TY - ELEC
AU - Kaloshin, Vadim
PY - 2018
TI - Can you hear the shape of a drum and deformational spectral rigidity of planar domains?
LA - eng
M3 - Moving Image
AB - M. Kac popularized the question {\em Can you hear the shape of a drum?}
Mathematically, consider a bounded planar domain $\Omega$ and the associated
Dirichlet problem $\Delta u + \lambda^2 u = 0$ with $u|_{\partial \Omega}$ = 0. The set of
$\lambda$s such that this equation has a solution, denoted $\mathcal{L}(\Omega)$
is called the Laplace spectrum of $\Omega.$
Does Laplace spectrum determine $\Omega$? In general, the answer is negative.
Consider the billiard problem inside ?. Call the length spectrum the
closure of the set of perimeters of all periodic orbits of the billiard. Due
to deep properties of the wave trace function, generically, the Laplace
spectrum determines the length spectrum. We show that any generic axis
symmetric planar domain with is dynamically spectrally rigid, i.e. can't be
deformed without changing the length spectrum. This partially answers a
question of P. Sarnak. This is joint works with J. De Simoi, A. Figalli, and
J. De Simoi, Q. Wei.
N2 - M. Kac popularized the question {\em Can you hear the shape of a drum?}
Mathematically, consider a bounded planar domain $\Omega$ and the associated
Dirichlet problem $\Delta u + \lambda^2 u = 0$ with $u|_{\partial \Omega}$ = 0. The set of
$\lambda$s such that this equation has a solution, denoted $\mathcal{L}(\Omega)$
is called the Laplace spectrum of $\Omega.$
Does Laplace spectrum determine $\Omega$? In general, the answer is negative.
Consider the billiard problem inside ?. Call the length spectrum the
closure of the set of perimeters of all periodic orbits of the billiard. Due
to deep properties of the wave trace function, generically, the Laplace
spectrum determines the length spectrum. We show that any generic axis
symmetric planar domain with is dynamically spectrally rigid, i.e. can't be
deformed without changing the length spectrum. This partially answers a
question of P. Sarnak. This is joint works with J. De Simoi, A. Figalli, and
J. De Simoi, Q. Wei.
UR - https://open.library.ubc.ca/collections/48630/items/1.0372059
ER - End of Reference