Distribution of Resonances in Scattering by Thin Barriers
Authors: Galkowski Jeffrey 
Published: April 2019

We study high energy resonances for the operators
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="minus normal upper Delta Subscript partial-differential normal upper Omega comma delta Baseline colon equals negative normal upper Delta plus delta Subscript partial-differential normal upper Omega Baseline circled-times upper V and minus normal upper Delta Subscript partial-differential normal upper Omega comma delta Sub Superscript prime Subscript Baseline colon equals negative normal upper Delta plus delta prime Subscript partial-differential normal upper Omega circled-times upper V partial-differential Subscript nu Baseline">

−

Δ

∂
Ω
,
δ

:=
−
Δ
+

δ

∂
Ω

⊗
V

and

−

Δ

∂
Ω
,

δ
′

:=
−
Δ
+

δ

∂
Ω

′

⊗
V

∂
ν

\begin{equation*} -\Delta _{\partial \Omega ,\delta }:=-\Delta +\delta _{\partial \Omega }\otimes V\quad \text {and}\quad -\Delta _{\partial \Omega ,\delta ’}:=-\Delta +\delta _{\partial \Omega }’\otimes V\partial _\nu \end{equation*}

where

Ω
⊂

R

d

\Omega \subset \mathbb {R}^d

is strictly convex with smooth boundary,

V
:

L
2

(
∂
Ω
)
→

L
2

(
∂
Ω
)

V:L^2(\partial \Omega )\to L^2(\partial \Omega )

may depend on frequency, and

δ

∂
Ω

\delta _{\partial \Omega }

is the surface measure on

∂
Ω

\partial \Omega

. These operators are model Hamiltonians for the quantum corrals studied in Aligia and Lobos (2005), Barr, Zaletel, and Heller (2010), and Crommie et al. (1995) and for leaky quantum graphs in Exner (2008). We give a quantum version of the Sabine Law (Sabine, 1964) from the study of acoustics for both

−

Δ

∂
Ω
,
δ

-\Delta _{\partial \Omega ,\delta }

and

−

Δ

∂
Ω
,

δ
′

-\Delta _{\partial \Omega ,\delta ’}

. It characterizes the decay rates (imaginary parts of resonances) in terms of the system’s ray dynamics. In particular, the decay rates are controlled by the average reflectivity and chord length of the barrier.

For

−

Δ

∂
Ω
,
δ

-\Delta _{\partial \Omega ,\delta }

with

Ω
\Omega

smooth and strictly convex, our results improve those given for general

∂
Ω

\partial \Omega

in Galkowski and Smith (2015) and are generically optimal. Indeed, we show that for generic domains and potentials there are infinitely many resonances arbitrarily close to the resonance free region found by our theorem. In the case of

−

Δ

∂
Ω
,

δ
′

-\Delta _{\partial \Omega ,\delta ’}

, the quantum Sabine law gives the existence of a resonance free region that converges to the real axis at a fixed polynomial rate. The size of this resonance free region is optimal in the case of the unit disk in

R

2

\mathbb {R}^2

. As far as the author is aware, this is the only class of examples that is known to have resonances converging to the real axis at a fixed polynomial rate but no faster.

The proof of our theorem requires several new technical tools. We adapt intersecting Lagrangian distributions from Melrose and Uhlman (1979) to the semiclassical setting and give a description of the kernel of the free resolvent as such a distribution. We also construct a semiclassical version of the Melrose–Taylor parametrix (Melrose and Taylor, unpublished manuscript) for complex energies. We use these constructions to give a complete microlocal description of the single, double, and derivative double layer operators in the case that

∂
Ω

\partial \Omega

is smooth and strictly convex. These operators are given respectively for

x
∈
∂
Ω

x\in \partial \Omega

by
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row 1st Column upper G left-parenthesis lamda right-parenthesis f left-parenthesis x right-parenthesis 2nd Column a m p semicolon colon equals integral Underscript partial-differential normal upper Omega Endscripts upper R 0 left-parenthesis lamda right-parenthesis left-parenthesis x comma y right-parenthesis f left-parenthesis y right-parenthesis d upper S left-parenthesis y right-parenthesis comma 2nd Row 1st Column ModifyingAbove upper N With tilde left-parenthesis lamda right-parenthesis f left-parenthesis x right-parenthesis 2nd Column a m p semicolon colon equals integral Underscript partial-differential normal upper Omega Endscripts partial-differential Subscript nu Sub Subscript y Subscript Baseline upper R 0 left-parenthesis lamda right-parenthesis left-parenthesis x comma y right-parenthesis f left-parenthesis y right-parenthesis d upper S left-parenthesis y right-parenthesis 3rd Row 1st Column partial-differential Subscript nu Baseline script upper D script l left-parenthesis lamda right-parenthesis f left-parenthesis x right-parenthesis 2nd Column a m p semicolon colon equals integral Underscript partial-differential normal upper Omega Endscripts partial-differential Subscript nu Sub Subscript x Subscript Baseline partial-differential Subscript nu Sub Subscript y Subscript Baseline upper R 0 left-parenthesis lamda right-parenthesis left-parenthesis x comma y right-parenthesis f left-parenthesis y right-parenthesis d upper S left-parenthesis y right-parenthesis period EndLayout">

G
(
λ
)
f
(
x
)

a
m
p
;
:=

∫

∂
Ω

R
0

(
λ
)
(
x
,
y
)
f
(
y
)
d
S
(
y
)

,

N
~

(
λ
)
f
(
x
)

a
m
p
;
:=

∫

∂
Ω

∂

ν
y

R
0

(
λ
)
(
x
,
y
)
f
(
y
)
d
S
(
y
)

∂

ν

D

ℓ
(
λ
)
f
(
x
)

a
m
p
;
:=

∫

∂
Ω

∂

ν
x

∂

ν
y

R
0

(
λ
)
(
x
,
y
)
f
(
y
)
d
S
(
y
)

.

\begin{align*} G(\lambda )f(x)&:=\int _{\partial \Omega }R_0(\lambda )(x,y)f(y)dS(y)\,,\\ \tilde {N}(\lambda )f(x)&:=\int _{\partial \Omega }\partial _{\nu _y}R_0(\lambda )(x,y)f(y)dS(y)\\ \partial _{\nu }\mathcal {D}\ell (\lambda )f(x)&:=\int _{\partial \Omega }\partial _{\nu _x}\partial _{\nu _y}R_0(\lambda )(x,y)f(y)dS(y)\,. \end{align*}

This microlocal description allows us to prove sharp high energy estimates on

G
G

,

N
~

\tilde {N}

, and

∂

ν

D

ℓ

\partial _{\nu }\mathcal {D}\ell

when

Ω
\Omega

is smooth and strictly convex, removing the log losses from the estimates for

G
G

in Galkowski and Smith (2015) and Han and Tacy (2015) and proving a conjecture from Appendix A in Han and Tacy (2015).