Modular index theory and spectral flow

模索引理论和谱流

• 批准号: 41224-2007
• 负责人: Phillips, John
• 金额: $ 1.75万
• 依托单位: University of Victoria
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2007
• 资助国家: 加拿大
• 起止时间: 2007-01-01 至 2008-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=361066
• 关键词: Modular; index; theory; spectral; flow

A "linear operator" is a mathematical object which (in many cases) can be represented by a square array (or"matrix") of numbers just like a spreadsheet. By a process called "matrix multiplication", such operators"operate" on vertical lists of numbers (called "vectors") to produce new vertical lists (new vectors). If A is the symbol for an operator and X is the symbol for a vertical list (vector), then we denote the newly transformedlist (vector) by AX. It is especially important to find particular vectors for the operator A, say Y, and particularnumbers (depending on A and Y) say 2 for example so that AY is 2 times Y, in symbols AY=2Y. In this case, we would say that 2 is in the "spectrum" of A. That is, the spectrum of A is the collection of all such numbers n for whichthere is some vector, say Z so that AZ=nZ. Of course different operators will usually have different spectrums.   Examples of such operators and their spectrums occur so often in scientific and economic applications thatstudents are routinely taught the rudiments of the subject in first year university (or even high school) in theguise of "Matrix Theory" or "Linear Algebra". At a deeper level, these operators are crucial to the understanding of quantum mechanics and all the myriad applications of that field.   I plan to continue my detailed study of how the spectrums of such operators change continuously as the operators themselves change continuously. This is the notion of "flowing spectrum" or "Spectral Flow". In particular, it is important to be able to calculate how much of the spectrum changes from negative to positive,without having to go through the hugely time-consuming process of calculating all the spectrums at each instant of the flow. My co-authors and I have been developing formulas which do this in many diverse cases.   WARNING: The above descriptions are highly over-simplified for the purposes of exposition.

“线性运算符”是一个数学对象，在许多情况下（在许多情况下）可以像电子表格一样由数字的平方数组（或“矩阵”）表示。通过称为“矩阵乘法”的过程，此类运算符在垂直列表（称为“向量”）上产生新的垂直列表（新向量）上。如果a是操作员的符号，而x是垂直列表（向量）的符号，则我们用AX表示新变换的清单（vector）。例如，为操作员A（例如y和特定的名字）找到特定的向量（取决于a和y）尤其重要，例如说2，以便y y y y y y y = 2y = 2y。在这种情况下，我们会说2位于A的“频谱”中，即A的频谱是所有此类数字n的集合，其中有一些向量，说z，以便az = nz。当然，不同的操作员通常会有不同的频谱。此类运营商及其频谱的例子经常出现在科学和经济应用中，以至于学生在“矩阵理论”或“线性代数”中通常会在学科（甚至是高中）中教授该学科的基础。在更深层次的层面上，这些操作员对于理解量子力学和该领域的所有无数应用至关重要。我计划继续详细研究此类运营商的频谱如何随着操作员本身的不断变化而不断变化。这是“流光谱”或“光谱流”的概念。特别是，重要的是要计算多少频谱从负面变为正变化，而不必经历大量耗时的过程来计算流量的每一个时刻的所有频谱。我和我的合着者一直在开发在许多潜水员情况下这样做的公式。   警告：上面的描述高度简化了，以进行说明。