STURM LIOUVILLE OPERATORS WITH MEASURE-VALUED COEFFICIENTS. 1. Introduction. + q(x)y(x) = zr(x)y(x)

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1 STUM LIOUVILLE OPEATOS WITH MEASUE-VALUED COEFFICIENTS JONATHAN ECKHADT AND GEALD TESCHL Abstrt. We give omprehensive tretment of Sturm Liouville opertors whose oeffiients re mesures inluding full disussion of self-djoint extensions nd boundry onditions, resolvents, nd Weyl Tithmrsh Kodir theory. We void previous tehnil restritions nd, t the sme time, extend ll results to lrger lss of opertors. Our opertors inlude lssil Sturm Liouville opertors, Sturm Liouville opertors with (lol nd non-lol) δ nd δ intertions or trnsmission onditions s well s eigenprmeter dependent boundry onditions, Krein string opertors, Lx opertors rising in the tretment of the Cmss Holm eqution, Jobi opertors, nd Sturm Liouville opertors on time sles s speil ses. Sturm Liouville problems (.) d dx ( p(x) dy dx (x). Introdution ) + q(x)y(x) = zr(x)y(x) hve long trdition (see e.g. the textbooks [43], [48], [49] nd the referenes therein) nd so hve their generliztions to mesure-vlued oeffiients. In ft, extensions to the se ( d (.2) dy y(t)dχ(t)) dϱ(x) dς(x) (x) + = zy(x) dte bk t lest to Feller [20] nd were lso dvoted in the fundmentl monogrph by Atkinson [5]. Here the derivtives on the left-hnd side hve to be understood s don Nikodým derivtives. We refer to the book by Mingrelli [34] for more detiled historil disussion. In ft, in those referenes, the mesure ς ws lwys ssumed to be bsolutely ontinuous, dς(x) = p(x) dx, suh tht y will t lest be ontinuous. We will not mke this restrition here sine it would exlude (e.g.) the se of δ intertions whih onstitute populr physil model. However, while the generliztion to mesure-vlued oeffiients hs been very suessful on the level of differentil equtions (see e.g. [5], [34], [47] nd the referenes therein), muh less is known bout the ssoited opertors in n pproprite Hilbert spe. First ttempts were mde by Feller nd lter omplemented by K [27] (f. lso Lnger [30] nd Bennewitz [6]). Agin, survey of these results nd further informtion n be found in the book of Mingrelli [34] Mthemtis Subjet Clssifition. Primry 34B20, 34L05; Seondry 34B24, 47A0. Key words nd phrses. Sturm Liouville opertors, mesure oeffiients, spetrl theory, strongly singulr potentils. J. d Anlyse Mth. 20, (203). eserh supported by the Austrin Siene Fund (FWF) under Grnt No. Y330.

2 2 J. ECKHADT AND G. TESCHL The se where only the potentil is llowed to be mesure is firly well treted sine it llows to inlude the se of point intertions whih is n importnt model in physis (see e.g. the monogrphs [], [2] s well s the reent results in [7] nd the referenes therein). More reently, Svhuk nd Shklikov [36] [39], Goriunov nd Mikhilets [22], [23] s well s Mikhilets nd Molybog [32], [33] were even ble to over the se where the potentil is the derivtive of n rbitrry L 2 funtion. However, note tht while this overs the se of δ intertions, it does not over the se of δ intertions whih re inluded in the present pproh. Moreover, sine we llow ll three oeffiients to be mesures, our pproh even inludes Shrödinger opertors with non-lol δ intertions on rbitrry sets of Lebesgue mesure zero s studied reently by Albeverio nd Nizhnik [3] nd Brshe nd Nizhnik [9]. This onnetion will be disussed in detil nd exploited in forthoming pper [6]. Finlly, the se where the weight oeffiient is mesure is known s Krein string nd hs lso ttrted onsiderble interest reently [44] [46] (see lso the monogrph [5]). However, while the theory developed by K nd extended by Mingrelli is quite generl, it still does exlude severl ses of interest. More preisely, the bsi ssumptions in Chpter 3 of Mingrelli [34] require tht the orresponding mesures hve no weight t finite boundry point. Unfortuntely, this ssumption exludes for exmple lssil ses like Jobi opertors on hlf-line. The reson for this ssumption is the ft tht otherwise the orresponding mximl opertor will be multi-vlued nd one hs to work within the frmework of multi-vlued opertors. This problem is lredy visible in the se of hlf-line Jobi opertors where the underlying Hilbert spe hs to be rtifiilly expnded in order to be ble to formulte pproprite boundry onditions [42]. In our se there is no nturl wy of extending the Hilbert spe nd the intrinsi pproh vi multi-vlued opertors is more nturl. Nevertheless, this multi-vluedness is not too severe nd orresponds to n t most two dimensionl spe whih n be removed to obtin single-vlued opertor, gin, ft well-known from Jobi opertors with finite endpoints. Finlly, this generl pproh will lso llow us to inlude lrge vriety of boundry onditions, inluding the se of eigenprmeter dependent boundry onditions. Moreover, the ft tht our differentil eqution is defined on lrger set thn the support of the mesure ϱ (whih determines the underlying Hilbert spe) is lso motivted by requirements from the pplitions we hve in mind. The most drsti exmple in this respet is the Sturm Liouville problem (.3) d dϱ(x) ( dy dx (x) + 4 y(t)dt) = zy(x) on whih rises in the Lx pir of the dispersionless Cmss Holm eqution [0], []. In se of the well-known one-pekon solution, ϱ is single Dir mesure nd the underlying Hilbert spe is one-dimensionl. However, the orresponding differentil eqution hs to be investigted on ll of, where the Cmss Holm eqution is defined. An pproprite spetrl theory for this opertor in the se where ϱ is n rbitrry mesure seems to be missing nd is one of the min motivtions for the present pper. Furthermore, there is of ourse nother reson why Sturm Liouville equtions with mesure-vlued oeffiients re of interest, nmely, the unifition of the ontinuous with the disrete se. While suh unifition lredy ws one of the min

3 STUM LIOUVILLE OPEATOS WITH MEASUE-VALUED COEFFICIENTS 3 motivtions in Atkinson [5] nd Mingrelli [34], it hs reently ttrted enormous ttention vi the introdution of the lulus on time sles [8]. In ft, given time sle T, the so-lled ssoited Hilger (or delt) derivtive is nothing but the don Nikodým derivtive with respet to the mesure ϱ, whih orresponds to the distribution funtion (x) = inf{y T y > x}. We refer to [7] for further detils nd to follow-up publition [8], where we will provide further detils on this onnetion. Finlly, our pproh lso inludes number of generliztions for Sturm Liouville problems whih hve been ttrting signifint interest in the pst. In prtiulr, our pproh overs boundry onditions depending polynomilly on the eigenprmeter s well s internl disontinuities (lso known s trnsmission onditions) s introdued by Hld [25] in his study of the inverse problem for the torsionl modes of the erth (f. [4]). As one of our entrl results we will develop singulr Weyl Tithmrsh Kodir theory for these opertors extending the reent work of Kostenko, Skhnovih, nd Teshl [29] (see lso Kodir [28], K [27], nd Gesztesy nd Zinhenko [2]). In prtiulr, we will over singulr settings where the Weyl Tithmrsh Kodir funtion is no longer Herglotz Nevnlinn funtion (or generlized Nevnlinn funtion). Agin this generl pproh is motivted by pplitions to the dispersionless Cmss Holm eqution, where the ssoited spetrl mesure n exhibit rbitrry growth. 2. Nottion Let (, b) be n rbitrry intervl nd µ be lolly finite omplex Borel mesure on (, b). By AC lo ((, b); µ) we denote the set of left-ontinuous funtions, whih re lolly bsolutely ontinuous with respet to µ. These re preisely the funtions f whih n be written in the form f(x) = f() + h(s)dµ(s), x (, b), where h L lo ((, b); µ) nd the integrl hs to be red s h(s)dµ(s), if x >, [,x) (2.) h(s)dµ(s) = 0, if x =, h(s)dµ(s), if x <. [x,) The funtion h is the don Nikodým derivtive of f with respet to µ. uniquely defined in L lo ((, b); µ) nd we write df dµ = h. It is Every funtion f whih is lolly bsolutely ontinuous with respet to µ is lolly of bounded vrition nd hene lso the right-hnd limits f(x+) = lim ε 0 f(x + ε), x (, b) of f exist everywhere. Also note tht some funtion f AC lo ((, b); µ) n only be disontinuous in some point, if µ hs mss in this point.

4 4 J. ECKHADT AND G. TESCHL In this respet we lso rell the integrtion by prts formul ([26, Theorem 2.67]) for two lolly finite omplex Borel mesures µ, ν on (, b) (2.2) β β F (x)dν(x) = F G β G(x+)dµ(x),, β (, b), where F, G re left-ontinuous distribution funtions of µ, ν respetively. 3. Sturm Liouville equtions with mesure-vlued oeffiients Let (, b) be n rbitrry intervl nd ϱ, ς nd χ be lolly finite omplex Borel mesures on (, b). We wnt to define liner differentil expression τ whih is informlly given by τf = d dϱ ( df dς + ) fdχ. In this setion, we will suessively dd ssumptions on our mesure oeffiients s soon s they re needed. All of them re inluded in Hypothesis 3.7 below, whih will then be in fore throughout the rest of this pper. However, up to now the only dditionl ssumptions we impose on our mesures is tht ς is supported on the whole intervl, i.e., supp(ς) = (, b). The mximl domin D τ of funtions suh tht τf mkes sense onsists of ll funtions f AC lo ((, b); ς) for whih the funtion (3.) df dς (x) + fdχ, x (, b) is lolly bsolutely ontinuous with respet to ϱ, i.e., there is some representtive of this funtion lying in AC lo ((, b); ϱ). As onsequene of the ssumption supp(ς) = (, b), this representtive is unique. We then set τf L lo ((, b); ϱ) to be the don Nikodým derivtive of this funtion with respet to ϱ. One esily sees tht this definition is independent of (, b) sine the orresponding funtions (3.) s well s their unique representtives only differ by n dditive onstnt. As usul, we denote the don Nikodým derivtive with respet to ς of some funtion f D τ by f [] = df dς L lo((, b); ς ). The funtion f [] is lled the first qusi-derivtive of f. We note tht the definition of τ is onsistent with lssil theory. Indeed, let ϱ, ς nd χ be lolly bsolutely ontinuous with respet to Lebesgue mesure, nd denote by r, p nd q the respetive densities i.e., ϱ(b) = B r(x)dx, ς(b) = B dx nd χ(b) = p(x) B q(x)dx for eh Borel set B. Then some funtion f lies in D τ if nd only if f s well s its qusi-derivtive f [] = pf re lolly bsolutely ontinuous (with respet to Lebesgue mesure). In this se τf(x) = r(x) ( d dx ( p(x) df dx (x) ) ) + q(x)f(x), x (, b)

5 STUM LIOUVILLE OPEATOS WITH MEASUE-VALUED COEFFICIENTS 5 is the usul Sturm Liouville differentil expression. Also note tht if we dd single point mss δ loted t to ς ϱ(b) = r(x)dx, ς(b) = p(x) dx + δ (B), nd χ(b) = q(x)dx, B then we obtin the following jump ondition B f(+) f() = f [] () t, nd hene this orresponds to δ intertion of strength. Similrly, dding point mss δ to χ we obtin δ intertion ssoited with the jump ondition f [] (+) f [] () = f(). Considering pieewise ontinuous weight funtion r(x) = p(x) with jump r(+) = β r() nd dding point mss r() β δ to χ we obtin trnsmission ondition f(+) = f(), f (+) βf () = f(). The more generl se where f is lso llowed to jump n be redue to this one by virtue of Liouville-type trnsformtion (f. emrk 2.4 in [4]). Moreover, our pproh lso inorportes quite generl boundry onditions inluding ses where the boundry onditions depend polynomilly on the eigenvlue prmeter (f. emrk 7.9). As nother speil se, hoosing the mesures ϱ(b) = δ n (B), ς(b) = n Z B dx nd χ(b) = q n δ n (B), p x n Z where p n 0, q n nd δ n is the Dir mesure in n Z, we obtin the usul Jobi differene expression. In ft, τf(n) t some point n Z is equl to the jump of the funtion p x f (x) + q n f(n), x in tht point nd hene 0 n<x τf(n) = p n (f(n) f(n )) p n (f(n + ) f(n)) + q n f(n). Now from the theory of liner mesure differentil equtions (see Appendix A for the required results) we get n existene nd uniqueness theorem for differentil equtions ssoited with τ. B Theorem 3.. Fix some rbitrry funtion g L lo ((, b); ϱ). unique solution f D τ of the initil vlue problem Then there is (3.2) (τ z)f = g with f() = d nd f [] () = d 2 for eh z C, (, b) nd d, d 2 C if nd only if (3.3) ϱ({x})ς({x}) = 0 nd χ({x})ς({x}) for ll x (, b). If in ddition g, d, d 2 nd z re rel, then the solution is rel.

6 6 J. ECKHADT AND G. TESCHL Proof. Some funtion f D τ is solution of (τ z)f = g with f() = d nd f [] () = d 2 if nd only if for eh x (, b) f(x) = d + f [] (x) = d 2 + f [] dς, fdχ (zf + g) dϱ. Now set ω = ς + χ + ϱ nd let m 2, m 2 nd f 2 be the don Nikodým derivtives of ς, χ zϱ nd gϱ with respet to ω. Then these equtions n for eh x (, b) be written s ( ) ( ) f(x) x ( ) ( ) d 0 m2 f x ( ) 0 f [] = + (x) m 2 0 f [] dω + dω. d 2 Hene the lim follows from Theorem A.2, sine (3.3) holds for ll x (, b) if nd only if ( ) ( ) 0 m2 (x) ς({x}) I + ω({x}) = m 2 (x) 0 χ({x}) zϱ({x}) is invertible for ll z C nd x (, b). Note tht if g L lo ((, b); ϱ) nd (3.3) holds for eh x (, b), then there is lso unique solution of the initil vlue problem (τ z)f = g with f(+) = d nd f [] (+) = d 2 for every z C, (, b), d, d 2 C by Corollry A.3. Beuse of Theorem 3., in the following we will lwys ssume tht the mesure ς hs no point msses in ommon with ϱ or χ, i.e., (3.4) ς({x})ϱ({x}) = ς({x})χ({x}) = 0 for ll x (, b). This ssumption is stronger thn the one needed in Theorem 3. but we will need it for the Lgrnge identity below. For f, g D τ we define the Wronski determinnt (3.5) W (f, g)(x) = f(x)g [] (x) f [] (x)g(x), x (, b). This funtion is lolly bsolutely ontinuous with respet to ϱ with d W (f, g) dϱ = g τf f τg. Indeed, this is simple onsequene of the following Lgrnge identity. Proposition 3.2. For eh f, g D τ nd, β (, b) we hve (3.6) β (g(x)τf(x) f(x)τg(x)) dϱ(x) = W (f, g)(β) W (f, g)(). Proof. By definition g is distribution funtion of the mesure g [] ς. Furthermore, the funtion f (x) = f [] (x) + fdχ, x (, b) f 2

7 STUM LIOUVILLE OPEATOS WITH MEASUE-VALUED COEFFICIENTS 7 is distribution funtion of τfϱ. Hene one gets from integrtion by prts β β g(t)τf(t)dϱ(t) = [f (t)g(t)] β t= f (t+)g [] (t)dς(t). We n drop the right-hnd limit in the integrl sine the disontinuities of f re null set with respet to ς by (3.4). Hene the integrl beomes β f (t)g [] (t)dς(t) = β t = g(β) β fdχ g [] (t)dς(t) β β fdχ gfdχ β f [] (t)g [] (t)dς(t) f [] (t)g [] (t)dς(t), where we performed nother integrtion by prts (nd used gin (3.4)). verifying the identity is n esy lultion. Now As onsequene of the Lgrnge identity, the Wronskin W (u, u 2 ) of two solutions u, u 2 D τ of (τ z)u = 0 is onstnt. Furthermore, we hve W (u, u 2 ) 0 u, u 2 linerly independent. Indeed, the Wronskin of two linerly dependent solutions vnishes obviously. Conversely, W (u, u 2 ) = 0 mens tht the vetors ( ) ( ) u (x) u2 (x) u [] (x) nd u [] 2 (x) re linerly dependent for eh x (, b). But beuse of uniqueness of solutions this implies the liner dependene of u nd u 2. For every z C we ll two linerly independent solutions of (τ z)u = 0 fundmentl system of (τ z)u = 0. From Theorem 3. nd the properties of the Wronskin, one sees tht fundmentl systems lwys exist. Proposition 3.3. Let z C nd u, u 2 be fundmentl system of the eqution (τ z)u = 0. Furthermore, let (, b), d, d 2 C, g L lo ((, b); ϱ). Then there exist, 2 C suh tht the solution f of is given by (τ z)f = g with f() = d nd f [] () = d 2 f(x) = u (x) + 2 u 2 (x) + u (x)u 2 (t) u (t)u 2 (x) g(t)dϱ(t), W (u, u 2 ) f [] (x) = u [] (x) + 2u [] 2 (x) + u [] (x)u 2(t) u (t)u [] 2 (x) g(t)dϱ(t), W (u, u 2 ) for eh x (, b). If u, u 2 is the fundmentl system with then = d nd 2 = d 2. Proof. We set u () = u [] 2 () = nd u[] () = u 2() = 0, h(x) = u (x) u 2 g dϱ u 2 (x) u g dϱ, x (, b).

8 8 J. ECKHADT AND G. TESCHL Integrtion by prts shows tht β u [] (x) u 2 g dϱ u [] 2 (x) u g dϱ dς(x) = = [ u (x) for ll, β (, b) with < β, hene u 2 g dϱ u 2 (x) h [] (x) = u [] (x) u 2 g dϱ u [] 2 (x) u g dϱ, Using gin integrtion by prts we get β u (x) u 2 g dϱ dχ(x) z β u (x) [ ( = u 2 g dϱ u dχ z β ( [ = u 2 g dϱ β ( = u [] (β) β ( u dχ z u [] u [] u 2 g dϱ dϱ(x) = ) ] β (x) u[] () x= )] β u dϱ ] β u g dϱ x= x (, b). x= ) u dϱ u 2 (x)g(x)dϱ(x) ) (x) u[] () u 2 (x)g(x)dϱ(x) u 2 g dϱ u [] () u 2 g dϱ for ll, β (, b) with < β. Now n esy lultion shows tht β h dχ β β zh + W (u, u 2 )g dϱ = h [] (β) h [] (). u 2 u [] g dϱ Hene h is solution of (τ z)h = W (u, u 2 )g nd therefore the funtion f given in the lim is solution of (τ z)f = g. Now if we hoose = W (f, u 2)() W (u, u 2 )() nd then f stisfies the initil onditions t. 2 = W (u, f)() W (u, u 2 )(), Another importnt identity for the Wronskin is the following Plüker identity. Proposition 3.4. For eh funtions f, f 2, f 3, f 4 D τ we hve 0 = W (f, f 2 )W (f 3, f 4 ) + W (f, f 3 )W (f 4, f 2 ) + W (f, f 4 )W (f 2, f 3 ). Proof. The right-hnd side is equl to the determinnt f f 2 f 3 f 4 f [] f [] 2 f [] 3 f [] 4 2 f f 2 f 3 f 4. f [] f [] 2 f [] 3 f [] 4

9 STUM LIOUVILLE OPEATOS WITH MEASUE-VALUED COEFFICIENTS 9 We sy τ is regulr t, if ϱ ((, ]), ς ((, ]) nd χ ((, ]) re finite for one (nd hene for ll) (, b). Similrly one defines regulrity for the right endpoint b. Finlly, we sy τ is regulr if τ is regulr t both endpoints, i.e., if ϱ, ς nd χ re finite. Theorem 3.5. Let τ be regulr t, z C nd g L ((, ); ϱ) for eh (, b). Then for every solution f of (τ z)f = g the limits f() := lim x f(x) nd f [] () := lim x f [] (x) exist nd re finite. For eh d, d 2 C there is unique solution of (τ z)f = g with f() = d nd f [] () = d 2. Furthermore, if g, d, d 2 nd z re rel, then the solution is rel. Similr results hold for the right endpoint b. Proof. The first prt of the theorem is n immedite onsequene of Theorem A.4. From Proposition 3.3 we infer tht ll solutions of (τ z)f = g re given by f(x) = u (x) + 2 u 2 (x) + f 0 (x), x (, b), where, 2 C, u, u 2 re fundmentl system of (τ z)u = 0 nd f 0 is some solution of (τ z)f = g. Now sine W (u, u 2 )() = u ()u [] 2 () u[] ()u 2() 0, there is extly one hoie for the oeffiients, 2 C suh tht the solution f stisfies the initil vlues t. If g, d, d 2 nd z re rel then u, u 2, nd f 0 n be hosen rel nd hene lso nd 2 re rel. Under the ssumptions of Theorem 3.5 one sees tht Proposition 3.3 remins vlid even in the se when = (respetively = b) with essentilly the sme proof. We now turn to nlyti dependene of solutions on the spetrl prmeter z C. These results will be needed in Setion 9. Theorem 3.6. Let g L lo ((, b); ϱ), (, b), d, d 2 C nd for eh z C let f z be the unique solution of (τ z)f = g with f() = d nd f [] () = d 2. Then f z (x) nd f z [] (x) re entire funtions of order t most /2 in z for every point x (, b). Moreover, for eh, β (, b) with < β we hve for some onstnts C, B. f z (x) + f [] z (x) Ce B z, x [, β], z C Proof. The nlytiity prt follows by pplying Theorem A.5 to the equivlent system from the proof of Theorem 3.. For the remining prt note tht beuse of Proposition 3.3 it suffies to onsider the se when g vnishes identilly. If we set for eh z C with z v z (x) = z f z (x) 2 + f [] z (x) 2, x (, b),

10 0 J. ECKHADT AND G. TESCHL n integrtion by prts shows tht for eh x (, b) ( ) v z (x) = v z () + z f z f [] dς + ( f z f [] z + f z [] z + f z [] Beuse of the elementry estimte f z f z ) dχ 2 f z (x)f z [] (x) z f z(x) 2 + f z [] (x) 2 z we get n upper bound for v z v z (x) v z () + v z (t) z dω(t), ( zf z f z [] + z f z [] = v z(x) z, x [, b), f z x (, b), ) dϱ. where ω = ς + χ + ϱ, s in the proof of Theorem 3.. Now n pplition of Lemm A. yields To the left-hnd side of we hve v z (x) v z ()e z dω, x [, b). v z (x+) v z () + x+ nd hene gin by the Gronwll Lemm A. whih is the required bound. v z (t) z dω(t), x (, ) v z (x+) v z ()e x+ z dω, x (, ), Under the ssumptions of Theorem 3.6 lso the right-hnd limits of f z nd their qusi-derivtives re entire funtions in z of order t most /2 with orresponding bounds. Moreover, the sme nlyti properties re true for the solutions f z of the initil vlue problem (τ z)f = g with f z (+) = d nd f [] z (+) = d 2. Indeed, this ft follows for exmple from the remrk fter the proof of Theorem A.5 in Appendix A. Furthermore, if, in ddition to the ssumptions of Theorem 3.6, τ is regulr t nd g is integrble ner, then the funtions z f z () nd z f z [] () re entire of order t most /2 s well nd the bound in Theorem 3.6 holds for ll x [, β]. Indeed, this follows sine the entire funtions z f z (x) nd z f z [] (x) re lolly bounded, uniformly in x (, ). Moreover, in this se the ssertions of Theorem 3.6 re vlid even if we tke = nd/or =, s the onstrution of the solution in the proof of Theorem 3.5 shows. The required bound is proven s in the generl se (hereby note tht ω is finite ner sine τ is regulr there). We gther the ssumptions mde on our mesure oeffiients so fr nd dd some new, ll of whih will be in fore in the rest of this pper (exept for Lemm 3.8). Therefore, we sy tht some intervl (, β) is gp of supp(ϱ) if it is ontined in the omplement of supp(ϱ) but the endpoints nd β lie in supp(ϱ).

11 STUM LIOUVILLE OPEATOS WITH MEASUE-VALUED COEFFICIENTS Hypothesis 3.7. The following ssumptions on our mesure oeffiients will be in fore throughout the rest of this pper: (i) The mesure ϱ is positive. (ii) The mesure χ is rel-vlued. (iii) The mesure ς is rel-vlued nd supported on the whole intervl; supp(ς) = (, b). (iv) The mesure ς hs no point msses in ommon with ϱ or χ, i.e., ς({x})χ({x}) = ς({x})ϱ({x}) = 0. (v) For eh gp (, β) of supp(ϱ) nd every funtion f D τ with the outer limits f( ) = f(β+) = 0 we hve f(x) = 0, x (, β). (vi) The mesure ϱ is supported on more thn one point. As onsequene of the rel-vluedness of the mesures, τ is rel differentil expression, i.e., f D τ if nd only if f D τ nd τf = (τf) in this se. Moreover, ϱ hs to be positive in order to obtin definite inner produt lter. Furthermore, ondition (v) in Hypothesis 3.7 is ruil for Proposition 3.9 nd Proposition 3.0 to hold. For exmple, if (, b) =, ϱ is supported on πz nd we hoose ς nd χ to be equl to the Lebesgue mesure, then the funtion f(x) = sin(x) belongs to D τ with τ f = 0, in ontrdistintion to Proposition 3.9. Upon utting this funtion off outside of the intervl (0, π), one lso sees tht Proposition 3.0 eses to hold in this se. However, ondition (v) is stisfied by lrge lss of mesures s the next lemm shows. Lemm 3.8. If for eh gp (, β) of supp(ϱ) the mesures ς (,β) nd χ (,β) re of one nd the sme sign, then (v) in Hypothesis (3.7) holds. Proof. Let (, β) be gp of supp(ϱ) nd f D τ with f( ) = f(β+) = 0. As in the proof of Proposition 3.2, integrtion by prts yields f(β) τf(β)ϱ({β}) = = β+ β+ τf(x)f(x) dϱ(x) f [] (x) 2 dς(x) + β+ f(x) 2 dχ(x). Now the left-hnd side vnishes sine either ϱ({β}) = 0 or f is ontinuous in β nd hene f(β) = f(β+) = 0. Thus f [] vnishes lmost everywhere with respet to ς, i.e., f [] vnishes in (, β) nd f is onstnt in (, β). Now sine f(β+) = f(β) + f [] (β)ς({β}), we infer tht f vnishes in (, β) The theory we re going to develop from now on is not pplible if the support of ϱ onsists of not more thn one point, sine in this se L lo ((, b); ϱ) is only one-dimensionl (nd hene ll solutions of (τ z)u = 0 re linerly dependent). In prtiulr, the essentil Proposition 3.9 does not hold in this se, whih is the min reson for ssumption (vi) in Hypothesis 3.7. Nevertheless, this se is importnt, in prtiulr for pplitions to the isospetrl problem of the Cmss Holm eqution. Hene we will tret the se when supp(ϱ) onsists of only one point seprtely in Appendix C. Our im is to introdue liner opertors in the Hilbert spe L 2 ((, b); ϱ), indued by the differentil expression τ. As first step we define liner reltion T lo of

12 2 J. ECKHADT AND G. TESCHL L lo ((, b); ϱ) into L lo ((, b); ϱ) by T lo = {(f, τf) f D τ } L lo((, b); ϱ) L lo((, b); ϱ). For brief introdution to the theory of liner reltions we refer to Appendix B nd the referenes ited there. Now, in ontrst to the lssil se, in generl D τ is not embedded in L lo ((, b); ϱ), i.e., T lo is multi-vlued. Insted we hve the following result, whih is importnt for our pproh. For lter use, we introdue the bbrevitions ϱ = inf supp(ϱ) nd β ϱ = sup supp(ϱ) for the endpoints of the onvex hull of the support of ϱ. Proposition 3.9. The liner mp is bijetive. D τ T lo f (f, τf) Proof. Clerly this mpping is liner nd onto T lo by definition. Now let f D τ suh tht f = 0 lmost everywhere with respet to ϱ. We will show tht f is of the form u (x), if x (, ϱ ], (3.7) f(x) = 0, if x ( ϱ, β ϱ ], b u b (x), if x (β ϱ, b), where, b C nd u, u b re the solutions of τu = 0 with u ( ϱ ) = u b (β ϱ +) = 0 nd u [] ( ϱ ) = u [] b (β ϱ+) =. Obviously we hve f(x) = 0 for ll x in the interior of supp(ϱ) nd points of mss of ϱ. Now if (, β) is gp of supp(ϱ), then sine, β supp(ϱ) we hve f( ) = f(β+) = 0 nd hene f(x) = 0, x [, β] by Hypothesis 3.7. Hene ll points x ( ϱ, β ϱ ) for whih possibly f(x) 0, lie on the boundry of supp(ϱ) suh tht there re monotone sequenes x +,n, x,n supp(ϱ) with x +,n x nd x,n x. Then for eh n N, we either hve f(x,n +) = 0 or f(x,n ) = 0, hene f(x ) = lim n f(x,n ) = lim n f(x,n+) = 0. Similrly one shows tht lso f(x+) = 0. Now sine f is solution of τu = 0 outside of [, β], it remins to show tht f( ϱ ) = f(β ϱ ) = 0. Therefore, ssume tht f is not ontinuous in ϱ, i.e., ς({ ϱ }) 0. Then f [] is ontinuous in ϱ nd hene f [] ( ϱ ) = 0. But this yields f( ϱ ) = f( ϱ +) f [] ( ϱ )ς({ ϱ }) = 0. Similrly, one shows tht f(β ϱ ) = 0 nd hene f is of the limed form. Furthermore, simple lultion yields (3.8) τf = {ϱ} b {βϱ}. Now in order to prove tht our mpping is one-to-one let f D τ be suh tht f = 0 nd τf = 0 lmost everywhere with respet to ϱ. By Theorem 3. it suffies to prove tht f() = f [] () = 0 t some point (, b). But this is vlid for ll points between ϱ nd β ϱ by the first prt of the proof.

13 STUM LIOUVILLE OPEATOS WITH MEASUE-VALUED COEFFICIENTS 3 In the following we will lwys identify the elements of the liner reltion T lo with funtions in D τ. Hene some element f T lo is lwys identified with some funtion f D τ, whih is n AC lo ((, b); ς) representtive of the first omponent of f (s n element of T lo ) nd τf L lo ((, b); ϱ) is the seond omponent of f (gin s n element of T lo ). In generl the reltion T lo is multi-vlued, i.e., mul(t lo ) = { g L lo((, b); ϱ) (0, g) T lo } {0}. In the formultion of the next result, we onsider the ondition tht ϱ hs no mss in ϱ s trivilly stisfied if ϱ = (sine ϱ lives on (, b) by definition) nd similrly t the other endpoint. Proposition 3.0. The multi-vlued prt of T lo is given by mul(t lo ) = spn { {ϱ}, {βϱ}}. In prtiulr, 0, if ϱ hs neither mss in ϱ nor in β ϱ, dim mul(t lo ) =, if ϱ hs either mss in ϱ or in β ϱ, 2, if ϱ hs mss in ϱ nd in β ϱ. Hene T lo is n opertor if nd only if ϱ hs neither mss in ϱ nor in β ϱ. Proof. Let (f, τf) T lo with f = 0 lmost everywhere with respet to ϱ. In the proof of Proposition 3.9 we sw tht suh n f is of the form (3.7) nd τf is liner ombintion of {ϱ} nd {βϱ} by (3.8). It remins to prove tht mul(t lo ) indeed ontins {ϱ} if ϱ hs mss in ϱ. Therefore, onsider the funtion { u (x), if x (, ϱ ], f(x) = 0, if x ( ϱ, b). One esily heks tht f lies in D τ nd hene (0, {ϱ}) = (f, τf) T lo. Similrly one shows tht {βϱ} indeed lies in mul(t lo ) if ϱ hs mss in β ϱ. Furthermore, note tht {ϱ} = 0 (respetively {βϱ} = 0) s funtions in L lo ((, b); ϱ) provided tht ϱ hs no mss in ϱ (respetively in β ϱ ). In ontrst to the lssil se one n not define proper Wronskin for elements in dom(t lo ), insted we define the Wronskin of two elements f, g of the liner reltion T lo s W (f, g)(x) = f(x)g [] (x) f [] (x)g(x), x (, b). The Lgrnge identity then tkes the form W (f, g)(β) W (f, g)() = β Furthermore, note tht by Theorem 3. we hve (g(x)τf(x) f(x)τg(x)) dϱ(x). (3.9) rn(t lo z) = L lo((, b); ϱ) nd dim ker(t lo z) = 2 for eh z C.

14 4 J. ECKHADT AND G. TESCHL 4. Sturm Liouville reltions In this setion we will restrit the differentil reltion T lo in order to obtin liner reltion in the Hilbert spe L 2 ((, b); ϱ) with slr produt f, g = b f(x)g(x) dϱ(x). First we define the mximl reltion T mx in L 2 ((, b); ϱ) by (4.) T mx = {(f, τf) T lo f L 2 ((, b); ϱ), τf L 2 ((, b); ϱ)}. In generl T mx is not n opertor. Indeed we hve mul(t mx ) = mul(t lo ), sine ll elements of mul(t lo ) re squre integrble with respet to ϱ. In order to obtin symmetri reltion we restrit the mximl reltion T mx to funtions with ompt support (4.2) T 0 = {(f, τf) f D τ, supp(f) ompt in (, b)}. Indeed, this reltion T 0 is n opertor s we will see lter. Sine τ is rel differentil expression, the reltions T 0 nd T mx re rel with respet to the nturl onjugtion in L 2 ((, b); ϱ), i.e., if f T mx (respetively f T 0 ), then lso f T mx (respetively f T 0 ), where the onjugtion is defined omponentwise. We sy some mesurble funtion f lies in L 2 ((, b); ϱ) ner (respetively ner b) if f lies in L 2 ((, ); ϱ) (respetively in L 2 ((, b); ϱ)) for ll (, b). Furthermore, we sy some f T lo lies in T mx ner (respetively ner b) if f nd τf both lie in L 2 ((, b); ϱ) ner (respetively ner b). One esily sees tht some f T lo lies in T mx ner (respetively b) if nd only if f lies in T mx ner (respetively b). Proposition 4.. Let τ be regulr t nd f lie in T mx ner. Then both limits f() := lim x f(x) nd f [] () := lim x f [] (x) exist nd re finite. A similr result holds t b. Proof. Under this ssumptions τf lies in L 2 ((, b); ϱ) ner nd sine ϱ is finite mesure ner we hve τf L ((, ); ϱ) for eh (, b). Hene the lim follows from Theorem 3.5. From the Lgrnge identity we now get the following lemm. Lemm 4.2. If f nd g lie in T mx ner, then the limit W (f, g )() := lim W (f, g )() exists nd is finite. A similr result holds t b. If f, g T mx, then (4.3) τf, g f, τg = W (f, g )(b) W (f, g )() =: W b (f, g ). Proof. If f nd g lie in T mx ner, then the limit of the left-hnd side in eqution (3.6) exists. Hene lso the limit in the lim exists. Now the remining prt follows by tking the limits nd β b.

15 STUM LIOUVILLE OPEATOS WITH MEASUE-VALUED COEFFICIENTS 5 If τ is regulr t nd f nd g lie in T mx ner, then we lerly hve In order to determine the djoint of T 0 W (f, g )() = f()g [] () f [] ()g(). T 0 = {(f, g) L 2 ((, b); ϱ) L 2 ((, b); ϱ) (u, v) T 0 : f, v = g, u }, s in the lssil theory, we need the following lemm (see [43, Lemm 9.3]). Lemm 4.3. Let V be vetor spe over C nd F,..., F n, F V, then n F spn {F,..., F n } ker F i ker F. Theorem 4.4. The djoint of T 0 is T mx. Proof. From Lemm 4.2 one immeditely gets T mx T 0. Indeed, for eh f T 0 nd g T mx we hve τf, g f, τg = lim β b W (f, g )(β) lim W (f, g )() = 0, sine W (f, g ) hs ompt support. Conversely, let (f, f 2 ) T0 nd f be solution of τ f = f 2. We expet tht (f f, 0) T lo. To prove this we will invoke Lemm 4.3. Therefore, we onsider liner funtionls b ( f(x)) l(g) = f(x) g(x)dϱ(x), g L 2 ((, b); ϱ), l j (g) = b i= u j (x) g(x)dϱ(x), g L 2 ((, b); ϱ), j =, 2, where u j re two solutions of τu = 0 with W (u, u 2 ) = nd L 2 ((, b); ϱ) is the spe of squre integrble funtions with ompt support. For these funtionls we hve ker l ker l 2 ker l. Indeed let g ker l ker l 2, then the funtion u(x) = u (x) u 2 (t)g(t)dϱ(t) + u 2 (x) b x u (t)g(t)dϱ(t), x (, b) is solution of τu = g by Proposition 3.3 nd hs ompt support sine g lies in the kernel of l nd l 2, hene u T 0. Then the Lgrnge identity nd the definition of the djoint yields b ( f(x) f(x) ) b τu(x)dϱ(x) = τu, f f(x) τu(x)dϱ(x) = u, f 2 b τ f(x) u(x)dϱ(x) = 0 nd hene g = τu ker l. Now pplying Lemm 4.3 there re, 2 C suh tht b ( f(x) f(x) ( ) + u (x) + 2 u 2 (x)) g(x)dϱ(x) = 0 for eh funtion g L 2 ((, b); ϱ). By definition of T lo we obviously hve ( f + u + 2 u 2, f 2 ) T lo. But the first omponent of this pir is equl to f, lmost everywhere with respet to ϱ beuse of ( ). Hene we lso hve (f, f 2 ) T lo nd therefore (f, f 2 ) T mx.

16 6 J. ECKHADT AND G. TESCHL By the preeding theorem T 0 is symmetri. The losure T min of T 0 is lled the miniml reltion, T min = T 0 = T 0 = T mx. In order to determine T min we need the following lemm on funtions in the mximl reltion T mx. Lemm 4.5. If f lies in T mx ner nd f b lies in T mx ner b, then there exists n f T mx suh tht f = f ner nd f = f b ner b (regrded s funtions in D τ ). Proof. Let u, u 2 be fundmentl system of τu = 0 with W (u, u 2 ) = nd let, β (, b) with < β suh tht the funtionls F j (g) = β u j gdϱ, g L 2 ((, b); ϱ), j =, 2 re linerly independent. This is possible sine otherwise u nd u 2 would be linerly dependent in L 2 ((, b); ϱ) nd hene lso in D τ by the identifition in Lemm 3.9. First we show tht there is some u D τ suh tht u() = f (), u [] () = f [] (), u(β) = f b (β) nd u [] (β) = f [] (β). Indeed, let g L 2 ((, b); ϱ) nd onsider the solution u of τu = g with the initil onditions u() = f () nd u [] () = f [] (). With Proposition 3.3 one sees tht u hs the desired properties if ( ) ( ) ( ) F2 (g) u (β) u 2 (β) fb (β) u (β) 2 u 2 (β) = F (g) u [] (β) u[] 2 (β) f [] b (β) u [] (β) 2u [] 2 (β), where, 2 C re the onstnts ppering in Proposition 3.3. But sine the funtionls F, F 2 re linerly independent, it is possible to hoose funtion g L 2 ((, b); ϱ) suh tht this eqution is vlid. Now the funtion f defined by f (x), if x (, ], f(x) = u(x), if x (, β], f b (x), if x (β, b), hs the limed properties. Theorem 4.6. The miniml reltion T min is given by (4.4) T min = {f T mx g T mx : W (f, g)() = W (f, g)(b) = 0}. Furthermore, T min is n opertor, i.e., dim mul(t min ) = 0. Proof. If f T min = T mx T mx we hve 0 = τf, g f, τg = W (f, g )(b) W (f, g )() for eh g T mx. Given some g T mx, there is g T mx suh tht g = g in viinity of nd g = 0 in viinity of b. Therefore, W (f, g)() = W (f, g )() W (f, g )() = 0. b

17 STUM LIOUVILLE OPEATOS WITH MEASUE-VALUED COEFFICIENTS 7 Similrly one sees tht W (f, g)(b) = 0 for eh g T mx. Conversely, if f T mx suh tht for eh g T mx, W (f, g)() = W (f, g)(b) = 0, then τf, g f, τg = W (f, g )(b) W (f, g )() = 0, hene f T mx = T min. In order to show tht T min is n opertor, let f T min with f = 0 lmost everywhere with respet to ϱ. If ϱ > nd ϱ({ ϱ }) 0, then f is of the form (3.7). From wht we lredy proved we know tht W (f, u )() = W (f, u 2 )() = 0 for eh fundmentl system u, u 2 of τu = 0. But W (f, u j )(x) is onstnt on (, ϱ ) nd hene we infer f( ϱ ) = f [] ( ϱ ) = 0. From this we see tht f vnishes on (, ϱ ). Similrly one proves tht f lso vnishes on (β ϱ, b), hene f = 0. For regulr differentil expressions we my hrterize the miniml opertor in terms of the boundry vlues of funtions f T mx. Corollry 4.7. If τ is regulr t nd f T mx, then we hve A similr result holds t b. f() = f [] () = 0 g T mx : W (f, g)() = 0. Proof. The lim follows from W (f, g)() = f()g [] () f [] ()g() nd the ft tht one finds g T mx with presribed initil vlues t. Indeed, one n tke g to oinide with some solution of τu = 0 ner. If the mesure ϱ hs no weight ner some endpoint, we get nother hrteriztion for funtions in T min in terms of their left-hnd (respetively right-hnd) limit t ϱ (respetively t β ϱ ). Corollry 4.8. If ϱ > nd f T mx, then we hve f( ϱ ) = f [] ( ϱ ) = 0 g T mx : W (f, g)() = 0. A similr result holds t b. Proof. The Wronskin of two funtions f, g whih lie in T mx ner is onstnt on (, ϱ ) by the Lgrnge identity. Hene we hve W (f, g)() = lim x ϱ f(x)g [] (x) f [] (x)g(x). Now the lim follows sine we my find some g whih lies in T mx ner, with presribed left-hnd limits t ϱ. Indeed, one my tke g to be suitble solution of τu = 0. Note tht ll funtions in T min vnish outside of ( ϱ, β ϱ ). In generl the opertor T min is, beuse of dom(t min ) = mul(t min) = mul(t mx ), not densely defined. On the other side, dom(t mx ) is lwys dense in the Hilbert spe L 2 ((, b); ϱ) sine dom(t mx ) = mul(t mx) = mul(t min ) = {0}. Next we will show tht T min lwys hs self-djoint extensions.

18 8 J. ECKHADT AND G. TESCHL Theorem 4.9. The defiieny indies of the miniml reltion T min re equl nd t most two, i.e., (4.5) n(t min ) := dim rn(t min i) = dim rn (T min + i) 2. Proof. The ft tht the dimensions re less thn two, is onsequene of the inlusion rn(t min ± i) = ker(t mx i) ker(t lo i). Now sine T min is rel with respet to the nturl onjugtion in L 2 ((, b); ϱ), we see tht the nturl onjugtion is onjugte-liner isometry from the kernel of T mx + i onto the kernel of T mx i nd hene their dimensions re equl. 5. Weyl s lterntive We sy τ is in the limit-irle (l..) se t, if for eh z C ll solutions of (τ z)u = 0 lie in L 2 ((, b); ϱ) ner. Furthermore, we sy τ is in the limit-point (l.p.) se t, if for eh z C there is some solution of (τ z)u = 0 whih does not lie in L 2 ((, b); ϱ) ner. Similrly one defines the l.. nd l.p. ses for the endpoint b. It is ler tht τ is only either in the l.. or in the l.p. se t some boundry point. The next lemm shows tht τ indeed is in one of these ses t eh endpoint. Lemm 5.. If there is z 0 C suh tht ll solutions of (τ z 0 )u = 0 lie in L 2 ((, b); ϱ) ner, then τ is in the l.. se t. A similr result holds t the endpoint b. Proof. Let z C nd u be solution of (τ z)u = 0. If u, u 2 re fundmentl system of (τ z 0 )u = 0 with W (u, u 2 ) =, then u nd u 2 lie in L 2 ((, b); ϱ) ner by ssumption. Therefore, there is some (, b) suh tht the funtion v = u + u 2 stisfies z z 0 v 2 dϱ 2. Sine u is solution of (τ z 0 )u = (z z 0 )u, we hve for eh x (, b) u(x) = u (x) + 2 u 2 (x) + (z z 0 ) (u (x)u 2 (t) u (t)u 2 (x)) u(t)dϱ(t) for some onstnts, 2 C by Proposition 3.3. Therefore, we hve u(x) Cv(x) + z z 0 v(x) x v(t) u(t) dϱ(t), where C = mx(, 2 ) nd furthermore, using Cuhy Shwrz u(x) 2 2C 2 v(x) z z 0 2 v(x) 2 v(t) 2 dϱ(t) Now n integrtion yields for eh s (, ) s ( u 2 dϱ 2C 2 v 2 dϱ + 2 z z 0 2 2C 2 v 2 dϱ + 2 s x u 2 dϱ x x (, ), u(t) 2 dϱ(t). ) 2 v 2 dϱ u 2 dϱ s

19 STUM LIOUVILLE OPEATOS WITH MEASUE-VALUED COEFFICIENTS 9 nd therefore s u 2 dϱ 4C 2 v 2 dϱ <. Sine s (, ) ws rbitrry, this yields the lim. As n immedite onsequene of Lemm 5. we obtin: Theorem 5.2 (Weyl s lterntive). Eh boundry point is either in the l.. se or in the l.p. se. Proposition 5.3. If τ is regulr t or if ϱ hs no weight ner, then τ is in the l.. se t. A similr result holds t the endpoint b. Proof. If τ is regulr t eh solution of (τ z)u = 0 n be ontinuously extended to. Hene u is in L 2 ((, b); ϱ) ner, sine ϱ is finite mesure ner. If ϱ hs no weight ner, eh solution lies in L 2 ((, b); ϱ) ner, sine every solution is lolly bounded. The set r(t min ) of points of regulr type of T min onsists of ll omplex numbers z C suh tht (T min z) is bounded opertor (not neessrily everywhere defined). ell tht dim rn(t min z) is onstnt on every onneted omponent of r(t min ) ([48, Theorem 8.]) nd thus for every z r(t min ). dim rn(t min z) = dim ker(t mx z ) = n(t min ) Lemm 5.4. For eh z r(t min ) there is non-trivil solution of the eqution (τ z)u = 0 whih lies in L 2 ((, b); ϱ) ner. A similr result holds for the endpoint b. Proof. Let z r(t min ) nd first ssume tht τ is regulr t b. If there were no solutions of (τ z)u = 0 whih lie in L 2 ((, b); ϱ) ner, we would hve ker(t mx z) = {0} nd hene n(t min ) = 0, i.e., T min = T mx. But sine there is n f T mx with f(b) = nd f [] (b) = 0, this is ontrdition to Theorem 4.6. In the generl se we tke some (, b) nd onsider the miniml opertor T in L 2 ((, ); ϱ) indued by τ (,). Then z is point of regulr type of T. Indeed, we n extend eh f dom(t ) by setting it equl to zero on (, b) nd obtin funtion f dom(t min ). For these funtions nd some positive onstnt C we hve (T z)f = (T min z)f C f = C f, where is the norm on L 2 ((, ); ϱ). Now sine the solutions of the eqution (τ (,) z)u = 0 re extly the solutions of (τ z)u = 0 restrited to (, ), the lim follows from wht we lredy proved. Corollry 5.5. If z r(t min ) nd τ is in the l.p. se t, then there is unique non-trivil solution of (τ z)u = 0 (up to slr multiples), whih lies in L 2 ((, b); ϱ) ner. A similr result holds for the endpoint b. Proof. If there were two linerly independent solutions in L 2 ((, b); ϱ) ner, τ would be in the l.. se t.

20 20 J. ECKHADT AND G. TESCHL Lemm 5.6. τ is in the l.p. se t if nd only if W (f, g)() = 0, f, g T mx. τ is in the l.. se t if nd only if there is n f T mx suh tht W (f, f )() = 0 nd W (f, g)() 0 for some g T mx. Similr results hold t the endpoint b. Proof. Let τ be in the l.. se t nd u, u 2 be rel fundmentl system of τu = 0 with W (u, u 2 ) =. Both, u nd u 2 lie in T mx ner. Hene there re f, g T mx with f = u nd g = u 2 ner nd f = g = 0 ner b. Then we hve nd W (f, g)() = W (u, u 2 )() = W (f, f )() = W (u, u )() = 0 sine u is rel. Now ssume τ is in the l.p. se t nd regulr t b. Then T mx is twodimensionl extension of T min, sine dim ker(t mx i) = by Corollry 5.5. Let v, w T mx with v = w = 0 in viinity of nd Then v(b) = w [] (b) = nd v [] (b) = w(b) = 0. T mx = T min + spn{v, w}, sine v nd w re linerly independent modulo T min nd do not lie in T min. Then for eh f, g T mx there re f 0, g 0 T min suh tht f = f 0 nd g = g 0 in viinity of nd therefore W (f, g)() = W (f 0, g 0 )() = 0. Now if τ is not regulr t b we tke some (, b). Then for eh f T mx the funtion f (,) lies in the mximl reltion indued by τ (,) nd the lim follows from wht we lredy proved. Lemm 5.7. Let τ be in the l.p. se t both endpoints nd z C\. Then there is no non-trivil solution of (τ z)u = 0 in L 2 ((, b); ϱ). Proof. If u L 2 ((, b); ϱ) is solution of (τ z)u = 0, then u is solution of (τ z )u = 0 nd both, u nd u lie in T mx. Now the Lgrnge identity yields for eh, β (, b) with < β β β W (u, u )(β) W (u, u )() = (z z ) uu dϱ = 2i Im(z) u 2 dϱ. As nd β b, the left-hnd side onverges to zero by Lemm 5.6 nd the right-hnd side onverges to 2i Im(z) u 2, hene u = 0. Theorem 5.8. The defiieny index of the miniml reltion is given by 0, if τ is in the l.. se t no boundry point, n(t min ) =, if τ is in the l.. se t extly one boundry point, 2, if τ is in the l.. se t both boundry points.

21 STUM LIOUVILLE OPEATOS WITH MEASUE-VALUED COEFFICIENTS 2 Proof. If τ is in the l.. se t both endpoints, ll solutions of (τ i)u = 0 lie in L 2 ((, b); ϱ) nd hene in T mx. Therefore, n(t min ) = dim ker(t mx i) = 2. In the se when τ is in the l.. se t extly one endpoint, there is (up to slr multiples) extly one non-trivil solution of (τ i)u = 0 in L 2 ((, b); ϱ), by Corollry 5.5. Now if τ is in the l.p. se t both endpoints, we hve ker(t mx i) = {0} by Lemm 5.7 nd hene n(t min ) = Self-djoint reltions We re interested in the self-djoint restritions of T mx (or equivlent the selfdjoint extensions of T min ). To this end rell tht we introdued the onvenient short-hnd nottion W b (f, g ) = W (f, g )(b) W (f, g )(), f, g T mx. Theorem 6.. Some reltion S is self-djoint restrition of T mx if nd only if (6.) S = {f T mx g S : W b (f, g ) = 0}. Proof. We denote the right-hnd side by S 0. First ssume S is self-djoint restrition of T mx. If f S, then 0 = τf, g f, τg = W b (f, g ) for eh g S, hene f S 0. Now if f S 0, then 0 = W b (f, g ) = τf, g f, τg for eh g S nd hene f S = S. Conversely, ssume tht S = S 0, then S is symmetri sine we hve τf, g = f, τg for eh f, g S. Now let f S T mx, then for eh g S nd hene f S 0 = S. 0 = τf, g f, τg = W b (f, g ) The im of this setion is to determine ll self-djoint restritions of T mx. If both endpoints re in the l.p. se, this is n immedite onsequene of Theorem 5.8. Theorem 6.2. If τ is in the l.p. se t both endpoints then T min = T mx is self-djoint opertor. Next we turn to the se when one endpoint is in the l.. se nd the other one is in the l.p. se. But before we do this, we need some more properties of the Wronskin. Lemm 6.3. Let v T mx suh tht W (v, v )() = 0 nd suppose there is n h T mx with W (h, v )() 0. Then for eh f, g T mx we hve (6.2) nd (6.3) W (f, v )() = 0 W (f, v )() = 0 W (f, v )() = W (g, v )() = 0 W (f, g)() = 0. Similr results hold t the endpoint b. Proof. Choosing f = v, f 2 = v, f 3 = h nd f 4 = h in the Plüker identity, we see tht lso W (h, v)() 0. Now let f = f, f 2 = v, f 3 = v nd f 4 = h, then the Plüker identity yields (6.2), wheres f = f, f 2 = g, f 3 = v nd f 4 = h yields (6.3).

22 22 J. ECKHADT AND G. TESCHL Theorem 6.4. Suppose τ is in the l.. se t nd in the l.p. se t b. Then some reltion S is self-djoint restrition of T mx if nd only if there is v T mx \T min with W (v, v )() = 0 suh tht (6.4) S = {f T mx W (f, v )() = 0}. A similr result holds if τ is in the l.. se t b nd in the l.p. se t. Proof. Beuse of n(t min ) = the self-djoint extensions of T min re preisely the one-dimensionl, symmetri extensions of T min. Hene some reltion S is self-djoint extension of T min if nd only if there is some v T mx \T min with W (v, v )() = 0 suh tht Hene we hve to prove tht S = T min + spn{v}. T min + spn{v} = {f T mx W (f, v )() = 0}. The subspe on the left-hnd side is inluded in the right one beuse of Theorem 4.6 nd W (v, v )() = 0. But if the subspe on the right-hnd side ws lrger, it would be equl to T mx nd hene would imply v T min. Two suh self-djoint restritions re distint if nd only if the orresponding funtions v re linerly independent modulo T min. Furthermore, v n lwys be hosen suh tht v is equl to some rel solution of (τ z)u = 0 with z in some viinity of. By Lemm 6.3 one sees tht ll these self-djoint restritions re rel with respet to the nturl onjugtion. In ontrst to the lssil theory, not ll of this self-djoint restritions of T mx re opertors. We will determine whih of them re multi-vlued in the following setion. It remins to onsider the se when both endpoints re in the l.. se. Theorem 6.5. Suppose τ is in the l.. se t both endpoints. Then some reltion S is self-djoint restrition of T mx if nd only if there re some v, w T mx, linerly independent modulo T min, with (6.5) suh tht (6.6) W b (v, v ) = W b (w, w ) = W b (v, w ) = 0, S = {f T mx W b (f, v ) = W b (f, w ) = 0}. Proof. Sine n(t min ) = 2, the self-djoint extensions of T min re preisely the twodimensionl, symmetri extensions of T min. Hene reltion S is self-djoint restrition of T mx if nd only if there re v, w T mx, linerly independent modulo T min, with (6.5) suh tht Therefore, we hve to prove tht S = T min + spn{v, w}. T min + spn{v, w} = {f T mx W b (f, v ) = W b (f, w ) = 0} = T, where we denote the subspe on the right-hnd side by T. Indeed the subspe on the left-hnd side is ontined in T by Theorem 4.6 nd (6.5). In order to prove tht it is lso not lrger, onsider the liner funtionls F v, F w on T mx defined by F v (f) = W b (f, v ) nd F w (f) = W b (f, w ) for f T mx.

23 STUM LIOUVILLE OPEATOS WITH MEASUE-VALUED COEFFICIENTS 23 The intersetion of the kernels of these funtionls is preisely T. Furthermore, these funtionls re linerly independent. Indeed, ssume, 2 C nd F v + 2 F w = 0, then for ll f T mx we hve 0 = F v (f) + 2 F w (f) = W b (f, v ) + 2 W b (f, w ) = W b (f, v + 2 w ). But by Lemm 4.5 this yields W (f, v + 2 w )() = W (f, v + 2 w )(b) = 0 for ll f T mx nd hene v + 2 w T min. Now sine v, w re linerly independent modulo T min, we get tht = 2 = 0. Now from Lemm 4.3 we infer tht ker F v ker F w nd ker F w ker F v. Hene there exist f v, f w T mx suh tht W(f b v, v ) = W(f b w, w ) = 0 but W(f b v, w ) 0 nd W(f b w, v ) 0. Both, f v nd f w do not lie in T nd re linerly independent. Hene T is t most two-dimensionl extension of the miniml reltion T min. In the se when τ is in the l.. se t both endpoints, we my divide the self-djoint restritions of T mx into two lsses. Indeed, we sy some reltion is self-djoint restrition of T mx with seprted boundry onditions if it is of the form (6.7) S = {f T mx W (f, v )() = W (f, w )(b) = 0}, where v, w T mx re suh tht W (v, v )() = W (w, w )(b) = 0 but W (h, v )() 0 W (h, w )(b) for some h T mx. Conversely, eh reltion of this form is selfdjoint restrition of T mx by Theorem 6.5 nd Lemm 4.5. The remining selfdjoint restritions re lled self-djoint restritions of T mx with oupled boundry onditions. From Lemm 6.3 one sees tht ll self-djoint restritions of T mx with seprted boundry onditions re rel with respet to the nturl onjugtion in L 2 ((, b); ϱ). In the se of oupled boundry onditions this is not the se in generl. Agin we will determine the self-djoint restritions whih re multi-vlued in the next setion. In this setion let w, w 2 T mx with (7.) 7. Boundry onditions W (w, w 2)() = nd W (w, w )() = W (w 2, w 2)() = 0, if τ is in the l.. se t nd (7.b) W (w, w 2)(b) = nd W (w, w )(b) = W (w 2, w 2)(b) = 0, if τ is in the l.. se t b. We will desribe the self-djoint restritions of T mx in terms of the liner funtionls BC, BC 2, BC b nd BC2 b on T mx, defined by BC (f) = W (f, w 2)() nd BC 2 (f) = W (w, f)() for f T mx, if τ is in the l.. se t nd BC b (f) = W (f, w 2)(b) nd BC 2 b (f) = W (w, f)(b) for f T mx, if τ is in the l.. se t b.

24 24 J. ECKHADT AND G. TESCHL Note tht if τ is in the l.. se t some endpoint, suh funtions w, w 2 T mx with (7.) (respetively with (7.b)) lwys exist. Indeed, one my tke them to oinide ner this endpoint with some rel solutions u, u 2 of (τ z)u = 0 with W (u, u 2 ) = for some z nd use Lemm 4.5. In the regulr se these funtionls my tke the form of point evlutions of the funtion nd its qusi-derivtive t the boundry point. Proposition 7.. Suppose τ is regulr t. Then there re w, w 2 T mx with (7.) suh tht the orresponding liner funtionls BC nd BC 2 re given by BC (f) = f() nd BC 2 (f) = f [] () for f T mx. A similr result holds t the endpoint b. Proof. Tke w, w 2 T mx to oinide ner with the rel solutions u, u 2 of τu = 0 with the initil onditions u () = u [] 2 () = nd u[] () = u 2() = 0. Moreover, lso if ϱ hs no weight ner some endpoint, we my hoose speil funtionls. Proposition 7.2. Suppose tht ϱ hs no weight ner, i.e., ϱ >. Then there re w, w 2 T mx with (7.) suh tht the orresponding liner funtionls BC nd BC 2 re given by BC (f) = f( ϱ ) nd BC 2 (f) = f [] ( ϱ ) for f T mx. A similr result holds t the endpoint b. Proof. Tke w, w 2 T mx to oinide ner with the rel solutions u, u 2 of τu = 0 with the initil onditions u ( ϱ ) = u [] 2 ( ϱ ) = nd u [] ( ϱ ) = u 2 ( ϱ ) = 0. Then sine the Wronskin is onstnt on (, ϱ ), we get nd for eh f T mx. BC (f) = W (f, u 2 )( ϱ ) = f( ϱ ) BC 2 (f) = W (u, f)( ϱ ) = f [] ( ϱ ) Using the Plüker identity one esily obtins the equlity W (f, g)() = BC (f)bc 2 (g) BC 2 (f)bc (g), f, g T mx for the Wronskin. Furthermore, for eh v T mx \T min with W (v, v )() = 0 but W (h, v )() 0 for some h T mx, one my show tht there is ϕ [0, π) suh tht for eh f T mx (7.2) W (f, v )() = 0 BC (f) os ϕ BC 2 (f) sin ϕ = 0. Conversely, if some ϕ [0, π) is given, then there is some v T mx \T min with W (v, v )() = 0 but W (h, v )() 0 for some h T mx suh tht (7.3) W (f, v )() = 0 BC (f) os ϕ BC 2 (f) sin ϕ = 0