Error bounds for Trotter-type formulas for self-adjoint operators
Error bounds for Trotter-type formulas for self-adjoint operators
复制标题
自伴随运算符 Trotter 型公式的误差界限
DOI：
10.1007/bf01087542
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发表时间：
1993
影响因子：
0.4
通讯作者：
D. L. Rogava
中科院分区：
文献类型：
--
作者：
D. L. Rogava
2如果c xjF= A”，则我们有codim（A81）~（F）> 2。因此，是单连通的，并且rr= id，即F= F。因此，F由元素a生成，其中X a包含除数。(c)现在我们证明，如果X a包含一个约数，那么a（a）是伪反射。设X~ CX a是X中余维数为1的分支.由于k [X]是UFD，我们有X~={x EX] p~（x）= 0}。显然，p是k [X]的齐次元素，且对每一g~ k [X]，ag_ge（Pa）（*）。现在简单的论证，如（a），表明Pa~ m2，我们可以将p~包含在m的生成元系统中。假设（Pa，g2，...，gs）是m的生成器系统。从（*）和（a）可以很容易地得出阿帕=~ po，其中~ 1 ol = 1，而且，~（a）= diag（~，t，.，1）。(d)现在我们可以使用Avramov的结果，该结果断言，如果（R，ra）是局部域，FC Aut R是有限群，使得（F）C GL（ra/ra）由伪反射生成，则R和RF具有许多”好”的共同性质[3]。特别地，这是对于作为”完全相交“的性质的情况。“因此，定理2被证明。[]4.定理1的证明。若~ DG是一个fce，则定理2可应用于X= V//G和F=~/G。在这种情况下，X/。F~-V//~，X是阶乘，是关于连通半单群作用的商，代数k [X]= k [V] G具有一个源于多项式代数k [IV]中自然分次的分次. []
2IF c xjF= A" and we have codim (A 8\)~/F)> _ 2. Hence, is simply-connected and rr= id, ie, F= F. Thus, F is generated by the elements a for which X a contains a divisor.(c) Now we prove that if X a contains a divisor, then a (a) is a pseudo-reflection. Let X~ CX a be a component of codimension 1 in X. Since k [X] is UFD, we have X~={x EX] p~(x)= 0}. Clearly, p, is a homogeneous element of k [X] and ag--ge (Pa)(*) for every g~ k IX]. Now simple arguments, as in (a), show that Pa~ m2 and we can include p~ in a system of generators of m. Suppose that (Pa, g2,..., gs) is a generator system of m. It easily follows from (*) and (a) that apa=~ po, where~ 1ol= 1, and, moreover,~(a)= diag (~, t,..., 1).(d) Now we can use the result of Avramov which asserts that if (R, ra) is a local domain and FC Aut R is a finite group such that a (F) C GL (ra/ra) is generated by pseudoreflections, then R and RF have many" nice" common properties [3]. In particular, this is the case for the property of being a" complete intersection." Thus, Theorem 2 is proved.[]4. Proof of Theorem 1. If~ DG is an fce, we can apply Theorem 2 to X= V//G and F=~/G. In this case X/. F~-V//~, X is factorial, being a quotient with respect to the action of a connected semisimple group, and the algebra k [X]= k [V] G possesses a grading originating from the natural grading in the polynomial algebra k IV].[]