An Introduction to Semiclassical and Microlocal Analysis by Andre Martinez

By Andre Martinez

"This booklet provides lots of the concepts utilized in the microlocal therapy of semiclassical difficulties coming from quantum physics. either the normal C[superscript [infinite]] pseudodifferential calculus and the analytic microlocal research are constructed, in a context that is still deliberately international in order that merely the appropriate problems of the idea are encountered. The originality lies within the indisputable fact that the most good points of analytic microlocal research are derived from a unmarried and straight forward a priori estimate. quite a few workouts illustrate the manager result of each one bankruptcy whereas introducing the reader to extra advancements of the speculation. purposes to the research of the Schrodinger operator also are mentioned, to additional the knowledge of recent notions or basic effects by way of putting them within the context of quantum mechanics. This booklet is aimed toward nonspecialists of the topic, and the single required prerequisite is a easy wisdom of the speculation of distributions.

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Additional info for An Introduction to Semiclassical and Microlocal Analysis

Example text

I [a, b] is of interest, as the reader who has studied the mean value theorem in differential calculus knows. 10 Product spaces. 7. Given two metric spaces (X, d x) and (Y, d y) we can define several metrics on X x Y. For points (x 1 , yl) and y = (x2, Y2) in X x Y let d1 ((x1, yl), (x2, Y2)) = dx(xl, x2) + dy(y1, Y2), d2((x1, yl), (x2, Y2)) = [dx(xJ, x2) 2 +dy(Yl, Y2) 2 p, 1 doc((xl, YJ), (x2, Y2)) = max{dx(xl, x2), dy(y1, Y2)}. 16). 7 any one of these deserves to be called a product metric. We shall see in the next chapter that they are all equivalent in a certain sense.

10 Product spaces. 7. Given two metric spaces (X, d x) and (Y, d y) we can define several metrics on X x Y. For points (x 1 , yl) and y = (x2, Y2) in X x Y let d1 ((x1, yl), (x2, Y2)) = dx(xl, x2) + dy(y1, Y2), d2((x1, yl), (x2, Y2)) = [dx(xJ, x2) 2 +dy(Yl, Y2) 2 p, 1 doc((xl, YJ), (x2, Y2)) = max{dx(xl, x2), dy(y1, Y2)}. 16). 7 any one of these deserves to be called a product metric. We shall see in the next chapter that they are all equivalent in a certain sense. The definition may be extended to the product of any finite number of metric spaces.

X ~ ---t : X 0 X x X of any set X is the ---t X x X of any metric Proof As before we use the metric d 1 on X x X defined by d1((x1, x2), (x~, x~)) = dx(x,, x;) + dx (x2, Let c > 0. Put 8 = c/2. Then whenever dx(x, x') < 8 we have dt(~(x), ~(x')) = d1((x, x), (x', x')) = dx(x, x') This establishes continuity of ~. x~). + dx(x, x')

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