Titchmarsh, Edward Charles

English

Titchmarsh Edward Charles

University of Szeged

16. On the discreteness of the spectrum of a differential equation. By E. C. TlTCHMARSH in Oxford. It was proved by WEYL1) that the spectrum associated with the diffe-rential equation 0 ) = 0 ( 0 ^ X < o o ) is discrete if q(x) is bounded in any finite interval and tends to infinity a s x->- oo. His proof is reproduced in my book Eigenfunction Expansions Associated with Second Order Differential Equations (Oxford, 1946), § 5 . 12. Other proofs have since been given2). The following is another simple proof. Let y (x, A) be the solution of (1) which satisfies a given boundary condition (2) (p(0, A) cos « + <px(0, A) sin a = 0 at x = 0. Then it can be proved as in § 5 . 12 of my book that, for every real I, either cpix^l) is L2 (in which case <p(x,l) and (fx(x, X) both tend to zero as x + oo), or <p(x, A)-»-00, or <p(x, A)-* — co. Consider any finite interval and denote the sub-sets of this interval where cp has the above properties by E0, Elr and E2 respectively. If I' belongs to Eu <p(x,l')-+ oc, yxx{x,l')-+oo (by (1)), and so q>x(x,l')-+ 00. Hence for some with q(%)>h', 0 and <px(%,V)>0. Hence also y A) > 0 and q>x(g, I ) > 0 if I—I' is sufficiently small. This, however, implies that cp(x,l) + 00. Hence any point of Ex is an interior point of an interval of points of Elt and so Ex is an open set. Similarly E2 is an open set. Hence E0 is a closed set. !) H. WEYC, Ober gewöhnliche Differentialgleichungen mit Singularitäten und die zugehörigen Entwicklungen willkürlicher Funktionen, Math. Annalen, 68 (1910), pp. 220—269. K. FRIEDRICHS, Spektraltheorie halbbeschränkter Operatoren und Anwendung auf die Spektralzerlegung vonDifferentialoperatoren.il., Math. Annalen, 1 0 9 (1934) , pp. 6 8 5 - 7 1 3 . Criteria for the discrete character of the spectra of ordinary differential operators, Studies and Essays presented to R. Courant (New York, 1948) , pp. 1 4 5 — 1 6 0 ; a proof by E . C. T I T C H -MARSH will be published in the Annali di Matematica. E. C. Titchmarsh: Spectrum of a differential equation. 17 The above argument also shows that, if V is a point of Elt then, for some r/>(§, /7/> 0, while <p(x, A) is steadily increasing for if I is in some interval Hence rp(x,l)^>m for \l — l ' \ I t follows as on p. 116 of my book that the function k(l) is constant throughout the interval \ l — X ' \ < i ] , and so in fact is constant throughout each interval of To prove the theorem, we have now to show that E0 consists at most of a finite number of points. Suppose on the contrary that there is a sequence of values of I tending to (but different from) a limit /t, such that these x's and fi all belong to E0. Let Xj be such that q ( x ) ^ > n + d (<f>0) for (such an x, exists if q{x)-+ oo). Let x 2 > x x be such that <je(x2, (i) + 0, and suppose e . g . that <p{xo, /.i) > 0 . As in § 5 . 1 2 of my book, this implies (since <p(x, fi) is L2) that (¡p(x, ¡i) decreases steadily to zero for x ^ x 2 , and in particular that cp(x, fi) > 0 for x ; > x 2 . Now rp(x, X) is a continuous function of both variables in any finite region (cf. § 1.5 of my book), and so q>(x, l)-+q>(x, fi) as uniformly over 0 ^ x < x 2 . Hence : lim \rp(x, X) <p(x, (i) c / x = {qp(x, fi)}2dx > 0, X u J J ' ••' 0 0 and so J cp(x, X).rp(x,fi) dx > 0 o if X is sufficiently near to /t. Also cp(x.2, X)-*tp(x2, ,«)>0 . and so (p(x2, A) > 0 if X is sufficiently near to ft. Since q(x) — / > 0 if x : > x , and X is sufficiently near to fi, this implies, as before, that cp(x, X)>0 for x > x 2 and X sufficiently near to ft. Hence 00 J' rp (x, X) rp (x, fi) dx > 0 •r2 if X is sufficiently near to fi. Altogether 0 0 •• (3) J <p(x, A) <p (x, fi)dx > 0 o . if X is sufficiently near to fi. This, however, is impossible; for on multiplying (1), and the corre-sponding equation with fi, by rp(x,fi), <jo(x,X), respectively and subtracting, 18 E. C. Titchmarsh : Spectrum of a differential equation. we obtain 0 {I —tu) <p(x, I) (f(x, n) = — {<p(x, A) <px(x, n) — (p(x, fi) cp,(x, A)}. Using (2) it follows that X (A — ( rp(x, A) <p(x, (I) dx = cp(X, A) cpx(X, fi) — q>(X, n) <pj:(X, A), o which tends to 0 as X-+00. Since it follows that 00' (4) j 9(x,/.)9(x, =•().. 0 • This gives a contradiction, and the theorem follows. (Received July 9, 19.49.) 

On the discreteness of the spectrum of a differential equation

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On the discreteness of the spectrum of a differential equation

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