We prove that the electronic densities of atomic and molecular eigenfunctions are real analytic in$R$^{3}away from the nuclei.

We use microlocal and paradifferential techniques to obtain$L$^{8}norm bounds for spectral clusters associated with elliptic second-order operators on two-dimensional manifolds with boundary. The result leads to optimal$L$^{$q$}bounds, in the range 2⩽q⩽∞, for$L$^{2}- normalized spectral clusters on bounded domains in the plane and, more generally, for two-dimensional compact manifolds with boundary. We also establish new sharp$L$^{$q$}estimates in higher dimensions for a range of exponents q̅_{n}⩽$q$⩽∞.

We show that smooth maps are$C$^{1}-dense among$C$^{1}volume-preserving maps.

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Results are obtained on the scattering theory for the Schrödinger equation $$i\partial _t u(t,x) = - \Delta _x u(t,x) + V(t,x)u(t,x) + F(u(t,x))$$ in spaces$L$^{$r$}($R$;$L$^{$q$}($R$^{$d$})) for a certain range of$r, q$, the so-called space-time scattering. In the linear case (i.e.$F$≡)) the relation with usual configuration space scattering is established.