Expand all Collapse all | Results 1 - 10 of 10 |
1. CMB 2013 (vol 56 pp. 827)
Erratum to ``Quantum Limits of Eisenstein Series and Scattering States'' This paper provides an erratum to Y. N. Petridis,
N. Raulf, and M. S. Risager, ``Quantum Limits
of Eisenstein Series and Scattering States.'' Canad. Math. Bull., published
online 2012-02-03, http://dx.doi.org/10.4153/CMB-2011-200-2.
Keywords:quantum limits, Eisenstein series, scattering poles Categories:11F72, 8G25, 35P25 |
2. CMB 2012 (vol 56 pp. 814)
Quantum Limits of Eisenstein Series and Scattering States We identify the quantum limits of scattering states
for the modular surface. This is obtained through the study of quantum
measures of non-holomorphic Eisenstein series away from the critical
line. We provide a range of stability for the quantum unique
ergodicity theorem of Luo and Sarnak.
Keywords:quantum limits, Eisenstein series, scattering poles Categories:11F72, 58G25, 35P25 |
3. CMB 2011 (vol 56 pp. 3)
Semiclassical Limits of Eigenfunctions on Flat $n$-Dimensional Tori We provide a proof of a conjecture by Jakobson, Nadirashvili, and
Toth stating
that on an $n$-dimensional flat torus $\mathbb T^{n}$, and the Fourier transform
of squares of the eigenfunctions $|\varphi_\lambda|^2$ of the Laplacian have
uniform $l^n$ bounds that do not depend on the eigenvalue $\lambda$. The proof
is a generalization of an argument by Jakobson, et al. for the
lower dimensional cases. These results imply uniform bounds for semiclassical
limits on $\mathbb T^{n+2}$. We also prove a geometric lemma that bounds the number of
codimension-one simplices satisfying a certain restriction on an
$n$-dimensional sphere $S^n(\lambda)$ of radius $\sqrt{\lambda}$, and we use it in
the proof.
Keywords:semiclassical limits, eigenfunctions of Laplacian on a torus, quantum limits Categories:58G25, 81Q50, 35P20, 42B05 |
4. CMB 2011 (vol 55 pp. 88)
Inequalities for Eigenvalues of a General Clamped Plate Problem
Let $D$ be a
connected bounded domain in $\mathbb{R}^n$. Let
$0<\mu_1\leq\mu_2\leq\dots\leq\mu_k\leq\cdots$ be the eigenvalues
of the following Dirichlet
problem:
$$
\begin{cases}\Delta^2u(x)+V(x)u(x)=\mu\rho(x)u(x),\quad x\in
D
u|_{\partial D}=\frac{\partial u}{\partial n}|_{\partial
D}=0,
\end{cases}
$$
where $V(x)$ is a nonnegative potential,
and $\rho(x)\in C(\bar{D})$ is positive.
We prove the following inequalities:
$$\mu_{k+1}\leq\frac{1}{k}\sum_{i=1}^k\mu_i+\Bigl[\frac{8(n+2)}{n^2}\Bigl(\frac{\rho_{\max}}
{\rho_{\min}}\Bigr)^2\Bigr]^{1/2}\times
\frac{1}{k}\sum_{i=1}^k[\mu_i(\mu_{k+1}-\mu_i)]^{1/2},
$$
$$\frac{n^2k^2}{8(n+2)}\leq
\Bigl(\frac{\rho_{\max}}{\rho_{\min}}\Bigr)^2\Bigl[\sum_{i=1}^k\frac{\mu_i^{1/2}}{\mu_{k+1}-\mu_i}\Bigr]
\times\sum_{i=1}^k\mu_i^{1/2}.
$$
Keywords:biharmonic operator, eigenvalue, eigenvector, inequality Category:35P15 |
5. CMB 2010 (vol 53 pp. 674)
Multiple Solutions for a Class of Neumann Elliptic Problems on Compact Riemannian Manifolds with Boundary |
Multiple Solutions for a Class of Neumann Elliptic Problems on Compact Riemannian Manifolds with Boundary
We study a semilinear elliptic problem on a compact Riemannian
manifold with boundary, subject to an inhomogeneous Neumann
boundary condition. Under various hypotheses on the nonlinear
terms, depending on their behaviour in the origin and infinity, we
prove multiplicity of solutions by using variational arguments.
Keywords:Riemannian manifold with boundary, Neumann problem, sublinearity at infinity, multiple solutions Categories:58J05, 35P30 |
6. CMB 2008 (vol 51 pp. 249)
On the Inner Radius of a Nodal Domain Let $M$ be a closed Riemannian manifold.
We consider the inner radius of a nodal domain for a large eigenvalue $\lambda$.
We give upper and lower bounds on the inner radius of the type
$C/\lambda^\alpha(\log\lambda)^\beta$. Our proof is based on
a local behavior of eigenfunctions discovered by Donnelly and
Fefferman and a Poincar\'{e} type inequality proved by Maz'ya.
Sharp lower bounds are known
only in dimension two. We give an account of this case too.
Categories:58J50, 35P15, 35P20 |
7. CMB 2006 (vol 49 pp. 358)
On the Principal Eigencurve of the $p$-Laplacian: Stability Phenomena We show that each point of the principal eigencurve of the
nonlinear problem
$$
-\Delta_{p}u-\lambda m(x)|u|^{p-2}u=\mu|u|^{p-2}u \quad
\text{in } \Omega,
$$
is stable (continuous) with respect to the exponent $p$ varying in
$(1,\infty)$; we also prove some convergence results
of the principal eigenfunctions corresponding.
Keywords:$p$-Laplacian with indefinite weight, principal eigencurve, principal eigenvalue, principal eigenfunction, stability Categories:35P30, 35P60, 35J70 |
8. CMB 2006 (vol 49 pp. 226)
The Spectrum and Isometric Embeddings of Surfaces of Revolution A sharp upper bound on the first $S^{1}$ invariant eigenvalue of the Laplacian
for $S^1$ invariant metrics on $S^2$ is used to find obstructions to the existence
of $S^1$ equivariant isometric embeddings of such metrics in $(\R^3,\can)$. As a
corollary we prove: If the first four distinct eigenvalues have even multiplicities
then the metric cannot be equivariantly, isometrically embedded in $(\R^3,\can)$. This
leads to generalizations of some classical results in the theory of surfaces.
Categories:58J50, 58J53, 53C20, 35P15 |
9. CMB 2004 (vol 47 pp. 407)
Multiplicity of Resonances in Black Box Scattering We apply the method of complex scaling to give a natural
proof of a formula relating the multiplicity of a resonance to the
multiplicity of a pole of the scattering matrix.
Category:35P25 |
10. CMB 2000 (vol 43 pp. 51)
Eigenfunction Decay For the Neumann Laplacian on Horn-Like Domains The growth properties at infinity for eigenfunctions corresponding to
embedded eigenvalues of the Neumann Laplacian on horn-like domains
are studied. For domains that pinch at polynomial rate, it is shown
that the eigenfunctions vanish at infinity faster than the reciprocal
of any polynomial. For a class of domains that pinch at an exponential
rate, weaker, $L^2$ bounds are proven. A corollary is that eigenvalues
can accumulate only at zero or infinity.
Keywords:Neumann Laplacian, horn-like domain, spectrum Categories:35P25, 58G25 |