematical Science, the University of Tokyo

Propagation of singularities for Schrödinger equations on manifolds with ends

Homogeneous wavefront sets

Radially homogeneous wavefront sets

Pseudodiﬀerential calculus

Proof of main theorem

Intrinsic L2 space
Let Cc∞ (M ; Ω1/2 ) be the space of compactly supported smooth
half-densities on M . Then we introduce an inner product on
Cc∞ (M ; Ω1/2 ) deﬁned asZ⟨u, v⟩ :=ũ(x)ṽ(x) dx
Mwhere u = ũ|dx|1/2 and v = ṽ|dx|1/2 locally. The intrinsic L2
space L2 (M ; Ω1/2 ) is deﬁned as the completion of Cc∞ (M ; Ω1/2 )by the inner product ⟨·, ·⟩.

Shota Fukushima (D3)

Graduate School of Mathematical Science, the University of Tokyo

Propagation of singularities for Schrödinger equations on manifolds with ends

Homogeneous wavefront sets

Radially homogeneous wavefront sets

Pseudodiﬀerential calculus

Proof of main theorem

Hamiltonian
We consider a Hamiltonian1
H = − △g + V (x)2where△g is the associated Laplacian with respect to a ﬁxed metricg on M ,V ∈ C ∞ (M ; R) is a potential function.
△g acts on half-density u = ũ|volg (x)|1/2 as△g (ũ|volg (x)|1/2 ) := (△g ũ)|volg (x)|1/2 .

Shota Fukushima (D3)

Graduate School of Mathematical Science, the University of Tokyo

Propagation of singularities for Schrödinger equations on manifolds with ends

Homogeneous wavefront sets

Radially homogeneous wavefront sets

Pseudodiﬀerential calculus

Proof of main theorem

MetricAssumption 2
The metric g is the formΨ∗ g(r, θ, dr, dθ) = c(r, θ)2 dr2 + h(r, θ, dθ)where c(r, θ) > 0 and h(r, θ, dθ) is a r-dependent metric on S.
C −1 f (r)2 h(1, θ, dθ) ≤ h(r, θ, dθ) ≤ Cf (r)2 h(1, θ, dθ) forsome constant C > 0 and a smooth function f : R → R+ with
c0 r−1 ≤ f ′ (r)/f (r) ≤ C(r ≥ 1)for some c0 > 1/2.
Shota Fukushima (D3)

Graduate School of Mathematical Science, the University of Tokyo

Propagation of singularities for Schrödinger equations on manifolds with ends

Homogeneous wavefront sets

Radially homogeneous wavefront sets

Pseudodiﬀerential calculus

Proof of main theorem

MetricAssumption 2 (continued)
For all multiindices α = (α0 , α′ ) ∈ Z≥0 × Zn−1≥0 , the estimates′|∂rα0 ∂θα (c(r, θ) − 1)| ≤ Cα r−1−µ ,
n−1
X

′

∂rα0 ∂θα hjk (r, θ)wj wk ≤ Cα h(r, θ, w) (∀w ∈ Tθ S),

j,k=1
′

|∂rα0 ∂θα V (r, θ)| ≤ Cα .

Shota Fukushima (D3)

Graduate School of Mathematical Science, the University of Tokyo

Propagation of singularities for Schrödinger equations on manifolds with ends

Homogeneous wavefront sets

Radially homogeneous wavefront sets

Pseudodiﬀerential calculus

Proof of main theorem

Classical free Hamiltonian∗ M : dual variable of (r, θ).
(ρ, η) ∈ T(r,θ)
The classical free Hamiltonian is1h0 (r, θ, ρ, η) =c(r, θ)−2 ρ2 + h∗ (r, θ, η) .
2∗
Here h (r, θ, η) is the dual metric∗h (r, θ, η) :=n−1
Xhjk (r, θ)ηj ηkj,k=1

where hjk (r, θ) is the inverse matrix of (hjk (r, θ))n−1j,k=1 deﬁned ash(r, θ, dθ) =n−1
Xhjk (r, θ)dθj dθk .

j,k=1
Shota Fukushima (D3)

Graduate School of Mathematical Science, the University of Tokyo

Propagation of singularities for Schrödinger equations on manifolds with ends

Homogeneous wavefront sets

Radially homogeneous wavefront sets

Pseudodiﬀerential calculus

Proof of main theorem

Classical analogue of Mourre estimate
Assumption 3{f ρ, h0 } ≥ 2f ′ (r)(h0 − Cr−1−µ )holds for all (r, θ, ρ, η) ∈ T ∗ E ∩ {r ≥ 1}.
Theorem 3
Let (r(t), θ(t), ρ(t), η(t)) be a nontrapping (r(t) → ∞ as t → ∞)classical orbit with respect to the Hamiltonian h0 . Then, under
Assumption 1–3,∃(ρ∞ , θ∞ , η∞ ) := lim (ρ(t), θ(t), η(t)) ∈ R+ × T ∗ S.
t→∞

Shota Fukushima (D3)

Graduate School of Mathematical Science, the University of Tokyo

Propagation of singularities for Schrödinger equations on manifolds with ends

Homogeneous wavefront sets

Radially homogeneous wavefront sets

Pseudodiﬀerential calculus

Proof of main theorem

Radially homogeneous wavefront sets
Deﬁnition 4
For u ∈ L2 (M ; Ω1/2 ), we deﬁne a radially homogeneous wavefront
set WFrh (u) ⊂ T ∗ E as follows: a point (x0 , ξ0 ) ∈ T ∗ E does not
in WFrh (u) if there exist a ∈ Cc∞ (T ∗ Rn ), χ ∈ C ∞ (M ) and apolar coordinate function
φ : U (⊂ E) → V = R+ × V ′ (⊂ R+ × S) near x0 such that
supp a ⊂ φ̃(T ∗ E) and a = 1 near φ̃(x0 , ξ0 ),χ(r, θ) = ∃χang (θ) for r ≫ 1, supp χ ⊂ U , and χ = 1 nearΨ−1 ([R, ∞) × {θ∞ }) for some R > 0,∥χφ∗ aw (ℏr, θ, ℏDr , ℏDθ )φ∗ (χu)∥L2 = O(ℏ∞ ).

Shota Fukushima (D3)

Graduate School of Mathematical Science, the University of Tokyo

Propagation of singularities for Schrödinger equations on manifolds with ends

Homogeneous wavefront sets

Radially homogeneous wavefront sets

Pseudodiﬀerential calculus

Proof of main theorem

Main theorem

Theorem 5 (F, arXiv:2201.09466 [math.AP])
Suppose Assumption 1–3. Let u ∈ L2 (M ; Ω1/2 ) and(x(t), ξ(t)) = Ψ−1 (r(t), θ(t), ρ(t), η(t)) is a nontrapping classicalorbit. Then, for any t0 > 0, (x(0), ξ(0)) ∈ WF(u) implies
Ψ−1 (ρ∞ t0 , θ∞ , ρ∞ , η∞ ) ∈ WFrh (e−it0 H u).

Shota Fukushima (D3)

Graduate School of Mathematical Science, the University of Tokyo

Propagation of singularities for Schrödinger equations on manifolds with ends

Homogeneous wavefront sets

Radially homogeneous wavefront sets

Pseudodiﬀerential calculus

Proof of main theorem

Symbol class
Deﬁnition 6m (T ∗ M ) ⊂ C ∞ (T ∗ M ) as
For m ∈ R, we deﬁne a symbol class Scyl
follows:

For polar coordinates (r, θ), the estimate′′|∂rα0 ∂θα ∂ρβ0 ∂ηβ a(r, θ, ρ, η)| ≤ Cαβ (1 + |ρ| + |η|)m−|β|holds for all multiindices α = (α0 , α′ ),β = (β0 , β ′ ) ∈ Z≥0 × Zn−1≥0 .
The conditions of usual Kohn-Nirenberg symbols are satisﬁed
on M \ E.

Shota Fukushima (D3)

Graduate School of Mathematical Science, the University of Tokyo

Propagation of singularities for Schrödinger equations on manifolds with ends

Homogeneous wavefront sets

Radially homogeneous wavefront sets

Pseudodiﬀerential calculus

Proof of main theorem

Quantization
We ﬁx
a ﬁnite atlas {φι : Uι → Vι }ι∈I ,a partition of unity {κι ∈ C ∞ (M )}ι∈I subordinate to theatlas,
a family of functions {χι ∈ C ∞ (M )}ι∈I such that
supp χι ⊂ Uι and χι = 1 near supp κι ,and deﬁneOpℏ (a)u :=Xχι φ∗ι (φ̃ι∗ a)w (x, ℏD)φ∗ (χι u)ι∈Im (T ∗ M ) and u ∈ C ∞ (M ; Ω1/2 ).
for a ∈ Scylc
Shota Fukushima (D3)

Graduate School of Mathematical Science, the University of Tokyo

Propagation of singularities for Schrödinger equations on manifolds with ends

Homogeneous wavefront sets

Radially homogeneous wavefront sets

Pseudodiﬀerential calculus

Proof of main theorem

Quantization
The index set I is decomposed into I = IK ∪ I∞ :{Uι }ι∈IK covers M \ E and κι , χι ∈ Cc∞ (Uι ), and{Uι }ι∈I∞ is a family of polar coordinates on E andκι , χι ∈ C ∞ (Uι ) depend only on θ near inﬁnity.
Basic properties as in the usual pseudodiﬀerential operators hold:
Calderón-Vaillancourt
Ptheorem:0 (T ∗ M ).
∥ Opℏ (a)∥L2 →L2 ≤ C |α|≤N |∂ α a| for a ∈ Scyl
Sharp Gårding inequality:
Re Opℏ (a) ≥ − Opℏ (b) + OL2 →L2 (ℏ∞ ) for any
0 (T ∗ M ) with Re a ≥ 0 and some b ∈ S 0 (T ∗ M ) witha ∈ Scyl
cylsupp b ⊂ supp a modulo O(ℏ∞ ).
Shota Fukushima (D3)

Graduate School of Mathematical Science, the University of Tokyo

Propagation of singularities for Schrödinger equations on manifolds with ends

Homogeneous wavefront sets

Radially homogeneous wavefront sets

Pseudodiﬀerential calculus

Proof of main theorem

Quantization

m1
m2
Composition: For a ∈ Scyl
(T ∗ M ) and b ∈ Scyl
(T ∗ M ),Opℏ (a) Opℏ (b) = Opℏ (c) + OL2 →L2 (ℏ∞ ),[Opℏ (a), Opℏ (b)] = iℏ Opℏ (c′ ) + OL2 →L2 (ℏ∞ )for some c = ab + OS m1 +m2 −1 (T ∗ M ) (ℏ) andcylc′ = {a, b} + OS m1 +m2 −2 (T ∗ M ) (ℏ) withcyl

supp c, supp c′ ⊂ supp(ab) modulo O(ℏ∞ ).

Shota Fukushima (D3)

Graduate School of Mathematical Science, the University of Tokyo

Propagation of singularities for Schrödinger equations on manifolds with ends

Homogeneous wavefront sets

Radially homogeneous wavefront sets

Pseudodiﬀerential calculus

Proof of main theorem

Remark: ΨDOs acting on half-densities
RemarkOn Euclidean spaces.
aw (x, ℏD)(ũ|dx|1/2 ) := (aw (x, ℏD)ũ)|dx|1/2 .
On curved spaces. Let g be a general metric on Rn andg(x) := det(gjk (x)). Then, noting that the natural identiﬁcationũ|g 1/2 dx|1/2 ≃ ũ, we haveaw (x, ℏD)(ũ|g 1/2 dx|1/2 ) = g −1/4 aw (x, ℏD)(ũg 1/4 )|g 1/2 dx|1/2 .

Shota Fukushima (D3)

Graduate School of Mathematical Science, the University of Tokyo

Propagation of singularities for Schrödinger equations on manifolds with ends

Homogeneous wavefront sets

Radially homogeneous wavefront sets

Pseudodiﬀerential calculus

Proof of main theorem

Construction of symbols

We follow the argument in Ito-Nakamura (Amer. J. Math., 2009).
Take χ ∈ Cc∞ (R with χ = 1 in [−1, 1] and χ = 0 outside [−2, 2].
For j = 0, 1, 2, . . ., we consider 
|r − r(t)||θ − θ(t)|ψ̃j (t, r, θ, ρ, η) :=χ
χ4δj t
δj − t−λ |ρ − ρ(t)||η − η(t)|×χχδj − t−λδj − t−λfor t ≥ T (≫ 1). δj > 0 and λ > 0 are chosen appropriately.

Shota Fukushima (D3)

Graduate School of Mathematical Science, the University of Tokyo

Propagation of singularities for Schrödinger equations on manifolds with ends

Homogeneous wavefront sets

Radially homogeneous wavefront sets

Pseudodiﬀerential calculus

Proof of main theorem

Construction of symbols
Take a function α ∈ C ∞ (R) with α = 0 in (−∞, T ) and α = 1 in
(T + 1, ∞), and we “extend” ψ̃j (t, . . .) (t ≥ T ) to ψj (t, . . .)(t ≥ 0) by the transport equation
!
∂ψj
∂ ψ̃j
+ {ψj , h0 } = α(t)+ {ψ̃j , h0 } ,∂t∂tψj (T + 1, r, θ, ρ, η) = ψ̃j (T + 1, r, θ, ρ, η).
Lemma 7−2
ψj belongs to Scyl
(T ∗ M ) and satisﬁes∂ψj+ {ψj , h0 } ≥ 0,∂t
Shota Fukushima (D3)∂ψj
+ {ψj , h0 } = OS 0 (T ∗ M ) (⟨t⟩−1 ).
cyl∂tGraduate School of Mathematical Science, the University of Tokyo

Propagation of singularities for Schrödinger equations on manifolds with ends

Homogeneous wavefront sets

Radially homogeneous wavefront sets

Pseudodiﬀerential calculus

Proof of main theorem

Construction of symbols
We deﬁne a symbol of the formã(ℏ; t, x, ξ) ∼ t∞X

cj ℏj ψj (t, x, ξ)

j=1

where cj ’s are positive constants, and considerAℏ (t) := Opℏ (ψ0 (ℏ−1 t))∗ Opℏ (ψ0 (ℏ−1 t)) + Opℏ (ã(ℏ; ℏ−1 t)).
we set∗Fk (t) := Opℏ (ψ0 (t)) Opℏ (ψ0 (t)) + tkX

cj ℏj ψj (t, x, ξ).

j=1

Shota Fukushima (D3)

Graduate School of Mathematical Science, the University of Tokyo

Propagation of singularities for Schrödinger equations on manifolds with ends

Homogeneous wavefront sets

Radially homogeneous wavefront sets

Pseudodiﬀerential calculus

Proof of main theorem

Heisenberg derivatives
Note that H = ℏ−2 (Opℏ (h0 ) + ℏ2 V + ℏ2 Vg /2) for some0 (T ∗ M ). We apply the sharp Gårding inequality forVg (r, θ) ∈ Scyl
F0 (t) and obtain:
Lemma 80 (T ∗ M ) such that
There exists a symbol b0 (ℏ; t, x, ξ) ∈ Scyl∂
F0 (t) − iℏ[F0 (t), H] ≥ −ℏ Opℏ (b0 (t)) + OL2 →L2 (ℏ∞ )∂twith supp b0 (t) ⊂ supp ψ0 (t) modulo O(ℏ∞ ).

Shota Fukushima (D3)

Graduate School of Mathematical Science, the University of Tokyo

Propagation of singularities for Schrödinger equations on manifolds with ends

Homogeneous wavefront sets

Radially homogeneous wavefront sets

Pseudodiﬀerential calculus

Proof of main theorem

Heisenberg derivative
If we take c1 ≫ 1, F1 (t) satisﬁes∂
F1 (t) − iℏ[F1 (t), H]∂t≥ −ℏ Opℏ (b0 (t)) + c1 ℏ Opℏ (ψ1 (t))|{z}
0 (T ∗ M )≥−ℏ2 Opℏ Scyl∂Opℏ (ψ1 (t)) − iℏ[Opℏ (ψ1 (t)), H] +OL2 →L2 (ℏ∞ )
∂t
|
{z
}


+ c1 tℏ0 (T ∗ M )≥−ℏ⟨t⟩−1 Opℏ Scyl≥ −ℏ2 Opℏ (b1 (t)) + OL2 →L2 (ℏ∞ )0 (T ∗ M ) with supp b (t) ⊂ supp ψ (t)for some b1 (ℏ; t) ∈ Scyl
1
1∞
modulo O(ℏ ).
Shota Fukushima (D3)

Graduate School of Mathematical Science, the University of Tokyo

Propagation of singularities for Schrödinger equations on manifolds with ends

Homogeneous wavefront sets

Radially homogeneous wavefront sets

Pseudodiﬀerential calculus

Proof of main theorem

Heisenberg derivative

We repeat the same procedure and obtain∂Fk (t) − iℏ[Fk (t), H] ≥ OL2 →L2 (ℏk+1 ).
∂t
Noting that Aℏ (t) − Fk (ℏ−1 t) = OL2 →L2 (ℏk ), we obtain∂Aℏ (t) − i[Aℏ (t), H] ≥ OL2 →L2 (ℏk )∂tfor any k ≥ 0.

Shota Fukushima (D3)

Graduate School of Mathematical Science, the University of Tokyo

Propagation of singularities for Schrödinger equations on manifolds with ends

Homogeneous wavefront sets

Radially homogeneous wavefront sets

Pseudodiﬀerential calculus

Proof of main theorem

Heisenberg derivative
We have⟨Aℏ (0)u, u⟩= Aℏ (t0 )e−it0 H u, e−it0 H uZ t0 
∂
−itH
−itH
−
Aℏ (t) − i[Aℏ (t), H] e
u, e
u dt
∂t
0
≤ Aℏ (t0 )e−it0 H u, e−it0 H u + O(ℏ∞ ).
By ⟨Aℏ (t0 )u, u⟩ = ∥ Opℏ (ψ0 (0))u∥2L2 and ψ0 (0) = 1 near(x(0), ξ(0)), we only have to proveAℏ (t0 )e−it0 H u, e−it0 H u = O(ℏ∞ ).
Shota Fukushima (D3)

Graduate School of Mathematical Science, the University of Tokyo

Propagation of singularities for Schrödinger equations on manifolds with ends

Homogeneous wavefront sets

Radially homogeneous wavefront sets

Pseudodiﬀerential calculus

Proof of main theorem

Estimate of expectation value
Roughly speaking, this estimate holds sinceAℏ (t0 ) = Opℏ (ψ0 (ℏ−1 t0 ))∗ Opℏ (ψ0 (ℏ−1 t0 )) + · · · and |ℏr − ℏr(ℏ−1 t0 )||θ − θ(ℏ−1 t0 )|ψj (ℏ−1 t0 ) = χχ4δj t0δj − (ℏ−1 t0 )−λ

 

−1
|ρ − ρ(ℏ t0 )||η − η(ℏ−1 t0 )|×χχ
δj − (ℏ−1 t0 )−λδj − (ℏ−1 t0 )−λ |ℏr − ρ∞ t0 ||θ − θ∞ |≈χχ
4δj t0δj |ρ − ρ∞ ||η − η∞ |
×χ
χ
δj
δj
for 0 < ℏ ≪ 1.
Shota Fukushima (D3)

Graduate School of Mathematical Science, the University of Tokyo

Propagation of singularities for Schrödinger equations on manifolds with ends