467 research outputs found
A note on low energy scattering for homogeneous long range potentials
We explicitly calculate the scattering matrix at energy zero for attractive,
radial and homogeneous long-range potentials. This proves a conjecture by
Derezinski and Skibsted.Comment: 3 page
Singular solutions to a semilinear biharmonic equation with a general critical nonlinearity
We consider positive solutions of the semilinear biharmonic equation
in with non-removable singularities at the origin. Under natural
assumptions on the nonlinearity , we show that is a
periodic function of and we classify all such solutions.Comment: To V. Maz'ya on the occasion of his 80th birthday; references adde
Two-term spectral asymptotics for the Dirichlet Laplacian on a bounded domain
Let -\Delta denote the Dirichlet Laplace operator on a bounded open set in
\mathbb{R}^d. We study the sum of the negative eigenvalues of the operator -h^2
\Delta - 1 in the semiclassical limit h \to 0+. We give a new proof that yields
not only the first term of the asymptotic formula but also the second term
involving the surface area of the boundary of the set. The proof is valid under
weak smoothness assumptions on the boundary.Comment: 10 pages; dedicated to Ari Laptev on the occasion of his 60th
birthda
Critical Lieb-Thirring bounds in gaps and the generalized Nevai conjecture for finite gap Jacobi matrices
We prove bounds of the form ∑_(e∈I⋂σ_d(H)) dist(e, σ_e(H)^(1/2) ≤ L^1 -norm of a perturbation, where I is a gap. Included are gaps in continuum one-dimensional periodic Schrödinger operators and finite gap Jacobi matrices, where we get a generalized Nevai conjecture about an L^(1)-condition implying a Szegő condition. One key is a general new form of the Birman-Schwinger bound in gaps
Spectral cluster bounds for orthonormal systems and oscillatory integral operators in Schatten spaces
We generalize the spectral cluster bounds of Sogge for the
Laplace-Beltrami operator on compact Riemannian manifolds to systems of
orthonormal functions. The optimality of these new bounds is also discussed.
These spectral cluster bounds follow from Schatten-type bounds on oscillatory
integral operators.Comment: 30 page
- …