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

## byL.A. Sakhnovich

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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) <_20_o.20_28_0.329_20_the20_first20_results20_connected20_with20_equation20_28_0.329_20_were20_obtained20_by20_d.20_bernoulli20_and20_l.20_euler.20_the20_investigation20_of20_this20_equation20_and20_of20_its20_various20_generalizations20_continues20_to20_be20_a20_very20_active20_field20_28_see2c_20_e.g.2c_20_5b_185d_2c_20_5b_195d_29_.20_the20_spectral20_theory20_of20_the20_equation20_28_0.329_20_has20_also20_found20_important20_applications20_in20_probability20_theory20_5b_205d_.20_example20_3.20_dirac-type20_systems20_of20_the20_form20_28_0.429_20_7d_20_where20_a28_x29_20_3d_20_a28_x29_2c_20_b28_x29_20_3d_20_b28_x29_2c_20_are20_also20_well20_studied.20_among20_the20_works20_devoted20_to20_the20_spectral20_theory20_of20_the20_system20_28_0.429_20_the20_well-known20_article20_of20_m.20_g.20_krein20_5b_485d_20_deserves20_special20_mention. o.="" _28_0.329_="" the="" first="" results="" connected="" with="" equation="" were="" obtained="" by="" d.="" bernoulli="" and="" l.="" euler.="" investigation="" of="" this="" its="" various="" generalizations="" continues="" to="" be="" a="" very="" active="" field="" _28_see2c_="" _e.g.2c_="" _5b_185d_2c_="" _5b_195d_29_.="" spectral="" theory="" has="" also="" found="" important="" applications="" in="" probability="" _5b_205d_.="" example="" 3.="" dirac-type="" systems="" form="" _28_0.429_="" _7d_="" where="" _a28_x29_="a(x)%2c" _b28_x29_="b(x)%2c" are="" well="" studied.="" among="" works="" devoted="" system="" well-known="" article="" m.="" g.="" krein="" _5b_485d_="" deserves="" special="">
Title:Spectral Theory of Canonical Differential Systems. Method of Operator IdentitiesFormat:PaperbackDimensions:202 pagesPublished:October 4, 2012Publisher:Springer-Verlag/Sci-Tech/TradeLanguage:English

The following ISBNs are associated with this title:

ISBN - 10:3034897391

ISBN - 13:9783034897396