PREFACE xi d1^ A for the Laplace-Beltrami operator A on a compact manifold. In §8 we use the results of §7 to obtain some theorems in spectral theory. In particular we obtain Hdrmander's asymptotic formula for the spectral function of a positive self- adjoint elliptic operator on a compact manifold, and a result of Chazarain- Duistermaat-Guillemin on the singularities of the Fourier transform of the spectral function. This result asserts, roughly speaking, that the knowledge of the eigenvalues of the Laplacian gives the lengths of the closed geodesies. To use the phraseology of Mark Kac, one can "hear" the lengths of the closed geodesies.2 (This result is connected with a physical observation of Hermholtz in his study of stringed instruments. Helmholtz was puzzled by the following problem: Why is it that a bowed string, which is a "forced vibration", should yield (approxi- mately) the same note as a plucked string, which is a "free vibration". The frequency of a forced vibration should be the same as the frequency of the forcing term (the bow). There must be some mechanism whereby the string triggers the bow to execute a forcing action at exactly the free frequency of the string.) Helmholtz examined the motion of strings using his 'vibrating micro- scope' (today we would use a stroboscope) and discovered the following result: The plucked string vibrates between the extreme positions as indicated: where the central line represents the rest position of the string. On the other hand, the instantaneous positions of the bowed string are of the form 2 The earliest reference that we were able to find in the scientific literature to this inverse problem is in a report to the British Association by Sir Arthur Schuster in 1882 on spectroscopy. Until that time, the primary function of the analysis of the spectra of atoms and molecules was to identify the chemicals in question. (Indeed the most striking results were those of Kirchoff in determining the chemistry of the sun's atmosphere by analysing the absorbtion lines of the solar spectrum.) In fact, the study of spectra was known as "spectrum analysis". In this report, Schuster suggested that the primary function of the study of spectra in the future would be to analyze the structure of atoms and molecules, and coined the name "spectroscopy" for this new science. He writes: "but we must not too soon expect the discovery of any grand and very general law, for the constitution of what we call a molecule is no doubt a very complicated one, and the difficulty of the problem is so great that were it not for the primary importance of the result which we may finally hope to obtain, all but the most sanguine might well be discouraged to engage in an inquiry which, even after many years of work, may turn out to have been fruitless. We know a great deal more about the forces which produce the vibrations of sound than about those which produce the vibrations of light. To find out the different tunes sent out by a vibrating system is a problem which may or may not be solvable in certain special cases, but it would baffle the most skillful mathematician to solve the inverse problem and to find out the shape of a bell by means of the sounds which it is capable of sending out. And this is the problem which ultimately spectroscopy hopes to solve in the case of light. In the meantime we must welcome with delight even the smallest step in the desired direction."
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