International Jour. of Diff. Eq. and Appl., 3, N1, (2001),
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1 Interntionl Jour. of Diff. Eq. nd Appl., 3, N1, (2001),
2 New proof of Weyl s theorem A.G. Rmm Mthemtics Deprtment, Knss Stte University, Mnhttn, KS , USA rmm@mth.ksu.edu rmm Abstrct Let lu = u + q(x)u, where q(x) is rel-vlued L 2 loc (0, ) function. H. Weyl hs proved in 1910 tht for ny z, Imz 0, the eqution (l z)w = 0, x > 0, hs solution w L 2 (0, ). We prove this clssicl result using new rgument. 1 Introduction Let lu = u + q(x)u, where q(x) L 2 loc is rel-vlued function. Fix n rbitrry complex number z, Imz > 0, nd consider the eqution lw zw = 0, x > 0 (1.1) H. Weyl proved [5] tht eqution (1.1) hs solution w L 2 (0, ), which is clled Weyl s solution. He gve the limit point-limit circle clssifiction of the opertor l: if eqution (1.1) hs only one solution w L 2 (0, ), then it is limit point cse, otherwise it is limit circle cse. Weyl s theory is presented in severl books, e.g. in [4], [3]. This theory is bsed on some limiting procedure b for the solutions to (1.1) on finite intervl (0, b). In [3] nice different proof is given for continuous q(x). The im of our pper is to give new method for proof of Weyl s result. Theorem 1.1. Eqution (1.1) hs solution w L 2 (0, ). key words: limit circle, limit point, Weyl s solution. Mth subject clssifiction: 34B25, 34B20
3 Let us outline the new pproch nd the steps of the proof. Since q(x) is rel-vlued function, symmetric opertor l 0 defined on liner dense subset C 0 (0, ) of H = L 2 (0, ) by the expression lu = u + q(x)u hs selfdjoint extension, which we denote by l. Therefore the resolvent (l z) 1 is bounded liner opertor on the Hilbert spce H = L 2 (0, ), (l z) 1 Imz 1. This opertor is n integrl opertor with the kernel G(x, y; z), which is distribution stisfying the eqution We will prove tht (l z)g(x, y; z) = δ(x y), G(x, y; z) = G(y, x; z). (1.2) 0 G(x, y; z) 2 dy c(x; z) x (0, ), Imz > 0, (1.3) where c(x; z) = const > 0. The kernel G(x, y; z), which is the Green function of the opertor l, cn be represented s G(x, y; z) = ϕ(y; z)w(x; z), x > y, (1.4) where w nd ϕ re linerly independent solution to (1.1), so tht w(x; z) 0. From (1.3) it follows tht w(x; z) L 2 (0, ). (1.5) A detiled proof is given in section 2. One my try to prove the existence of Weyl s solution s follows: tke n h L 2 loc (0, ), h = 0 for x > R, h 0, nd let W := W (x, z) := (l z) 1 h, Imz > 0. Then W solves (1.1) for x > R nd W L 2 (0, ) since l is selfdjoint opertor in H. However, one hs to prove then tht W does not vnish identiclly for x > R, nd this will be the cse not for n rbitrry h with the bove properties. In our pper the role of h is plyed by the delt-function, nd since ϕ(y; z) nd w in (1.4) re linerly independent solutions of (1.1), one concludes tht w does not vnish identiclly. 2 Proofs Lemm 2.1. If q(x) L 1 loc (0, ) nd q(x) is rel-vlued, then symmetric opertor l 0 u := u + q(x)u, D(l 0 ) = { u : u C0 (0, ), l 0 u H := L 2 (0, ) } is defined on liner dense in H subset, nd dmits selfdjoint extension l. 3
4 Proof. This result is known: the density of the domin of definition of the symmetric opertor l 0 mentioned in Lemm 1 nd the existence of selfdjoint extension re proved in [2]. The defect indices of l 0 re (1,1) or (2,2), so tht by von Neumnn extension theory l 0 hs selfdjoint extensions (see [2]). Actully we ssume in the Appendix tht q L 2 loc (0, ), in which cse the conclusion of Lemm 2.1 is obvious: C 0 (0, ) is the liner dense subset in H on which l 0 is defined. Let l be selfdjoint extension of l 0, (l z) 1 be its resolvent, Imz > 0, nd G(x, y; z) be the resolvent s kernel (in the sense of distribution theory) of (l z) 1, G(x, y; z) = G(y, x; z). Lemm 2.2. For ny fixed x [0, ) one hs ( 0 ) 1 G(x, y; z) 2 2 dy c, c = c(x; z) = const > 0. (2.1) Proof. Let h C 0 (0, ) nd u := (l z) 1 h, so Let us prove tht: u(x; z) = 0 G(x, y; z)h(y)dy, (l z)u = h. (2.2) u(x; z) c(x; z) h, (2.3) where x [0, ) is n rbitrry fixed point, c(x) = const > 0, h := h L 2 (0, ), (u, v) := (u, v) L 2 (0, ). If (2.3) is proved, then (G(x, y; z), h) c(x; z) h. (2.4) From (2.4) the desired conclusion (2.1) follows immeditely by the Riesz theorem bout liner functionls in H. To complete the proof, one hs to prove estimte (2.3). This estimte follows from the inequlity: u C(D1 ) c ( ) u + q(x)u zu L 2 (D 2 ) + u L 2 (D 2 ) c (1 + 1 Imz ) h, (2.5) where c = c(d 1, D 2 ) = const > 0, D 1 D 2, D 2 [0, ), D 1 is strictly inner open subintervl of D 2. Indeed, since l is selfdjoint, (2.2) implies: Moreover u h Imz. (2.6) u + qu zu = h, (2.7) 4
5 so, using (2.6), one gets: u L 2 (D 2 ) + u + qu zu L 2 (D 2 ) h ( Imz + h ) h, (2.8) Imz From (2.5), (2.6) nd (2.8) one gets (2.3). Let us finish the proof by proving (2.5). In fct, inequlity (2.5) is prticulr cse of the well-known elliptic estimtes (see e.g. [1, pp ]), but n elementry proof of (2.5) is given below in the Appendix. Lemm 2 is proved. Proof of Theorem 1.1 Eqution (1.2) implies tht G(x, y; z) = ϕ(x; z)w(y; z), y x, where w(y; z) solves (1.1), nd the function ϕ(x; z) is lso solution to (1.1). Inequlity (2.1) implies w L 2 (0, ) if Imz > 0. Theorem 1.1 is proved. To mke this pper self-contined we give n elementry proof of inequlity (2.5) in the Appendix. This proof llows one to void reference to the elliptic inequlities [1], the proof of which in [1] is long nd complicted (in [1] the multidimensionl elliptic equtions of generl form re studied, which is the reson for the complicted rgument in [1]). Appendix: An elementry proof of inequlity (2.5). Since u(x) is Cloc 1 (0, ) it is sufficient to prove (2.5) ssuming tht D 1 = (, b) nd b is rbitrrily smll. Let η(x) C0 (, b) be cut-off function, 0 η 1, η(x) = 1 in ( + δ, b δ), 0 < δ < b, η(x) = 0 in neighborhoods of points nd b. 4 Let v = ηu. Then (2.2) implies: lv = ηh 2η u η u, v() = v () = 0. Thus nd Here v(x) = x v = qv zv ηh + η u + 2η u, (x s)v (s)ds c 1 [ qv + z v ] ds + c 2, (A.1) h ds + c 2 u ds + c 2 u ds. (A.2) c 1 = b, c 2 = mx x b [ η(x) + η + 2 η ]. 5
6 If b is sufficiently smll, then c 1 Therefore (A.1) implies ( q + z ) dx mx v(x) < γ mx v(x), 0 < γ < 1. x b x b mx x b v(x) c 3 [ ] h L 2 (,b) + u L 2 (,b) = u L 2 (,b), (A.3) where c 3 = c 3 (, b; z). From (A.3) nd (2.6) it follows tht inequlity (2.5) holds, provided tht: u L 2 (,b) c h + δ u L. (A.4) The lst estimte is proved s follows. Multiply (2.2) by ηu (the br stnds for complex conjugte nd η is cut-off function, η C0 (, b) nd integrte over (, b) to get u 2 ηdx = u uη dx + ηhudx + z η u 2 dx One hs, using the inequlity uv ε u 2 + v 2, ε > 0, 4ε I 1 c (ε u ε ) u 2, c = mx η, q u 2 ηdx := I 1 + I 2 + I 3 + I 4. I 2 + I 3 c ( h u + u 2) c 1 h 2, where (2.6) ws used, I 4 qu u q L 2 u L u. Thus, if < 1 < b 1 < b, where η = 1 on [ 1, b 1 ], one gets 1 1 u 2 dx C ( h 2 + u L h ) δ u 2 L + C h 2, (A.5) where C = C(ε, z,, b, δ) = const > 0, 0 < δ cn be chosen rbitrrily smll. Inequlity (A.5) implies (A.4). Inequlity (2.5) is proved. References [1] Gilbrg, D., Trudinger, N., Elliptic prtil differentil equtions of second order, Springer, New York, 1983 [2] Nimrk, M., Liner differentil opertors, Ungr, New York,
7 [3] Reed, M., Simon, B., Methods of modern mthemticl physics, vol.2, Acd. Press, New York, [4] Titchmrsh E., Eigenfunction expnsions ssocited with second-order differentil equtions, Oxford, Clrendon Press, [5] Weyl, H., Uber gewöhnliche Differentilgleichungen mit Singulritäten nd die zugehörigen Entwicklungen willkürliche Funktionen, Mth. Ann., 68, (1910),
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