Perturbation of translation invariant positivity preserving semigroups on
Perturbation of translation invariant positivity preserving semigroups on
复制标题
平移不变正性保留半群的扰动
DOI：
10.1090/s0002-9947-1978-0470750-2
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发表时间：
1978
影响因子：
1.3
通讯作者：
A. Sloan
中科院分区：
文献类型：
--
作者：
I. Herbst;A. Sloan
建立了L2（R ', d ' x）上平移不变保正半群的奇异局部摄动理论。证明了一个强大的近似定理，它允许处理一类非常一般的奇异摄动。给出了半群e~ H核的局部奇异性的估计。导出了特征函数展开式。讨论了本征函数和广义本征函数的局部奇异性。用涉及-A奇异摄动的例子说明了结果。我的介绍。在L2（RN, dNx）上生成保正平移不变半群的算子//0和势V的和是本研究的主题。在§11中更充分地讨论了这类H0。这里我们只注意到量子力学中与非相对能动能量和相对能动能量相对应的算符也包括在内。一般来说，所考虑的势太过奇异，不能作为运算符，而以形式给出，因此H0 + V必须定义为形式和。在§111和§IV中给出了势的详细描述。用于研究算子和的摄动程序的成功令人印象深刻（20）。对于形式和，这样的分析比较困难，因为形式的函数通常是未定义的。分析H0 + V的函数/的一种方法是：(1)近似Kby有界函数V '；(2)显示f（H0 + V”）近似于/(//0 + V)；(3)通过(2)直接分析f（H0 + V”）来分析f（H0 + V）。这种(1)和(2)的程序是由Kato（20）和Faris（10）开发的，使用/(x) = (x + x) -1。它采用单调收敛参数，因此只有当势V可以写成一个相当一般的非负函数V+和一个非正函数V_（它是H0的一个小形式扰动）的和时才适用。截断V+和K_得到函数V+ n和V_ '它们绝对被整数n限定，然后，对于所有
The theory of singular local perturbations of translation invariant positivity preserving semigroups on L2(R", d"x) is developed. A powerful approximation theorem is proved which allows the treatment of a very general class of singular perturbations. Estimates on the local singularities of the kernels of the semigroups, e~'H, are given. Eigenfunction expansions are derived. The local singularities of the eigenfunction and generalized eigenfunctions are discussed. Results are illustrated with examples involving singular perturbations of —A. I. Introduction. The sum of an operator, //0, which generates a positivity preserving translation invariant semigroup on L2(RN, dNx) and a potential V is the subject of the present work. In §11 the class of such H0's is discussed more fully. Here we only remark that the operators corresponding to nonrel- ativistic and relativistic energy in quantum mechanics are included. The potentials considered are, in general, too singular to be operators and are given as forms, so that H0 + V must be defined as a form sum. A detailed description of the potentials is given in §111 and IV. The success of the perturbation program for the investigation of operator sums is impressive (20). For form sums such an analysis is more difficult because functions of forms are generally undefined. One technique for analyzing functions,/, of H0 + V is to: (1) approximate Kby bounded functions V"; (2) show f(H0 + V") approximates/(//0 + V); and (3) analyze f(H0 + V) by (2) and a direct analysis of f(H0 + V"). Such a procedure for (1) and (2) was developed by Kato (20) and Faris (10), using/(x) = (x + X)-1. It employs monotone convergence arguments and so is applicable only when the potential V can be written as the sum of a rather general nonnegative function V+ and a nonpositive function V_ which is a small form perturbation of H0. One truncates V+ and K_ to obtain functions V+ n and V_ " which are absolutely bounded by the integer n. Then, for all