Read e-book online Direct Methods of Qualitative Spectral Analysis of Singular PDF

By I. M Glazman

ISBN-10: 070650562X

ISBN-13: 9780706505627

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

Example text

Proof. From Theorem 21 it follows that the operator A2+1 = (A - if) (A+if) defined on a manifold dense in His closed. It also follows that Def (As+1) DA, = Def (A - if) 'lA+ Def (A+if) `DA = m+n. We obtain therefore that the operator A2 is closed and defined on a manifold dense in H. 'A,=m+n, Q. D. This theorem may be easily generalized to any arbitrary natural power of a symmetric operator with a finite defect number. In particular, for any symmetric operator with equal defect numbers we obtain the formula Def An=nDefA.

In order to construct the above-mentioned e-network we choose for a given 8, a number 7 such that for x > i we have lik(x) (k = 1. 2.... , 82 Pt (x) < n). This is possible because of equation (8). Setting n T (K1.!. f), = I f 1 , 7 , , l ' I f)n-k) (x) I2 dx. k-1 0 we have (K1f, f)=(K1f f)T+Y f k-1 T n < (K1f, f),+ 82 Fj f Pk (x) I f(n-k) (x) I2 dX < k-1 T <(K1f. f)T+81(Af, f)<(K1f. f),+M282. (10) The following inequality which follows from (9) holds for all f E F 7 f I f(n) (x) I2 dx < M. (11) 0 From inequality (11) and the conditions fck) (0) = 0 (k = 0, 1.....

Thus, for all the intervals w situated to the right of the point N=NN, we shall have A(w)>µ. This shows that the condition is necessary. Conversely, suppose that for arbitrarily chosen µ> 0 we have A(w)> for all the intervals w lying to the right of the point N=Nµ. By virtue of Lemma 6 we then have for all the N-finite functions of ZL that (LNY, Y) > (µ - s) (Y. Y) 39 It follows from this inequality that C(LN)fl (- oo, -f)=0 and by virtue of the decomposition principle also that C(L)fl-cc. µ- i) =0 hence for µL - oo we obtain that C(L)=0.

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


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