Pure and Applied Analysis Articles (Project Euclid)
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The latest articles from Pure and Applied Analysis on Project Euclid, a site for mathematics and statistics resources.enusCopyright 2019 Cornell University LibraryEuclidL@cornell.edu (Project Euclid Team)Mon, 04 Feb 2019 11:33 ESTMon, 04 Feb 2019 11:33 ESThttps://projecteuclid.org/collection/euclid/images/logo_linking_100.gifProject Euclid
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Semiclassical resolvent estimates for bounded potentials
https://projecteuclid.org/euclid.paa/1549297977
<strong>Frédéric Klopp</strong>, <strong>Martin Vogel</strong>. <p><strong>Source: </strong>Pure and Applied Analysis, Volume 1, Number 1, 125.</p><p><strong>Abstract:</strong><br/>
We study the cutoff resolvent of semiclassical Schrödinger operators on [math] with bounded compactly supported potentials [math] . We prove that for real energies [math] in a compact interval in [math] and for any smooth cutoff function [math] supported in a ball near the support of the potential [math] , for some constant [math] , one has
∥
χ
(
−
h
2
Δ
+
V
−
λ
2
)
−
1
χ
∥
L
2
→
H
1
≤
C
e
C
h
−
4
∕
3
log
1
∕
h
.
This bound shows in particular an upper bound on the imaginary parts of the resonances [math] , defined as a pole of the meromorphic continuation of the resolvent [math] as an operator [math] : any resonance [math] with real part in a compact interval away from [math] has imaginary part at most
Im
λ
≤
−
C
−
1
e
C
h
−
4
∕
3
log
1
∕
h
.
This is related to a conjecture by Landis: The principal Carleman estimate in our proof provides as well a lower bound on the decay rate of [math] solutions [math] to [math] with [math] . We show that there exists a constant [math] such that for any such [math] , for [math] sufficiently large, one has
∫
B
(
0
,
R
+
1
)
∖
B
(
0
,
R
)
¯

u
(
x
)

2
d
x
≥
M
−
1
R
−
4
∕
3
e
−
M
∥
V
∥
∞
2
∕
3
R
4
∕
3
∥
u
∥
2
2
.
</p>projecteuclid.org/euclid.paa/1549297977_20190204113302Mon, 04 Feb 2019 11:33 ESTThe quantum Sabine law for resonances in transmission problems
https://projecteuclid.org/euclid.paa/1549297978
<strong>Jeffrey Galkowski</strong>. <p><strong>Source: </strong>Pure and Applied Analysis, Volume 1, Number 1, 27100.</p><p><strong>Abstract:</strong><br/>
We prove a quantum version of the Sabine law from acoustics describing the location of resonances in transmission problems. This work extends the work of the author to a broader class of systems. Our main applications are to scattering by transparent obstacles, scattering by highly frequencydependent delta potentials, and boundary stabilized wave equations. We give a sharp characterization of the resonancefree regions in terms of dynamical quantities. In particular, we relate the imaginary part of resonances, or generalized eigenvalues, to the chord lengths and reflectivity coefficients for the ray dynamics, thus proving a quantum version of the Sabine law.
</p>projecteuclid.org/euclid.paa/1549297978_20190204113302Mon, 04 Feb 2019 11:33 ESTDispersive estimates for the wave equation on Riemannian manifolds of bounded curvature
https://projecteuclid.org/euclid.paa/1549297979
<strong>Yuanlong Chen</strong>, <strong>Hart F. Smith</strong>. <p><strong>Source: </strong>Pure and Applied Analysis, Volume 1, Number 1, 101148.</p><p><strong>Abstract:</strong><br/>
We prove spacetime dispersive estimates for solutions to the wave equation on compact Riemannian manifolds with bounded curvature tensor, where we assume that the metric tensor is of [math] regularity for some [math] , which ensures that the curvature tensor is welldefined in the weak sense. The estimates are established for the same range of Lebesgue and Sobolev exponents that hold in the case of smooth metrics. Our results are for bounded time intervals, so by finite propagation velocity they hold also on noncompact manifolds under appropriate uniform geometry conditions.
</p>projecteuclid.org/euclid.paa/1549297979_20190204113302Mon, 04 Feb 2019 11:33 EST