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- W73929230 abstract "R esonances are most readily associated with musical instruments or with the Tacoma bridge disaster. The latter is described in many physics and ODE books, and at the Ontario Science Center one can even find a model allowing one to find the destructive resonant frequency. The resonances I would like to write about are closely related but have their origins in quantum or electromagnetic scattering. To introduce them in a rough way, let us first recall the notion of eigenvalues. Eigenvalues of self-adjoint operators describe, among other things, the energies of bound states, states that exist forever if unperturbed. These do exist in real life; for instance, we can tell the composition of stars from our knowledge of atomic spectra. In most situations, however, states do not exist forever, and a more accurate model is given by a decaying state that oscillates at some rate. The decay might be caused by damping or by a possibility of escape to infinity. To describe these more realistic states, we use resonances. They have a very long tradition in mathematical physics, but they also appear naturally in pure mathematics. The last ten years brought many new ideas and new results into the subject. Old problems concerning the proximity of resonances to the real axis, their relation to quasi-modes, and their distribution for scattering by convex bodies have been solved. Upper bounds for counting functions of resonances have become well understood, and the new area of lower bounds has become active. New directions were opened by considering resonances in geometry, where in fact they make a very natural appearance. The purpose of this article is to motivate the study of resonances and to survey the recent advances. The focus of the presentation is very personal, and many aspects of the enormous subject of resonances are bound to be neglected. Allusion to some other aspects of the subject is made at various places in the text, and some references, or pointers to lists of references, are included." @default.
- W73929230 created "2016-06-24" @default.
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- W73929230 date "1999-01-01" @default.
- W73929230 modified "2023-09-26" @default.
- W73929230 title "RESONANCES IN PHYSICS AND GEOMETRY" @default.
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