Further Development of Trotter Product Formulas with Problems on Path Integrals

在Commun.Math.Phys. 2001的两篇论文中，Ichinose与Hideo Tamura，Hiroshi Tamura和V.A.Zagrebnov证明了两个非负自伴算子之和的Trotter乘积公式的范数收敛性，并给出了最优误差界。目前的研究已经做了超越这一结果，记住它的关系路径积分和例子/薛定谔算子理论的应用，并带来了以下结果。（1）Trotter乘积公式在范数意义下的进一步发展--已知Trotter乘积公式的范数收敛性对两个自伴算子的形式和一般不成立。然而，在J. Functional Analysis 2004中，Ichinose和H. Neidhardt和V.A.扎格雷布诺夫这一条件是否可以放宽是一个悬而未决的问题。出乎意料的是，一之濑和田村秀夫也发现，在Lett。数学物理2004，正规收敛的酉积公式，以保持狄拉克和相对论薛定谔算子与非平凡标量势。（2）路径积分和Trotter积的积分核的收敛问题：藤原成功地改进了高维振荡积分的定相法的误差估计，并用它确定了费曼路径积分的半经典近似的第二项。在这方面，Trotter乘积公式被认为给出了一种时间切片近似。Trotter乘积公式的良好范数收敛性可能暗示了积分核的收敛性，正如Ichinose和Hideo Tamura所预期的那样。事实上，我们在2004年的两篇论文中证明了这一点。PDE和J. Reine Angew.数学（3）Zeno乘积公式：Ichinose在2005年证明了P. Exner的一个中间结果。

In two papers in Commun.Math.Phys. 2001, Ichinose, with Hideo Tamura, Hiroshi Tamura and V.A.Zagrebnov, proved norm convergence of Trotter product formula for the operator sum of two nonnegative selfadjoint operators with optimal error bound. The present research has been done to go beyond this result, keeping in mind its relation to path integral and examples/applications of the theory of Schrodinger operators, and brought the following results.(1)Further development of Trotter product formula in norm-It is known that the norm convergence of Trotter product formula does not hold in general for the form sum of two selfadjoint operators. However, it was proved in J. Functional Analysis 2004 with some condition by Ichinose together with H. Neidhardt and V.A. Zagrebnov. It is an open problem whether the condition may be relaxed. To be unexpected,Ichinose and Hideo Tamura also discovered, in Lett. Math. Phys. 2004, norm convergence of the unitary product formula to hold for the Dirac and the relativistic SchrOdinger operator with nontrivial scalar potentials.(2)Problems of path integral and of convergence of integral kernels of the Trotter product: Fujiwara succeeded to improve the error estimate of the stationary phase method of the oscillatory integral in large dimensions, and used it to determine the second term of the semiclassical approximation to Feynman path integral. In this connection, Trotter product formula is thought to give a kind of time-sliced approximation. The good norm-convergence of Trotter product formula might suggest convergence of the integral kernels, as Ichinose and Hideo Tamura anticipated. In fact, we proved it 2004 in two papers in Commn. PDE and J. Reine Angew. Math.(3)Zeno product formula: Ichinose proved 2005 an intermediate result with P.Exner.