Monotonicity and bounds on Bessel functions.

I survey my recent results on monotonicity with respect to order of general Bessel functions, which follow from a new identity and lead to best possible uniform bounds. Application may be made to the ‘spreading of the wave packet’ for a free quantum particle on a lattice and to estimates for perturbative expansions. On my arrival as a graduate student at Berkeley in September 1964, I was amused to see a Volkswagen Beetle with Schrödinger’s equation written on it drive past. (I don’t recall if it was the time-dependent or time-independent equation.) As I stood in line to enroll, a table off to the side with a ‘Free Speech’ banner caught my eye. Soon were to begin the student demonstrations which culminated in Vietnam war protests. I managed to complete the typing of my thesis in 1969 even as tear gas wafted in through the open window. I had asked Eyvind Wichmann if he would supervise my Ph.D. studies, and after checking that Emilio Segrè had given a good report on my oral examination, he agreed to take me on. I’d like to thank Eyvind for helping to make my stay at Berkeley a successful one.