On Schrödinger semigroups generated by universal Malliavin calculus

Abstract

Using Malliavin’s calculus, it is proved that the generator of the one-parameter unitary semigroup of Schrödinger type on the complex Hilbert space \(L^2_\mathbb {C}(\mathbb {R}^n,\gamma )\) equipped with the Gaussian measure \(\gamma \) on \(\mathbb {R}^n\) takes the form \(\sum _j^n(\mathfrak {h}_2(\phi _{\jmath })+1)\), where \(\mathfrak {h}_2(\phi _{\jmath })\) are second-order Hermite polynomials of independent random variables \(\phi _\jmath \), generated by an orthonormal basis in \(\mathbb {R}^n\) using the Paley-Wiener maps. The Weyl-Schrödinger unitary irreducible representation of Heisenberg matrix group \(\mathbb {H}_{2n+1}\) and the Segal-Bargmann transform are essentially used. By applying the inverse Gauss transform, it is found that this representation of \(\mathbb {H}_{2n+1}\) can be fully described by complex Weyl pairs, generated using the multiplication operator with a real Gaussian variable on \(\mathbb {R}^n\).