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Spectral properties of a class of quadratic operator pencils

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  • 6.


    V. N. Tulovskii and M. A. Shubin, "The asymptotic distribution of the eigenvalues of pseudodifferential operators in l~n, w Mat. Sb., 92, 571-588 (1973). V. I. Feigin, WAsymptotic distribution of eigenvalues for hypoelliptic systems in Rn," Mat. Sb. (N. S.), 9_99, No. 4, 594-614 (1976).



    A . I. M i los lavsk i i


    UDC 517.43

    A series of dynamical problems of systems carrying a moving distributed load leads to the study of stability of solutions of the equation

    ~--~-~ B~+Aw = 0, (i) dt 2 - - d t

    where w = w(t) is a function with values in a Hilbert space H. Concerning the operators in Eq. (1) we will as- sume the following.

    a) The operator A has the form A = A+ + A1, where the operator A+ is positive self-adjoint and its in- verse A+ I is completely continuous. The symmetr ic operator A 1 is subordinate to A+ in the sense that A1 = DAI+/2, where D is a bounded operator.

    b) The operator B is skew-symmetr ic and subordinate to A+ in the sense that B = CAI+/2, where C is a bounded operator.

    For Eq. (1) we consider the Cauchy problem

    d~_ t=o (2) w (0) =: wo, = wl.

    Definition 1. By a solution of the Cauehy problem (1), (2) on an interval (a, b), where -~ _< a _< 0 -< b -< ~, we mean a twice continuously differentiable function w (t) ~ D (A) such that A1/2w(t) is differentiable and w satisfies relations (1) and (2).

    THEOREM 1. For any w0 ~D (A), w, ~D (A~) there exists a unique solution of the Cauehy problem (1), (2).

    Definition 2. By the stability of the solutions of Eqs. (I), (2) on the positive semiaxis (on the axis) we understand the inequality

    I~-f~ 2+ (A+w, w) ~ C(llwlll~ + (A+wo, Wo))

    on the semiaxis (on the axis) with a constant C independent of t, w0, and wl.

    In seeking the solutions of Eq. (1) in the form w (t) = e ~t z, :: ~ H, we arr ive at the problem of eigenvalues of the quadratic operator pencil L(~):

    L (~) x = (A -q- )~ B -q- k~I) x = 0. (3)

    THEOREM 2. The spectrum g(L) of the pencil L(k) consists of isolated points of finite algebraic multi- plicity; these points either lie on the imaginary axis or are distributed symmetr ical ly with respect to this axis and are in the disk I hl -< 0.5[IDll.

    Let {kj} be the eigenvalues of L(k) among which there are no eigenvalues symmetr ic with respect to the imaginary axis. We denote by p(h) the algebraic multiplicity of a not purely imaginary eigenvalue X. If h is a

    purely imaginary eigenvalue, then by p (h) we denote p(~)=~ [di/2], where dl, d2 , . . . , d r are the nonsimple

    Ukrainian Correspondence Polytechnic Institute. Translated from Funktsional'nyi Analiz i Ego Prilo- zheniya, Vol. 15, No. 2, pp. 81-82, April?-June, 1981. Original article submitted June 2, 1980.

    142 0016-2663/81/1502- 0142 $07.50 1981 Plenum l:'ublishing Corporation

  • e lementary d iv isors cor respond ing to X.* Then we have the inequality

    where z is the number of nonpos i t ive e igenva lues of A , counted wi th multiplicity.

    The sys tem of e igenvectors and assoc ia ted vectors of L(~) fo rms a bivariate R iesz basis in H @ //, equ ipped with the norm Jl []+,

    II x II-,- ~ = It z~. II ~ + 11 A+V% I1 "% z = (Xl , x2), x i ~ H , x 2 (~ D (A~) .

    Definition 3. The pencil L(X) is said to be stable if all of its eigenvalues are purely imaginary and their algebraic multiplicity is equal to 1.

    THEOREM 3. The stability of solutions of Eqs. (1), (2) (whether on the semiaxis or on the axis) is equiv- alent to the stability of the pencil L(k).

    THEOREM 4 (Generalization of the Thompson-Tare Theorem). Suppose the following conditions are satisfied:

    I) The operator A is self-adjoint and semibounded from below; the operator A -I exists and A -I ~ op (for the definition of the ideal ~p of completely continuous operators, cf. [1]).

    2) The operator B can be written in the form B = G(A + ~I) ~, where 0 -< ~ < 1/2, G is abounded0perator and the constant ? is chosen so that A + ?I is positive.

    3) The complex Hilbert space H is the complexification of a real Hilbert space and the operators A and B are real.

    Then the pencil L(I) has an even (odd) number of positive eigenvalues, considering algebraic multiplici- ties, according as the operator A has an even (odd) number of negative eigenvalues, considering multiplicities.

    If conditions a), b), and 3) are satisfied, A is invertible and has an odd number of negative eigenvalues, then L(I) has at least one positive eigenvalue and the solutions of Eqs. (1), (2) are unstable on the positive semiax is .

    THEOREM 5. Suppose condit ions a), b), and 3) a re satisfied and A is invertible. Fur thermore , assume that the penci l L(~) has posit ive e igenva lue X 0. Then we have the inequality

    I ~ I (t + II C*A-I clJ)-~< ~ < I ~ I,

    where At(;%, j is that negative eigenvalue of A which is the smallest (largest) in absolute value and A, C = ( I - P)BP, where P is the orthogonal projection corresponding to the negative portion of the spectrum of A.

    The proofs of Theorems 2, 3, and 4 use results of [1, 4, and 5].

    We il lustrate the abstract theorems by the example of an equation derived by Feodos'ev [2]. The equa- tion of small transversal vibrations of a pipeline secured with joints, carrying the flow of an ideal incompres- sible fluid, has the form

    O~w v~ O~w ~ o*x _ a~W _ O, (4) ~ + ~ ~- ~PV a-Z~ ~- KK -

    in dimensionless variables (v and fl are positive constants). It is easy to see that Eqs. (4), (5) can be written in the form (1), where A acts according to the rule

    .4 y = y iv + v~y. (6)

    on functions u ~ tv~ [0, l] satisfying the condition

    The operator B acts accord ing to the rule

    g (0) = y(1) = u"(o) = y- 0) = 0. (7)

    BU = 2~vU'. (8)

    *We have in mind the e lementary d iv isors of the l inear operator cor respond ing to L(~) and act ing in the space H@H.


  • It is easy to verify that conditions a), b) and the hypotheses of Theorems 4 and 5 are satisfied.

    Movchan [3] has proved that for small velocities of the fluid (0 -< v < ,~) the solutions of Eqs. (4), (5) are stable and for v = ~n, n ~- 1, 2, . . . they are unstable on the positive semiaxis.

    In particular, Theorems 2, 3, and 4 imply that the spectrum of the pencil (3), (6)-(8) is symmetr ic with respect to the real and imaginary axes and for ~n < v < ~(n + 1), u = 0, 1, 2, . . . the number of eigenvalues of the pencil lying in the open right half-space does not exceed n and the not purely imaginary eigenvalues lie in a disk of radius r =0.5v 2. For ~(2n-1)