Semiclassical resonances generated bycrossings of classical trajectories
Himeji Conference on Partial Differential Equations
2021. 3. 4.
Kenta Higuchi, Ritsumeikan University D3 (from April)0. Aim
Consider a coupled 1D semiclassical Schrödinger operator

Q. Are there any resonances generated by crossings of two nontrapping classical trajectories?

A. -Yes!

We will give an example.

Contents
1. Background:-Non-trapping condition and resonance free domain
2. Result:-Existence and asymptotics-Related work
3. Method:-Microlocal formulae near crossings (Fujiié-Martinez-Watanabe: 2021)-Microlocal Cauchy problem (Bony-Fujiié-Ramond-Zerzeri: 2019)1. Backgrounds 1-1. non-trapping condition: the classicalHamiltonian associated to the (scalar)
Schrödinger operator: non-trapping energy foras

iffor any
from: the Hamiltonian flow starts
at the time

An example of a potential 𝑉 𝑥and energies 𝐸 , 𝐸 , 𝐸 . Only 𝐸 is
non-trapping1-2. resonance free domain for scalar operators(A1)

can be extended to a bounded holomorphic function on
for some: the set of resonances of, that is, eigenvalues of.

Theorem (Helffer-Sjöstrand 1986)(A1) and is a non-trapping energy for2. Result 2-1. assumptions
(A2)

±→±

satisfying.

(singleton) for each(A3)
only at.
(A4)
extended holomorphically to, where(: non-trapping forcan beThis behaves like aperiodic trajectory!

)

2-2. Existence and asymptotics of resonances: “periodic trajectory” with period: action of .
/Theorem (H 2021)
Under (A1-4),
Moreover,
Remark For,

forin.

ℂ𝑆 𝐸 = ℎ 2𝑛 + 1 𝜋 − arg 𝑊 − 𝑖ℎlog

1𝜋𝑊+ logℎ(𝑉 0 − 𝑉 0 ) 𝐸.

2-3. remarks
1. The assumptions can be relaxed as follows: holomorphic ononly aton
at finitely many points (andfor some0.
there).

2. This theorem optimizes the previous result (H 2020):
Under (A1-4),
Theorem (H 2021)
Under (A1-4),
Moreover,forforin.

Quantization Conditionℂ/2-4. related work Resonances of coupled Schrödinger operators:
Martinez (1994), Nakamura (1995), Baklouti (1998), Ashida (2018), etc.

 Resonances in the presence of crossings of classical trajectories:
Fujiié-Martinez-Watanabe (2016,2017,2021).

is trapping for[Resonance of, non-trapping for] [Eigenvalue of(Fujiié-Martinez-Watanabe 2021)][in terms of].

3. Method 3-1. outline
We prove the theorem in 4 steps.
1. Apply the microlocal connection formulae due to Fujiié-Martinez-Watanabe(2021) to the microlocal Cauchy problem along .
2. A contradiction argument on monodromy shows a resolvent estimatefor.
3. Let. We obtainwithanalytically depending onon

4. Since the RHS
has a unique simple pole in, so does(by the residue theorem), that is, a resonance of.
Note Steps 2-4 are applications of the one of Bony-Fujiié-Ramond-Zerzeri(2019).

3-2. microlocal terminologies
1. For

2. For3.

, defines.t.

on

, say

by

microlocally nearif

microlocally on

4. On each connected component of,
the space of microlocal solutions is spanned by a WKB-type solution.

3-3. microlocal Cauchy problem (1)Put
where(0, 𝐸 ) = 𝛾 0

Onthe space of microlocal solutions

is spanned by a WKB-type solution(𝑥 , 0) = 𝛾 𝑡 + 𝑡 /2( )∫( , ),( , )/(𝑥 , 0) = 𝛾 𝑡 /2(0, − 𝐸 ) = 𝛾 𝑡//

3-3. microlocal Cauchy problem (2)Takeand put
Continue microlocal solution near(negative)near

Lemma (positive direction)
near
near

𝜌 =𝛾 𝑎(0, − 𝐸 ) = 𝛾 𝑡,

in two directions (positive and negative).
near
near .

near,nearwith(0, 𝐸 ) = 𝛾 0(𝑥 , 0) = 𝛾 𝑡 + 𝑡 /2

to,/Recall/.

Continue given microlocal solution by𝜌 =𝛾 𝑏
• Along each path: propagation of singularities• Turning points □: Maslov’s theory(𝑥 , 0) = 𝛾 𝑡 /2
• Crossing points 〇: reduction to a normalform (Fujiié-Martinez-Watanabe: 2021)

3-4. Resolvent estimate
Proposition
on ℂIn particular, there is no resonances.

Proof by a contradiction argument
1. Suppose
such thatas,.
2. By Lemma (microlocal Cauchy problem), we have
3. Then,.

,on

implies(contradiction!).
Remark We apply the localization theorems due to Bony-FujiiéRamond-Zerzeri (2019) in the step 3.

.

3-5. Inhomogeneous Cauchy problem
Proposition 2
where : WKB function (given in 3-3),∗
supported near1.

2. PutThen
3.

near.

satisfies

for,,,

𝜒=1
𝜓=1,.

, wherenear (by Lemma).
near withshows Proposition 2.

supp 𝜒supp 𝜓References
1.
2.
3.
4.
5.
6.
7.
8.
9.

S. Ashida; Molecular predissociation resonances below an energy level crossing, Asymptot. Anal.
107(3-4) (2018), 135-167.
H. Baklouti; Asymptotique des largeurs de résonances pour un modèle d’effet tunnel microlocal,
Ann. Inst. H. Poincaré Phys. Theor. 68(2) (1998) 179-228.
J.-F. Bony, S. Fujiié, T. Ramond, M. Zerzeri; Resonances for homoclinic trapped sets, Astérisque 405,
Société Mathématique de France, 2018.
S. Fujiié, A. Martinez, T. Watanabe; Widths of resonances above an energy-level crossing, J. Funct.
Anal. 280 (2021), no. 6.
B. Helffer, J. Sjöstrand; Résonances en limite semi-classique, Mémoires de la Société Mathématiquede France 24-25.
A. Martinez; Estimates on complex interactions in phase space, Math. Nachr. 167 (1994) 203-254.
S. Nakamura; On an example of phase-space tunneling, Ann. Inst. H. Poincaré Phys. Theor. 63(2)(1995) 211-229.
K. Higuchi; Resonance free domain for a system of Schrödinger operators with energy-levelcrossings, Rev. Math. Phys. (2021),
https://www.worldscientific.com/doi/pdf/10.1142/S0129055X21500070.
K. Higuchi; Résonances semiclassiques engendrées par des croisements de trajectoires classiques,Preprint.