Anisotropic Lavine's formula and symmetrised time delay in scattering theory

We consider, in quantum scattering theory, symmetrised time delay defined in terms of sojourn times in arbitrary spatial regions symmetric with respect to the origin. For potentials decaying more rapidly than $\mathrm{\mid }x{\mathrm{\mid }}^{-4}$ at infinity, we show the existence of symmetrised time delay, and prove that it satisfies an anisotropic version of Lavine’s formula. The importance of an anisotropic dilations-type operator is revealed in our study.