]> 7.5.1 Spectral properties of the system operator

7.5.1 Spectral properties of the system operator

It follows directly from Definition 2.3.3 that the system operator (1.11) is self–adjoint, i.e., 𝒜 = 𝒜. Observe that boundary conditions in (1.11) are strictly regular. To be more precise, the case 1 of Corollary 2.9.1 holds. Since 𝒜 is self–adjoint, its spectrum consists of real eigenvalues, and the corresponding normalized eigenvectors form an orthonormal basis of H. To find this basis we solve the eigenproblem for 𝒜,

𝒜q = λq,λ ,q D(𝒜),q0

which by (1.11) takes the form

aq(θ) R aq(θ) =λq(θ), 0 θ 1 q(1) =R 1q(1) q(0) =R 0q(0) (7.20)

Solving (7.20) we get the desired orthonormal basis of eigenvectors {en}n,

en = hn hn H,hn(θ) = R0 μn sin(μnθ) + cos(μnθ),0 θ 1 (7.21)

where μn are positive solutions of the equation

(μ2 + R 0R1) sin μ = (R0 R1)μ cos μ (7.22)

Moreover,

hnH2 =01h n2(θ)dθ = 1 2 + R02 2μn2 + μn2 R 02 2μn3 cos μn sin μn + R0 μn2 sin 2μ n (7.23)

The eigenvalues of 𝒜 express by the formula

λn = aμn2 R a Ra < 0 (7.24)

By Theorem 2.3.4, 𝒜 generates a linear analytic EXS semigroup {et𝒜} t0 which in the basis {en}n has the following representation

et𝒜q = n=1eλntq,e nHen (7.25)

Exercise 7.5.1. Confirm (7.24) by showing that the operator 𝒜 + RaI is nonnegative. Indeed, for q D(𝒜) we have

q,𝒜qH =01q(θ) aq(θ) R aq(θ) dθ =

= aq2(1)R 1 aq2(0)R 0 01a q(θ) 2dθ R a q2 R a qH2 .