By L.A. Sakhnovich
ISBN-10: 3034887132
ISBN-13: 9783034887137
ISBN-10: 3034897391
ISBN-13: 9783034897396
The spectral concept of standard differential operators L and of the equations (0.1) Ly= AY attached with such operators performs a huge position in a few difficulties either in physics and in arithmetic. allow us to supply a few examples of differential operators and equations, the spectral thought of that is good built. instance 1. The Sturm-Liouville operator has the shape (see [6]) 2 d y (0.2) Ly = - dx + u(x)y = Ay. 2 In quantum mechanics the Sturm-Liouville operator L is called the one-dimen sional Schrodinger operator. The behaviour of a quantum particle is defined by way of spectral features of the operator L. instance 2. The vibrations of a nonhomogeneous string are defined by means of the equa tion (see [59]) p(x) ~ o. (0.3) the 1st effects attached with equation (0.3) have been got via D. Bernoulli and L. Euler. The research of this equation and of its quite a few generalizations remains to be a really lively box (see, e.g., [18], [19]). The spectral thought of the equation (0.3) has additionally came across vital functions in chance concept [20]. instance three. Dirac-type platforms of the shape (0.4) } the place a(x) = a(x), b(x) = b(x), also are good studied. one of the works dedicated to the spectral thought of the procedure (0.4) the well known article of M. G. KreIn [48] merits distinctive mention.
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Additional info for Spectral Theory of Canonical Differential Systems. Method of Operator Identities
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JOO 7r -00 f(t) dt, x - t is unitary and the operator-valued functions L(x), F1(x), F2(x) are bounded implies that the operators S and T are bounded. A straightforward calculation shows that AS-SA = iII1 JIIi , TA-AT=if 1 Jri. 26) Denote by fh C SJ the closure of the set of all vectors of the form g(x) = Fl(X)f(x), where f(x) E SJ. Also, let SJ2 = SJ e SJ1. The subspaces SJ1 and 5)2 are invariant under the operators A, Sand T. 22) that Sf -L(x)f E SJl for all f(x) E SJ. , SJ1 is an invariant subspace of the operator S.
1. For Z D(A, B). 9) Proof. 7). 12) where 01 is a Hilbert space (dim01 ::; 00). I>k, Wk E {01,fl}, k = 1,2. 1. 2) of Chapter 1 as k = 1,2. 18) have the following solutions: k = 1,2. 20) The reflection and transmission coefficients in radiative transfer problems and the scattering diagram in diffraction problems can be expressed in terms of p(z, (). The analytic structure of p(z, () is characterized by the following theorem. 2. 16) are satisfied and that z, ( E V(A,B). 21) Proof. 17), p(z, () = [Cl(Z) - Cl(()]a2((~ ~ ~dl(Z) - dl (()]b 2() .
By the reflection principle, WB(Z) = JWA(z)J = I + iJIIi(A* - zI)-lr 1. 2), we obtain B=A*, II;=iJIIi, r;=iJq. 5). 2) of Chapter 1. 6), we obtain the relation WA(Z)JWA (() = J - i(z - ()Jq(A - ZE)-18(A* - (E)-lr1J. 8) Now assume that the vectors h j (1 ::; j ::; m ::; 00) form a basis in ~. For any natural number n ::; m and an arbitrary choice of points Zk E V(A, A*) such that Zk i= Zl, the quadratic form r r r ) _ . 9) is real. ) is real for f E j), that is, 8 = 8*. This completes the proof of the theorem.
Spectral Theory of Canonical Differential Systems. Method of Operator Identities by L.A. Sakhnovich
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