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By I. M Glazman

ISBN-10: 070650562X

ISBN-13: 9780706505627

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Additional info for Direct Methods of Qualitative Spectral Analysis of Singular Differential Operators

Example text

Y(k) (a) = Y(k) (P) = 0 Let a1= a, and P1 be any number larger than p. 40 It follows from variational principles that the problem with the same differential operation l and boundary conditions (k-0, I. I)=0 should have an eigenvalue k, < 0. Let the corresponding eigenfunction be fi(x). This function has n-fold zeros at the points a, and P, and (D [f,]=a,(f,, f,)<0. Applying Lemma 5 of No. 10, we find a finite function cP, E'I)L with the same support [a,, P,1, for which we also have (Ly1, Cfl) = 4' [p>] < 0.

It would be incorrect to say that the types of the points of C(L) are the same as the types of the points of C(L), but if we require, in addition to (7), the fulfillment of the inequality q(x)>0 for sufficiently large x, then from Theorem 7D' the conservation of the kind of the points of C(L) of type (1) (No. 3, page 14) easily follows. The question of whether the above theorem is also correct for the second order differential operation was posed for the first time in / 103 (1)/. There instead of condition (7) only the coincidence of the extreme left points of the sets C(L) and C(Ldis established.

Lis called the minimal unclosed differential operator, defined by the operation (1). Its closure L will be called the minimal differential operator defined by the operation (1). As is known (see /7/), the defect number of this operator satisfies the inequality a < Def L < 2n. Any self -adjoint extension L of the operator L will be called a self -adjoint operator produced by the operation 1. By virtue of Theorem 4, all the self-adjoint operators, gen- erated by a given operation (1) have the same continuous spectrum C(Z)=C(L).

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Direct Methods of Qualitative Spectral Analysis of Singular Differential Operators by I. M Glazman


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