Estimates on Strang’s formula. (Estimations sur la formule de Strang.) (French. Abridged English version) Zbl 0827.47034

Dia, Boun Oumar; Schatzman, Michelle

C. R. Acad. Sci., Paris, Sér. I 320, No. 7, 775-779 (1995).

Summary: Let \(a\), \(b\), \(a_0\), \(b_0\) be positive functions of class \(C^\infty\) on \(\mathbb{T}^2= (\mathbb{R}/ \mathbb{Z})^2\). The operators \[ A= {\textstyle {\partial \over {\partial x}} a {\partial \over {\partial x}}}- a_0 \qquad \text{and} \qquad B= {\textstyle {\partial \over {\partial y}} b {\partial \over {\partial y}}}- b_0 \] generate holomorphic semi-groups in \(L^2 (\mathbb{T}^2)\) denoted by \(e^{tA}\) and \(e^{tB}\). We prove that there exists a constant \(c\) such that \[ |e^{t(A+ B)}- e^{t a/2} e^{tB} e^{t A/2} |_{{\mathcal L} (L^2 (\mathbb{T}^2))}\leq ct. \] This estimate implies stability for a class of extrapolations of Strang’s formula.

MSC:

47D06	One-parameter semigroups and linear evolution equations
47A60	Functional calculus for linear operators

Keywords:

holomorphic semi-groups; extrapolations of Strang’s formula