The Li-Yau semiclassical lower bound for the sum of the first $N$ eigenvalues of the Dirichlet–Laplacian is extended to Dirichlet– Laplacians with constant magnetic fields. Our method involves a new diamagnetic inequality for constant magnetic fields.


The Diffusion Monte Carlo method is devoted to the computation of electronic ground-state energies of molecules. In this paper, we focus on implementations of this method which consist in exploring the configuration space with a fixed number of random walkers evolving according to a stochastic differential equation discretized in time. We allow stochastic reconfigurations of the walkers to reduce the discrepancy between the weights that they carry. On a simple one-dimensional example, we prove...

We give sufficient conditions for the discreteness of the spectrum of differential operators of the form $Au=-\mathrm{△}u+\left(\nabla F,\nabla u\right)$ in $L_{\mu }^2\left({\mathbb{R}}^n\right)$ where $d\mu \left(x\right)=e^{-F\left(x\right)}dx$ and for Schrödinger operators in $L^2\left({\mathbb{R}}^n\right)$. Our conditions are also necessary in the case of polynomial coefficients.

In [2] Kenig, Ruiz and Sogge proved$${\parallel u\parallel }_{L^{\frac{2n}{n-2}}\left({ℝ}^n\right)}\lesssim {\parallel Lu\parallel }_{L^{\frac{2n}{n+2}}\left({ℝ}^n\right)}$$provided $n\ge 3$, $u\in C_0^{\infty }\left({ℝ}^n\right)$ and $L$ is a second order operator with constant coefficients such that the second order coefficients are real and nonsingular. As a consequence of [3] we state local versions of this inequality for operators with $C^2$ coefficients. In this paper we show how to apply these local versions to the absence of embedded eigenvalues for potentials in $L^{\frac{n+1}{2}}$ and variants thereof.