Regular singular asymptotics

✍ Scribed by Jochen Brüning; Robert Seeley

• Publisher: Elsevier Science
• Year: 1985
• Tongue: English
• Weight: 594 KB
• Volume: 58
• Category: Article
• ISSN: 0001-8708
• DOI: 10.1016/0001-8708(85)90114-8

✦ Synopsis

Various asymptotic expansions reduce to an integral of the form J m u(x, xz) dx, ZjKJ 0 where 0(x, c) has an expansion as c -+ ~II in terms of the functions CVn Urn, a complex, m integer 20. For instance, the singular differential operator which is the Friedrichs extension in L*(O, XJ) of A = 4:+x-32, w!l where u is constant > -$ has resolvent with a symmetric kernel involving Bessel functions, ~~YY'*Lw ~"~Y~~~ .xG y, v = (u + ;y*. Callias [3] treats this for u > 3, in which case A is essentially self-adjoint. The resolvent does not have finite trace, but it does have a "distributional trace" given by the integral . for q5 in the Schwartz space Y(R'). This is an example of (O.l), and the function cr(x, c) = x4(x) Z"(c) J&(c) has an expansion in terms of [-I, [-*,... as [ + cc in any open sector iarg ci c 7r/2 -s. This example is developed in Section 3 below, and Section 4 generalizes it to the case where the term u in (0.2) is a Cw function with u(O) > -i and lu(x)j < C(1 +x). This is an 133 oool-8708/85 s7.50