# Spectral Theory of Canonical Differential Systems. Method of Operator Identities

Part of the Operator Theory: Advances and Applications book series (OT, volume 107)

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Part of the Operator Theory: Advances and Applications book series (OT, volume 107)

The spectral theory of ordinary differential operators L and of the equations (0.1) Ly= AY connected with such operators plays an important role in a number of problems both in physics and in mathematics. Let us give some examples of differential operators and equations, the spectral theory of which is well developed. Example 1. The Sturm-Liouville operator has the form (see [6]) 2 d y (0.2) Ly = - dx + u(x)y = Ay. 2 In quantum mechanics the Sturm-Liouville operator L is known as the one-dimen sional Schrodinger operator. The behaviour of a quantum particle is described in terms of spectral characteristics of the operator L. Example 2. The vibrations of a nonhomogeneous string are described by the equa tion (see [59]) p(x) ~ o. (0.3) The first results connected with equation (0.3) were obtained by D. Bernoulli and L. Euler. The investigation of this equation and of its various generalizations continues to be a very active field (see, e.g., [18], [19]). The spectral theory of the equation (0.3) has also found important applications in probability theory [20]. Example 3. Dirac-type systems of the form (0.4) } where a(x) = a(x), b(x) = b(x), are also well studied. Among the works devoted to the spectral theory of the system (0.4) the well-known article of M. G. KreIn [48] deserves special mention.

differential operator mechanics operator quantum mechanics schrödinger equation spectral theory

- DOI https://doi.org/10.1007/978-3-0348-8713-7
- Copyright Information Birkhäuser Verlag 1999
- Publisher Name Birkhäuser, Basel
- eBook Packages Springer Book Archive
- Print ISBN 978-3-0348-9739-6
- Online ISBN 978-3-0348-8713-7
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