Schrödinger-type semigroups intertwined by Weyl pairs on abstract Wiener spaces

Abstract

It is proven that Schrödinger-type problem \(w'_t=\text {i}\mathfrak {A} w\), \(w(0)=f\), \({(t>0)}\) in the Gaussian Hilbert space \(L^2_\mathbb {C}(X,\mathcal {B},\gamma )\) has the unique solution \({e}^{\text {i}t\mathfrak {A}}f=\frac{1}{\sqrt{4\pi t}}{\mathop {\mathbb {E}}}_xe ^{-\frac{1}{4t}\Vert x\Vert _X^2}\mathcal {W}_{\text {i}x}f\), where the semigroup \({e}^{\text {i}t\mathfrak {A}}\) is irreducible-intertwined via Weyl pairs \(\left\{ \mathcal {W}_{\text {i}x}:x\in X\right\} \) with the shift and multiplication coordinate groups on the space \(\mathcal {H}^2_\mathbb {C}\) of Hilbert-Schmidt analytic functionals on \({H\oplus \text {i}H}\). The expectation \({\mathop {\mathbb {E}}}f={\int f\,d\gamma }\) is defined by Gaussian measure \(\gamma \) on a real separable Banach space X, using Gross’s theory of an abstract Wiener space \(\jmath :H\looparrowright X\) with the reproducing Hilbert space H. It is established the explicit formula for Hamiltonian \(\mathfrak {A}\) in the form of a closure of sums \({\sum [\mathfrak {h}_2(\phi _j)+\mathbb {1}_j]}\) with the 2nd-degree Hermite polynomial \(\mathfrak {h}_2\) from Gaussian variables \(\phi _j\) and number operators \(\mathbb {1}_j\) generated by the basis \((\mathfrak {e}_j)\subset H\) in the probability space \((X,\mathcal {B},\gamma )\) with Borel’s field \(\mathcal {B}\) created by \(\jmath \). The Jackson inequalities with explicit constants for best approximations of \(\mathfrak {A}\) are established.