We present a sparse grid/hyperbolic cross discretization for many-particle problems. It involves the tensor product of a one-particle multilevel basis. Subsequent truncation of the associated series expansion then results in a sparse grid discretization. Here, depending on the norms involved, different variants of sparse grid techniques for many-particle spaces can be derived that, in the best case, result in complexities and error estimates which are independent of the number of particles. Furthermore...

We study the spectral projection associated to a barrier-top resonance for the semiclassical Schrödinger operator. First, we prove a resolvent estimate for complex energies close to such a resonance. Using that estimate and an explicit representation of the resonant states, we show that the spectral projection has a semiclassical expansion in integer powers of $h$, and compute its leading term. We use this result to compute the residue of the scattering amplitude at such a resonance. Eventually, we...

2000 Mathematics Subject Classification: 35P20, 35J10, 35Q40.We give a complete pointwise asymptotic expansion for the Spectral Shift Function for Schrödinger operators that are perturbations of the Laplacian on Rn with slowly decaying potentials.

We study various statistics related to the eigenvalues and eigenfunctions of random Hamiltonians in the localized regime. Consider a random Hamiltonian at an energy $E$ in the localized phase. Assume the density of states function is not too flat near $E$. Restrict it to some large cube $\Lambda$. Consider now $I_{\Lambda }$, a small energy interval centered at $E$ that asymptotically contains infintely many eigenvalues when the volume of the cube $\Lambda$ grows to infinity. We prove that, with probability one in the large volume...

We discuss spectral and scattering theory of the discrete laplacian limited to a half-space. The interesting properties of such operators stem from the imposed boundary condition and are related to certain phenomena in surface physics.

What is called the “Semi-classical trace formula” is a formula expressing the smoothed density of states of the Laplace operator on a compact Riemannian manifold in terms of the periodic geodesics. Mathematical derivation of such formulas were provided in the seventies by several authors. The main goal of this paper is to state the formula and to give a self-contained proof independent of the difficult use of the global calculus of Fourier Integral Operators. This proof is close in the spirit of...

We characterize geometric properties of Banach spaces in terms of boundedness of square functions associated to general Schrödinger operators of the form ℒ = -Δ + V, where the nonnegative potential V satisfies a reverse Hölder inequality. The main idea is to sharpen the well known localization method introduced by Z. Shen. Our results can be regarded as alternative proofs of the boundedness in H¹, $L^p$ and BMO of classical ℒ-square functions.

We prove an estimate of the kind $\parallel q_1-q_2{\parallel }_{L^{\infty }}\le C\varphi (\parallel A_{q_1}-A_{q_2}{\parallel }_{R,3/2-1/2})$, where $A_{q_i}(\omega ,\theta )$, $i=1,2$ is the scattering amplitude related to the compactly supported potential $q_i\left(x\right)$ at a fixed energy level $k=$ const., $\varphi \left(t\right)=(-lnt{)}^{-\delta }$, $0\<\delta \<1$ and $\parallel ·{\parallel }_{R,3/2-1/2}$ is a suitably defined norm.

In questo articolo studiamo problemi di Dirichlet singolari, lineari e semilineari, della forma ${\left|x\right|}^2\mathrm{\Delta }u=f\left(u\right)$ in $\mathrm{\Omega }$, $u=0$ su $\partial \mathrm{\Omega }$, dove $\mathrm{\Omega }$ è un dominio in ${\mathbb{R}}^2$ e $f\left(u\right)=\lambda u$ o $f\left(u\right)=\lambda u+{\left|u\right|}^{p-2}u$ con $p>2$ (o nonlinearità più generali). In tali problemi bidimensionali emergono alcune difficoltà a causa della non validità della disuguaglianza di Hardy in ${\mathbb{R}}^2$ e a causa delle invarianze dell'equazione $-{\left|x\right|}^2\mathrm{\Delta }u=f\left(u\right)$. Pertanto opportune condizioni su $\lambda$ e $\mathrm{\Omega }$ sono necessarie al fine di garantire l'esistenza di una soluzione positiva. Per esempio, se ${\mathrm{\Gamma }}_0$ è una curva non costante...

We present a novel approach for bounding the resolvent of $$H=-\Delta +i(A·\nabla +\nabla ·A)+V=:-\Delta +L1$$ for large energies. It is shown here that there exist a large integer $m$ and a large number ${\lambda }_0$ so that relative to the usual weighted $L^2$-norm, $$\parallel {\left(L{(-\Delta +(\lambda +i0))}^{-1}\right)}^m\parallel <\frac{1}{2}2$$ for all $\lambda >{\lambda }_0$. This requires suitable decay and smoothness conditions on $A,V$. The estimate (2) is trivial when $A=0$, but difficult for large $A$ since the gradient term exactly cancels the natural decay of the free resolvent. To obtain (2), we introduce a conical decomposition of the resolvent and then sum over...