The Erwin Schrödinger International Boltzmanngasse 9 Institute for Mathematical Physics A-1090 Wien, Austria

ESI The Erwin Schrödinger Interntionl Boltzmnngsse 9 Institute for Mthemticl Physics A-1090 Wien, Austri Weyl Titchmrsh Theory for Sturm Liouville Opertors with Distributionl Potentils Jonthn Eckhrdt Fritz Gesztesy Roger Nichols Gerld Teschl Vienn, Preprint ESI 2381 2012 My 7, 2013 Supported by the Austrin Federl Ministry of Eduction, Science nd Culture Avilble online t http://www.esi.c.t

WEYL TITCHMARSH THEORY FOR STURM LIOUVILLE OPERATORS WITH DISTRIBUTIONAL POTENTIALS JONATHAN ECKHARDT, FRITZ GESZTESY, ROGER NICHOLS, AND GERALD TESCHL Abstrct. We systemticlly develop Weyl Titchmrsh theory for singulr differentil opertors on rbitrry intervls, b R ssocited with rther generl differentil expressions of the type τf = 1 p[f +sf] +sp[f +sf]+qf, r where the coefficients p, q, r, s re rel-vlued nd Lebesgue mesurble on,b, with p 0, r > 0.e. on,b, nd p 1, q, r, s L 1 loc,b;dx, nd f is supposed to stisfy f AC loc,b, p[f +sf] AC loc,b. In prticulr, this setup implies tht τ permits distributionl potentil coefficient, including potentils in H 1 loc,b. We study mximl nd miniml Sturm Liouville opertors, ll self-djoint restrictions of the mximl opertor T mx, or equivlently, ll self-djoint extensions of the miniml opertor T min, ll self-djoint boundry conditions seprted nd coupled ones, nd describe the resolvent of ny self-djoint extension of T min. In ddition, we chrcterize the principl object of this pper, the singulr Weyl Titchmrsh Kodir m-function corresponding to ny self-djoint extension with seprted boundry conditions nd derive the corresponding spectrl trnsformtion, including chrcteriztion of spectrl multiplicities nd miniml supports of stndrd subsets of the spectrum. We lso del with principl solutions nd chrcterize the Friedrichs extension of T min. Finlly, in the specil cse where τ is regulr, we chrcterize the Krein von Neumnn extension of T min nd lso chrcterize ll boundry conditions tht led to positivity preserving, equivlently, improving, resolvents nd hence semigroups. Contents 1. Introduction 2 2. The Bsics on Sturm Liouville Equtions 5 3. Sturm Liouville Opertors 8 4. Weyl s Alterntive 11 5. Self-Adjoint Reliztions 14 6. Boundry Conditions 17 7. The Spectrum nd the Resolvent 21 8. The Weyl Titchmrsh Kodir m-function 24 9. The Spectrl Trnsformtion 27 10. The Spectrl Multiplicity 32 2010 Mthemtics Subject Clssifiction. Primry 34B20, 34B24, 34L05; Secondry 34B27, 34L10, 34L40. Key words nd phrses. Sturm Liouville opertors, distributionl coefficients, Weyl Titchmrsh theory, Friedrichs nd Krein extensions, positivity preserving nd improving semigroups. Opuscul Mth. 33, 467 563 2013. Reserch supported by the Austrin Science Fund FWF under Grnt No. Y330. 1

2 J. ECKHARDT, F. GESZTESY, R. NICHOLS, AND G. TESCHL 11. Non-Principl Solutions, Boundedness from Below, nd the Friedrichs Extension 36 12. The Krein von Neumnn Extension in the Regulr Cse 53 13. Positivity Preserving nd Improving Resolvents nd Semigroups in the Regulr Cse 57 Appendix A. Sesquiliner Forms in the Regulr Cse 67 References 74 1. Introduction The prime motivtion behind this pper is to develop Weyl Titchmrsh theory for singulr Sturm Liouville opertors on n rbitrry intervl, b R ssocited with rther generl differentil expressions of the type τf = 1 p[f +sf] +sp[f +sf]+qf. 1.1 r Here the coefficients p, q, r, s re rel-vlued nd Lebesgue mesurble on,b, with p 0, r > 0.e. on,b, nd p 1, q, r, s L 1 loc,b;dx, nd f is supposed to stisfy f AC loc,b, p[f +sf] AC loc,b, 1.2 with AC loc,b denoting the set of loclly bsolutely continuous functions on,b. The expression f [1] = p[f +sf] will subsequently be clled the first qusiderivtive of f. One notes tht in the generl cse 1.1, the differentil expression is formlly given by τf = 1 pf [ + ps +ps 2 +q ] f. 1.3 r Moreover, in the specil cse s 0 this pproch reduces to the stndrd one, tht is, one obtins, τf = 1 pf +qf. 1.4 r In prticulr, in the cse p = r = 1 our pproch is sufficiently generl to include rbitrry distributionl potentil coefficients from H 1 loc,b = W 1,2 loc,b s the term s 2 cn be bsorbed in q, nd thus even in this specil cse our setup is slightly more generl thn the pproch pioneered by Svchuk nd Shklikov [140], who defined the differentil expression s τf = [f +sf] +s[f +sf] s 2 f, f,[f +sf] AC loc,b. 1.5 Oneobservestht inthis cseq cnbebsorbedinsbyvirtueofthetrnsformtion s s x q. Their pproch requires the dditionl condition s 2 L 1 loc,b;dx. Moreover, since there re distributions in H 1 loc,b which re not mesures, the opertors discussed here re not specil cse of Sturm Liouville opertors with mesure-vlued coefficients s discussed, for instnce, in [41]. We emphsize tht similr differentil expressions hve lredy been studied by Bennewitz nd Everitt [21] in 1983 see lso [42, Sect. I.2]. While some of their discussion is more generl, they restrict their considertions to compct intervls nd focus on the specil cse of left-definite setting. An extremely thorough nd systemtic investigtion, including even nd odd higher-order opertors defined in terms of pproprite qusi-derivtives, nd in the generl cse of mtrixvlued coefficients including distributionl potentil coefficients in the context of Schrödinger-type opertors ws presented by Weidmnn [157] in 1987. In fct, the generl pproch in [21] nd [157] drws on erlier discussions of qusi-derivtives

WEYL TITCHMARSH THEORY 3 in Shin [148] [150], Nimrk [127, Ch. V], nd Zettl [158]. Still, it ppers tht the distributionl coefficients treted in [21] did not ctch on nd subsequent uthors referring to this pper mostly focused on the vrious left nd right-definite spects developed therein. Similrly, it seems likely tht the extrordinry generlity exerted by Weidmnn [157] in his tretment of higher-order differentil opertors obscured the fct tht he lredy delt with distributionl potentil coefficients bck in 1987. There were ctully erlier ppers deling with Schrödinger opertors involving strongly singulr nd oscillting potentils which should be mentioned in this context, such s, Betemn nd Chdn [15], [16], Combescure [28], Combescure nd Ginibre [27], Person [131], Rofe-Beketov nd Hristov [134], [135], nd more recent contribution treting distributionl potentils by Herczyński [72]. In ddition, the cse of point interctions s prticulr distributionl potentil coefficients in Schrödinger opertors received enormous ttention, too numerous to be mentioned here in detil. Hence, we only refer to the stndrd monogrphs by Albeverio, Gesztesy, Høegh-Krohn, nd Holden [2] nd Albeverio nd Kursov [5], nd some of the more recent developments in Albeverio, Kostenko, nd Mlmud [4], Kostenko nd Mlmud [101], [102]. We lso mention the cse of discontinuous Schrödinger opertors originlly considered by Hld [69], motivted by the inverse problem for the torsionl modes of the erth. For recent development in this direction we refer to Shhriri, Jodyree Akbrfm, nd Teschl [147]. It ws not until 1999 tht Svchuk nd Shklikov[140] strted new development for Sturm Liouville resp., Schrödinger opertors with distributionl potentil coefficients in connection with res such s, self-djointness proofs, spectrl nd inverse spectrl theory, oscilltion properties, spectrl properties in the non-selfdjoint context, etc. In ddition to the importnt series of ppers by Svchuk nd Shklikov [140] [146], we lso mention other groups such s Albeverio, Hryniv, nd Mykytyuk [3], Bk nd Shklikov [17], Ben Amr nd Shklikov [18], Ben Amor nd Remling [19], Dvies [32], Djkov nd Mitygin [33] [36], Eckhrdt nd Teschl [41], Fryer, Hryniv, Mykytyuk, nd Perry [45], Gesztesy nd Weikrd [55], Goriunov nd Mikhilets [61], [62], Goriunov, Mikhilets, nd Pnkrshkin [63], Hryniv [73], Kppeler nd Möhr [90], Kppeler, Perry, Shubin, nd Toplov [91], Kppeler nd Toplov [92], Hryniv nd Mykytyuk [74] [81], Hryniv, Mykytyuk, nd Perry [82] [83], Kto [95], Korotyev [99], [100], Mz y nd Shposhnikov [113, Ch. 11], Mz y nd Verbitsky [114] [117], Mikhilets nd Molybog [118] [122], Mirzoev nd Sfnov [123], Mykytyuk nd Trush [126], Sdovnichy [138], [139]. It should be mentioned tht some of the ttrction in connection with distributionl potentil coefficients in the Schrödinger opertor clerly stems from the low-regulrity investigtions of solutions of the Korteweg de Vries KdV eqution. We mention, for instnce, Buckmster nd Koch [24], Grudsky nd Rybkin [68], Kppeler nd Möhr [90], Kppeler nd Toplov [93], [94], nd Rybkin [137]. The cse of strongly singulr potentils t n endpoint nd the ssocited Weyl Titchmrsh Kodir theory for Schrödinger opertors cn lredy be found in the seminl pper by Kodir [98]. A gp in Kodir s pproch ws lter circumvented by Kc [87]. The theory did not receive much further ttention until it ws independently rediscovered nd further developed by Gesztesy nd Zinchenko [56]. This soon led to systemtic development of Weyl Titchmrsh theory for strongly singulr potentils nd we mention, for instnce, Eckhrdt [37], Eckhrdt

4 J. ECKHARDT, F. GESZTESY, R. NICHOLS, AND G. TESCHL nd Teschl [40], Fulton [49], Fulton nd Lnger [50], Fulton, Lnger, nd Luger [51], Kostenko, Skhnovich, nd Teschl [103], [104], [105], [106], nd Kursov nd Luger [109]. In contrst, Weyl Titchmrsh theory in the presence of distributionl potentil coefficients, especilly, in connection with 1.1 resp., 2.2 hs not yet been developed in the literture, nd it is precisely the purpose of this pper to ccomplish just tht under the full generlity of Hypothesis 2.1. Applictions to inverse spectrl theory will be given in [39]. It remins to briefly describe the content of this pper: Section 2 develops the bsics of Sturm Liouville equtions under our generl hypotheses on p, q, r, s, including the Lgrnge identity nd unique solvbility of initil vlue problems. Mximl nd miniml Sturm Liouville opertors re introduced in Section 3, nd Weyl s lterntive is described in Section 4. Self-djoint restrictions of the mximl opertor, or equivlently, self-djoint extensions of the miniml opertor, re the principl subject of Section 5, nd ll self-djoint boundry conditions seprted nd coupled ones re described in Section 6. The resolvent of ll self-djoint extensions nd some of their spectrl properties re discussed in Section 7. The singulr Weyl Titchmrsh Kodir m-function corresponding to ny self-djoint extension with seprted boundry conditions is introduced nd studied in Section 8, nd the corresponding spectrl trnsformtion is derived in Section 9. Clssicl spectrl multiplicity results for Schrödinger opertors due to Kc [85], [86] see lso Gilbert [59] nd Simon [151] re extended to our generl sitution in Section 10. Section 11 dels with vrious pplictions of the bstrct theory developed in this pper. More specificlly, we prove simple nlogue of the clssic Sturm seprtion theorem on the seprtion of zeros of two rel-vlued solutions to the distributionl Sturm Liouville eqution τ λu = 0, λ R, nd show the existence of principl solutions under certin sign-definiteness ssumptions on the coefficient p ner n endpoint of the bsic intervl,b. When τ λ is non-oscilltory t n endpoint, we present sufficient criterion on r nd p for τ to be in the limit-point cse t tht endpoint. This condition dtes bck to Hrtmn [70] in the specil cse p = r = 1, s = 0, nd ws subsequently studied by Rellich [133] in the cse s = 0. This section concludes with detiled chrcteriztion of the Friedrichs extension of T 0 in terms of non-principl solutions, closely following seminl pper by Klf [88] lso in the cse s = 0. In Section 12 we chrcterize the Krein von Neumnn self-djoint extension of T min by explicitly determining the boundry conditions ssocited to it. In our finl Section 13, we derive the qudrtic form ssocited to ech self-djoint extension of T min, ssuming τ is regulr on,b. We then combine this with the Beurling Deny criterion to present chrcteriztion of ll positivity preserving resolvents nd hence semigroups ssocited with self-djoint extensions of T min in the regulr cse. In prticulr, this result confirms tht the Krein von Neumnn extension does not generte positivity preserving resolvent or semigroup. We ctully go step further nd prove tht the notions of positivity preserving nd positivity improving re equivlent in the regulr cse. We lso mention tht n entirely different pproch to Schrödinger opertors ssumed to be bounded from below with mtrix-vlued distributionl potentils, bsed on supersymmetric considertions, hs been developed simultneously in[38]. Finlly, we briefly summrize some of the nottion used in this pper: The Hilbert spces used in this pper re typiclly of the form L 2,b;rxdx with

WEYL TITCHMARSH THEORY 5 sclr product denoted by, r liner in the first fctor, ssocited norm 2,r, nd corresponding identity opertor denoted by I r. Moreover, L 2 c,b;rxdx denotes the spce of squre integrble functions with compct support. In ddition, we use the Hilbert spce L 2 R;dµ for n pproprite Borel mesure µ on R with sclr product nd norm bbrevited by, µ nd 2,µ, respectively. Next, let T be liner opertor mpping subspce of Hilbert spce into nother, with domt, rnt, nd kert denoting the domin, rnge, nd kernel i.e., null spce of T. The closure of closble opertor S is denoted by S. The spectrum, essentil spectrum, point spectrum, discrete spectrum, bsolutely continuous spectrum, nd resolvent set of closed liner opertor in the underlying Hilbert spce will be denoted by σ, σ ess, σ p, σ d, σ c, nd ρ, respectively. The Bnch spces of liner bounded, compct, nd Hilbert Schmidt opertors in seprble complex Hilbert spce re denoted by B, B, nd B 2, respectively. The orthogonl complement of subspce S of the Hilbert spce H will be denoted by S. The symbol SL 2 R will be used to denote the specil liner group of order two over R, tht is, the set of ll 2 2 mtrices with rel entries nd determinnt equl to one. At lst, we will use the bbrevitions iff for if nd only if,.e. for lmost everywhere, nd supp for the support of functions throughout this pper. 2. The Bsics on Sturm Liouville Equtions In this section we provide the bsics of Sturm Liouville equtions with distributionl potentil coefficients. Throughout this pper we mke the following set of ssumptions: Hypothesis 2.1. Suppose,b R nd ssume tht p, q, r, s re Lebesgue mesurble on,b with p 1, q, r, s L 1 loc,b;dx nd rel-vlued.e. on,b with r > 0 nd p 0.e. on,b. Assuming Hypothesis 2.1 nd introducing the set, D τ = { g AC loc,b g [1] = p[g +sg] AC loc,b }, 2.1 the differentil expression τ considered in this pper is of the type, τf = 1 f [1] +sf [1] +qf L 1 r loc,b;rxdx, f D τ. 2.2 The expression f [1] = p[f +sf], f D τ, 2.3 will be clled the first qusi-derivtive of f. Given some g L 1 loc,b;rxdx, the eqution τ zf = g is equivlent to the system of ordinry differentil equtions f s p 1 f 0 f [1] = q zr s f [1]. 2.4 rg From this, we immeditely get the following existence nd uniqueness result. Theorem 2.2. For ech g L 1 loc,b;rxdx, z C, c,b, nd d 1, d 2 C there is unique solution f D τ of τ zf = g with fc = d 1 nd f [1] c = d 2. If, in ddition, g, d 1, d 2, nd z re rel-vlued, then the solution f is rel-vlued.

6 J. ECKHARDT, F. GESZTESY, R. NICHOLS, AND G. TESCHL For ech f,g D τ we define the modified Wronski determinnt Wf,gx = fxg [1] x f [1] xgx, x,b. 2.5 The Wronskin is loclly bsolutely continuous with derivtive Wf,g x = [gxτfx fxτgx]rx, x,b. 2.6 Indeed, this is consequence of the following Lgrnge identity, which is redily proved using integrtion by prts. Lemm 2.3. For ech f, g D τ nd α,β,b we hve β α [gxτfx fxτgx] rxdx = Wf,gβ Wf,gα. 2.7 As consequence, one verifies tht the Wronskin Wu 1,u 2 of two solutions u 1, u 2 D τ of τ zu = 0 is constnt. Furthermore, Wu 1,u 2 0 if nd only if u 1, u 2 re linerly independent. In fct, the Wronskin of two linerly dependent solutions vnishes obviously. Conversely, Wu 1,u 2 = 0 mens tht for c,b there is K C such tht Ku 1 c = u 2 c nd Ku [1] 1 c = u[1] 2 c, 2.8 where we ssume, without loss of generlity, tht u 1 is nontrivil solution i.e., not vnishing identiclly. Now by uniqueness of solutions this implies the liner dependence of u 1 nd u 2. Lemm 2.4. Let z C, u 1, u 2 be two linerly independent solutions of τ zu = 0 nd c,b, d 1,d 2 C, g L 1 loc,b;rxdx. Then there exist c 1, c 2 C such tht the solution u of τ zf = g with fc = d 1 nd f [1] c = d 2, is given for ech x,b by fx = c 1 u 1 x+c 2 u 2 x+ u 1x Wu 1,u 2 u x 2x u 1 tgtrtdt, Wu 1,u 2 c x f [1] x = c 1 u [1] 1 x+c 2u [1] u[1] 1 2 x+ x Wu 1,u 2 u[1] 2 x Wu 1,u 2 x c u 1 tgtrtdt. c x c u 2 tgtrtdt u 2 tgtrtdt 2.9 2.10 If u 1, u 2 is the fundmentl system of solutions of τ zu = 0 stisfying u 1 c = u [1] 2 c = 1 nd u[1] 1 c = u 2c = 0, then c 1 = d 1 nd c 2 = d 2. We omit the strightforwrd clcultions underlying the proof of Lemm 2.4. Another importnt identity for the Wronskin is the well-known Plücker identity: Lemm 2.5. For ll f 1,f 2,f 3,f 4 D τ one hs 0 = Wf 1,f 2 Wf 3,f 4 +Wf 1,f 3 Wf 4,f 2 +Wf 1,f 4 Wf 2,f 3. 2.11 We sy τ is regulr t, if p 1, q, r, nd s re integrble ner. Similrly, we sy τ is regulr t b if these functions re integrble ner b. Furthermore, we sy τ is regulr on,b if τ is regulr t both endpoints nd b.

WEYL TITCHMARSH THEORY 7 Theorem 2.6. Let τ be regulr t, z C, nd g L 1,c;rxdx for ech c,b. Then for every solution f of τ zf = g the limits f = lim x fx nd f [1] = lim x f [1] x 2.12 exist nd re finite. For ech d 1, d 2 C there is unique solution of τ zf = g with f = d 1 nd f [1] = d 2. Furthermore, if g, d 1, d 2, nd z re rel, then the solution is rel. Similr results hold for the right endpoint b. Proof. This theorem is n immedite consequence of the corresponding result for the equivlent system 2.4. Under the ssumptions of Theorem 2.6 one lso infers tht Lemm 2.4 remins vlid even in the cse when c = resp., c = b. We now turn to nlytic dependence of solutions on the spectrl prmeter z C. Theorem 2.7. Let g L 1 loc,b;rxdx, c,b, d 1,d 2 C nd for ech z C let f z be the unique solution of τ zf = g with fc = d 1 nd f [1] c = d 2. Then f z x nd f z [1] x re entire functions of order 1 /2 in z for ech x,b. Moreover, for ech α,β,b with α < β we hve for some constnts C, B R. f z x + f [1] z x Ce B z, x [α,β], z C, 2.13 Proof. The nlyticity prt follows from the corresponding result for the equivlent system. For the remining prt, first note tht becuse of Lemm 2.4 it suffices to consider the cse when g vnishes identiclly. Now if we set for ech z C with z 1 v z x = z f z x 2 + f [1] z x 2, x,b, 2.14 n integrtion by prts shows tht for ech x,b v z x = v z c + x c x c x Employing the elementry estimte c 2 [ z f z t 2 f [1] z t 2] stdt [ z pt 2Re f z tf z t [1] 1 +qt ] dt 2Re zf z tf z [1] t rtdt. 2 f z xf z [1] x z f zx 2 + f z [1] x 2 z we obtin n upper bound for v z : v z x v z c+2 x c 2.15 = v zx z, x,b, 2.16 v z t z ωtdt, x,b, 2.17 where ω = p 1 + q + r + s. Now n ppliction of the Gronwll lemm yields v z x v z ce 2 z x c ωtdt, x,b. 2.18

8 J. ECKHARDT, F. GESZTESY, R. NICHOLS, AND G. TESCHL If, in ddition to the ssumptions of Theorem 2.7, τ is regulr t nd g is integrble ner, then the limits f z nd f z [1] re entire functions of order 1 /2 nd the bound in Theorem 2.7 holds for ll x [,β]. Indeed, this follows since the entire functions f z x nd f z [1] x, x,c re loclly bounded, uniformly in x,c. Moreover, in this cse the ssertions of Theorem 2.7 re vlid even if we tke c = nd/or α =. 3. Sturm Liouville Opertors In this section, we will introduce opertors ssocited with our differentil expression τ in the Hilbert spce L 2,b;rxdx with sclr product f,g r = fxgxrxdx, f, g L 2,b;rxdx. 3.1 First, we define the mximl opertor T mx in L 2,b;rxdx by T mx f = τf, 3.2 f domt mx = { g L 2,b;rxdx g D τ, τg L 2,b;rxdx }. In order to obtin symmetric opertor, we restrict the mximl opertor T mx to functions with compct support by T 0 f = τf, f domt 0 = {g domt mx g hs compct support in,b}. 3.3 Since τ is rel differentil expression, the opertors T 0 nd T mx re rel with respect to the nturl conjugtion in L 2,b;rxdx. We sy some mesurble function f lies in L 2,b;rxdx ner resp., ner b if f lies in L 2,c;rxdx resp., in L 2 c,b;rxdx for ech c,b. Furthermore, we sy some f D τ lies in domt mx ner resp., ner b if f nd τf both lie in L 2,b;rxdx ner resp., ner b. One redily verifies tht some f D τ lies in domt mx ner resp., b if nd only if f lies in domt mx ner resp., b. Lemm 3.1. If τ is regulr t nd f lies in domt mx ner, then the limits f = lim x fx nd f [1] = lim x f [1] x 3.4 exist nd re finite. Similr results hold t b. Proof. Under the ssumptions of the lemm, τf lies in L 2,b;rxdx ner nd since rxdx is finite mesure ner we hve τf L 1,c;rxdx for ech c,b. Hence, the clim follows from Theorem 2.6. The following lemm is consequence of the Lgrnge identity. Lemm 3.2. If f nd g lie in domt mx ner, then the limit Wf,g = lim α Wf,gα 3.5 exists nd is finite. A similr result holds t the endpoint b. If f, g domt mx, then τf,g r f,τg r = Wf,gb Wf,g =: W b f,g. 3.6

WEYL TITCHMARSH THEORY 9 Proof. If f nd g lie in domt mx ner, the limit α of the left-hnd side in eqution 2.7 exists. Hence, the limit in the clim exists s well. Now the remining prt follows by tking the limits α nd β b. If τ is regulr t nd f nd g lie in domt mx ner, then we clerly hve Wf,g = fg [1] f [1] g. 3.7 In order to determine the djoint of T 0 we will rely on the following lemm see, e.g., [153, Lemm 9.3] or [156, Theorem 4.1]. Lemm 3.3. Let V be vector spce over C nd F 1,...,F n,f liner functionls defined on V. Then n F spn{f 1,...,F n } iff kerf j kerf. 3.8 Theorem 3.4. The opertor T 0 is densely defined nd T 0 = T mx. Proof. If we set T 0 = { f1,f 2 L 2,b;rxdx 2 g domt 0 : f 1,T 0 g r = f 2,g r }, 3.9 then from Lemm 3.2 one immeditely sees tht the grph of T mx is contined in T 0. Indeed, for ech f domtmx nd g domt 0 we infer j=1 τf,g r f,τg r = lim β b Wf,gβ lim α Wf,gα = 0, 3.10 since Wf,g hs compct support. Conversely, let f 1, f 2 L 2,b;rxdx such tht f 1,T 0 g r = f 2,g r for ech g domt 0 nd f be solution of τf = f 2. In order to prove tht f 1 f is solution of τu = 0, we will invoke Lemm 3.3. Therefore, consider the liner functionls lg = l j g = f 1 x fxgxrxdx, g L 2 c,b;rxdx, 3.11 u j xgxrxdx, g L 2 c,b;rxdx, j = 1,2, 3.12 whereu j retwosolutionsofτu = 0withWu 1,u 2 = 1ndL 2 c,b;rxdxisthe spce of squre integrble functions with compct support. For these functionls we hve kerl 1 kerl 2 kerl. Indeed, let g kerl 1 kerl 2, then the function ux = u 1 x x u 2 tgtrtdt+u 2 x x u 1 tgtrtdt, x,b, 3.13 is solution of τu = g by Lemm 2.4 nd hs compct support since g lies in the kernels of l 1 nd l 2, hence u domt 0. Then the Lgrnge identity nd the property of f 1,f 2 yield [f 1 x fx]τuxrxdx = τu,f 1 r = u,f 2 r fxτuxrxdx τfxuxrxdx = 0, 3.14

10 J. ECKHARDT, F. GESZTESY, R. NICHOLS, AND G. TESCHL hence g = τu kerl. Now pplying Lemm 3.3 there re c 1, c 2 C such tht [f 1 x fx+c 1 u 1 x+c 2 u 2 x]gxrxdx = 0, 3.15 for ech g L 2 c,b;rxdx. Hence, obviously f 1 D τ nd τf 1 = τf = f 2, tht is, f 1 domt mx nd T mx f 1 = f 2. But this shows tht T 0 ctully is the grph of T mx, which shows tht T 0 is densely defined with djoint T mx. Indeed, if T 0 werenotdenselydefined, therewouldexist0 h L 2,b;rxdx domt 0. Consequently, if f 1,f 2 T 0, then f1,f 2 +h T 0, contrdicting the fct tht T 0 is the grph of n opertor. The opertor T 0 is symmetric by the preceding theorem. The closure T min of T 0 is clled the miniml opertor, T min = T 0 = T 0 = T mx. 3.16 InordertodetermineT min weneedthefollowing lemmonfunctions indomt mx. Lemm 3.5. If f lies in domt mx ner nd f b lies in domt mx ner b, then there exists n f domt mx such tht f = f ner nd f = f b ner b. Proof. Let u 1, u 2 be fundmentl system of τu = 0 with Wu 1,u 2 = 1 nd let α,β,b, α < β such tht the functionls F j g = β α u j xgxrxdx, g L 2,b;rxdx, j = 1,2, 3.17 re linerly independent. First we will show tht there is some u D τ such tht uα = f α, u [1] α = f [1] α, uβ = f b β, u [1] β = f [1] β. 3.18 Indeed, let g L 2,b;rxdx nd consider the solution u of τu = g with initil conditions uα = f α nd u [1] α = f [1] α. 3.19 With Lemm 2.4 one sees tht u hs the desired properties if 1 F2 g u1 β u 2 β fb β c 1 u 1 β c 2 u 2 β = F 1 g u [1] 1 β u[1] 2 β f [1] b β c 1u [1] 1 β c 2u [1] 2 β, 3.20 where c 1, c 2 C re the constnts ppering in Lemm 2.4. But since the functionls F 1, F 2 re linerly independent, we my choose g L 2,b;rxdx such tht this eqution is vlid. Now the function f defined by f x, x,α, fx = ux, x α,β, 3.21 f b x, x β,b, hs the climed properties. b Theorem 3.6. The miniml opertor T min is given by T min f = τf, f domt min = {g domt mx h domt mx : Wg,h = Wg,hb = 0}. 3.22

WEYL TITCHMARSH THEORY 11 Proof. If f domt min = domt mx domt mx, then 0 = τf,g r f,τg r = Wf,gb Wf,g, g domt mx. 3.23 Given some g domt mx, there is g domt mx such tht g = g in vicinity of nd g = 0 in vicinity of b. Therefore, Wf,g = Wf,g Wf,g = 0. Similrly, one obtins Wf,gb = 0 for ech g domt mx. Conversely, if f domt mx such tht for ech g domt mx, Wf,g = Wf,gb = 0, then τf,g r f,τg r = Wf,gb Wf,g = 0, 3.24 hence f domt mx = domt min. Forregulrτ on,bwemychrcterizetheminimlopertorbytheboundry vlues of the functions f domt mx s follows: Corollry 3.7. If τ is regulr t nd f domt mx, then f = f [1] = 0 iff g domt mx : Wf,g = 0. 3.25 A similr result holds t b. Proof. The clim follows from Wf,g = fg [1] f [1] g nd the fct tht one finds g domt mx with prescribed initil vlues t. Indeed, one cn tke g to coincide with some solution of τu = 0 ner. Next we will show tht T min lwys hs self-djoint extensions. Theorem 3.8. The deficiency indices nt min of the miniml opertor T min re equl nd t most two, tht is, nt min = dim rn T min i = dim rn T min +i 2. 3.26 Proof. The fct tht the dimensions re less thn two follows from rn T min ±i = kert mx i, 3.27 becuse there re t most two linerly independent solutions of τ ± iu = 0. Moreover, equlity is due to the fct tht T min is rel with respect to the nturl conjugtion in L 2,b;rxdx. 4. Weyl s Alterntive We sy τ is in the limit-circle l.c. cse t, if for ech z C ll solutions of τ zu = 0 lie in L 2,b;rxdx ner. Furthermore, we sy τ is in the limit-point l.p. cse t if for ech z C there is some solution of τ zu = 0 which does not lie in L 2,b;rxdx ner. Similrly, one defines the l.c. nd l.p. cses t the endpoint b. It is cler tht τ is only either in the l.c. or in the l.p. cse t some boundry point. The next lemm shows tht τ indeed is in one of these cses t ech endpoint, which is known s Weyl s lterntive. Lemm 4.1. If there is z 0 C such tht ll solutions of τ z 0 u = 0 lie in L 2,b;rxdx ner, then τ is in the l.c. cse t. A similr result holds t the endpoint b.

12 J. ECKHARDT, F. GESZTESY, R. NICHOLS, AND G. TESCHL Proof. Let z C nd u be solution of τ zu = 0. If u 1, u 2 re fundmentl system of τ z 0 u = 0 with Wu 1,u 2 = 1, then u 1 nd u 2 lie in L 2,b;rxdx ner by ssumption. Therefore, there is some c,b such tht the function v = u 1 + u 2 stisfies z z 0 c vt 2 rtdt 1 2. 4.1 Since u is solution of τ z 0 u = z z 0 u we hve for ech x,b, ux = c 1 u 1 x+c 2 u 2 x+z z 0 x c u 1 xu 2 t u 1 tu 2 xutrtdt, 4.2 for some c 1, c 2 C by Lemm 2.4. Hence, with C = mx c 1, c 2, one estimtes ux Cvx+ z z 0 vx nd furthermore, using Cuchy Schwrz, c c ux 2 2C 2 vx 2 +2 z z 0 2 vx 2 vt 2 rtdt Now n integrtion yields for ech s,c, c s nd therefore, ut 2 rtdt c c 2C 2 vt 2 rtdt+2 z z 0 2 c 2C 2 vt 2 rtdt+ 1 2 c s c s x x vt ut rtdt, x, c, 4.3 c x ut 2 rtdt. 4.4 2 c vt 2 rtdt ut 2 rtdt s ut 2 rtdt, 4.5 c ut 2 rtdt 4C 2 vt 2 rtdt <. 4.6 Since s,c ws rbitrry, this yields the clim. In prticulr, if τ is regulr t n endpoint, then τ is in the l.c. cse there since ech solution of τ zu = 0 hs continuous extension to this endpoint. With rt min we denote the set of ll points of regulr type of T min, tht is, ll z C such tht T min z 1 is bounded opertor not necessrily everywhere defined. Recll tht dimrnt min z is constnt on every connected component of rt min [156, Theorem 8.1] nd thus dim rn T min z = dimkert mx z = nt min for every z rt min. Lemm 4.2. For ech z rt min there is nontrivil solution of τ zu = 0 which lies in L 2,b;rxdx ner. A similr result holds t the endpoint b. Proof. First ssume tht τ is regulr t b. If there were no solution of τ zu = 0 which lies in L 2,b;rxdx ner, we would hve kert mx z = {0} nd hence nt min = 0, tht is, T min = T mx. But since there is n f domt mx with fb = 1 nd f [1] b = 0, 4.7 this is contrdiction to Theorem 3.6. For the generl cse pick some c,b nd consider the miniml opertor T c in L 2,c;rxdx induced by τ,c. Then z is point of regulr type of T c. Indeed,

WEYL TITCHMARSH THEORY 13 we cn extend ech f c domt c with zero nd obtin function f domt min. For these functions nd some positive constnt C, T c zf c L2,c;rxdx = T min zf 2,r C f 2,r = C f c L2,c;rxdx. 4.8 Now since the solutions of τ,c zu = 0 re exctly the solutions of τ zu = 0 restricted to, c, the clim follows from wht we lredy proved. Corollry 4.3. If z rt min nd τ is in the l.p. cse t, then there is unique nontrivil solution of τ zu = 0 up to sclr multiples, which lies in L 2,b;rxdx ner. A similr result holds t the endpoint b. Proof. If there were two linerly independent solutions in L 2,b;rxdx ner, τ would be l.c. t. Lemm 4.4. τ is in the l.p. cse t if nd only if Wf,g = 0, f, g domt mx. 4.9 τ is in the l.c. cse t if nd only if there is f domt mx such tht Wf,f = 0 nd Wf,g 0 for some g domt mx. 4.10 Similr results hold t the endpoint b. Proof. Let τ be in the l.c. cse t nd u 1, u 2 be rel fundmentl system of τu = 0 with Wu 1,u 2 = 1. Both, u 1 nd u 2 lie in domt mx ner. Hence, there re f, g domt mx with f = u 1 nd g = u 2 ner nd f = g = 0 ner b. Consequently, we obtin Wf,g = Wu 1,u 2 = 1 nd Wf,f = Wu 1,u 1 = 0, 4.11 since u 1 is rel. Now ssume τ is in the l.p. cse t nd regulr t b. Then domt mx is two-dimensionl extension of domt min, since dimkert mx i = 1 by Corollry 4.3. Let v, w domt mx with v = w = 0 in vicinity of nd Then vb = w [1] b = 1 nd v [1] b = wb = 0. 4.12 domt mx = domt min +spn{v,w}, 4.13 since v nd w re linerly independent modulo domt min nd they do not lie in domt min. Then for ech f, g domt mx there re f 0, g 0 domt min such tht f = f 0 nd g = g 0 in vicinity of nd therefore, Wf,g = Wf 0,g 0 = 0. 4.14 Now if τ is not regulr t b we pick some c,b. Then for ech f domt mx, f,c lies in the domin of the mximl opertor induced by τ,c nd the clim follows from wht we lredy proved. Lemm 4.5. Let τ be in the l.p. cse t both endpoints nd z C\R. Then there is no nontrivil solution of τ zu = 0 in L 2,b;rxdx.

14 J. ECKHARDT, F. GESZTESY, R. NICHOLS, AND G. TESCHL Proof. If u L 2,b;rxdx is solution of τ zv = 0, then u is solution of τ zw = 0 nd both u nd u lie in domt mx. Now the Lgrnge identity yields Wu,uβ Wu,uα = z z β α ut 2 rtdt = 2iImz β α ut 2 rtdt. 4.15 If α nd β b, the left-hnd side converges to zero by Lemm 4.4 nd the right-hnd side converges to 2iImz u 2,r, hence u 2,r = 0. Theorem 4.6. The deficiency indices of the miniml opertor T min re given by 0, if τ is l.c. t no boundry point, nt min = 1, if τ is l.c. t exctly one boundry point, 4.16 2, if τ is l.c. t both boundry points. Proof. If τ is in the l.c. cse t both endpoints, ll solutions of τ iu = 0 lie in L 2,b;rxdx nd hence in domt mx. Therefore, nt min = dimkert mx i = 2. In the cse when τ is in the l.c. cse t exctly one endpoint, there is up to sclr multiples exctly one nontrivil solution of τ iu = 0 in L 2,b;rxdx, by Corollry 4.3. Now if τ is in the l.p. cse t both endpoints, we hve kert mx i = {0} by Lemm 4.5 nd hence nt min = 0. 5. Self-Adjoint Reliztions We re interested in the self-djoint restrictions of T mx or equivlently the selfdjoint extensions of T min. To this end, recll tht we introduced the convenient short-hnd nottion Wf,g b = Wf,gb Wf,g, f, g domt mx. 5.1 Theorem 5.1. Some opertor S is self-djoint restriction of T mx if nd only if Sf = τf, f doms = { f domt mx g doms : Wf,g b = 0 }, 5.2 Proof. We denote the right-hnd side of 5.2 by doms 0. First ssume S is self-djoint restriction of T mx. If f doms then 0 = τf,g r f,τg r = W b f,g 5.3 for ech g doms so tht f doms 0. Now if f doms 0, then 0 = W b f,g = τf,g r f,τg r 5.4 for ech g doms, hence f doms = doms. Conversely, ssume doms = doms 0. Then S is symmetric since τf,g r = f,τg r forechf,g doms. Nowletf doms domt min = domt mx, then 0 = τf,g r f,τg r = W b f,g, 5.5 for ech g doms. Hence, f doms 0 = doms, nd it follows tht S is self-djoint. The im of this section is to determine ll self-djoint restrictions of T mx. If both endpoints re in the l.p. cse this is n immedite consequence of Theorem 4.6.

WEYL TITCHMARSH THEORY 15 Theorem 5.2. If τ is in the l.p. cse t both endpoints then T min = T mx is self-djoint opertor. Next we turn to the cse when one endpoint is in the l.c. cse nd the other one is in the l.p. cse. But before we do this, we need some more properties of the Wronskin. Lemm 5.3. Let v domt mx such tht Wv,v = 0 nd suppose there is n h domt mx with Wh,v 0. Then for ech f, g domt mx we hve nd Wf,v = 0 if nd only if Wf,v = 0 5.6 Wf,v = Wg,v = 0 implies Wf,g = 0. 5.7 Similr results hold t the endpoint b. Proof. Choosing f 1 = v, f 2 = v, f 3 = h nd f 4 = h in the Plücker identity, we infer tht lso Wh,v 0. Now let f 1 = f, f 2 = v, f 3 = v nd f 4 = h, then the Plücker identity yields 5.6, wheres f 1 = f, f 2 = g, f 3 = v nd f 4 = h yields 5.7. Theorem 5.4. Suppose τ is in the l.c. cse t nd in the l.p. cse t b. Then some opertor S is self-djoint restriction of T mx if nd only if there is v domt mx \domt min with Wv,v = 0 such tht Sf = τf, f doms = {g domt mx Wg,v = 0}. 5.8 A similr result holds if τ is in the l.c. cse t b nd in the l.p. cse t. Proof. Since nt min = 1, the self-djoint extensions of T min re precisely the one-dimensionl, symmetric extensions of T min. Hence some opertor S is selfdjoint extension of T min if nd only if there is v domt mx \domt min with Wv,v = 0 such tht Hence, we hve to prove tht Sf = τf, f doms = domt min +spn{v}. 5.9 domt min +spn{v} = {g domt mx Wg,v = 0}. 5.10 The subspce on the left-hnd side is included in the right one becuse of Theorem 3.6 nd Wv,v = 0. On the other hnd, if the subspce on the right-hnd side were lrger, then it would coincide with domt mx nd, hence, would imply v domt min. Two self-djoint restrictions re distinct if nd only if the corresponding functions v re linerly independent modulo T min. Furthermore, v cn lwys be chosen such tht v is equl to some rel solution of τ zu = 0 with z R in some vicinity of. It remins to consider the cse when both endpoints re in the l.c. cse. Theorem 5.5. Suppose τ is in the l.c. cse t both endpoints. Then some opertor S is self-djoint restriction of T mx if nd only if there re v, w domt mx, linerly independent modulo domt min, with W b v,v = W b w,w = W b v,w = 0 5.11

16 J. ECKHARDT, F. GESZTESY, R. NICHOLS, AND G. TESCHL such tht Sf = τf, f doms = { g domt mx W b g,v = W b g,w = 0 }. 5.12 Proof. Since nt min = 2 the self-djoint restrictions of T mx re precisely the twodimensionl, symmetric extensions of T min. Hence, n opertor S is self-djoint restriction of T mx if nd only if there re v, w domt mx, linerly independent modulo domt min, with 5.11 such tht Sf = τf, f doms = domt min +spn{v,w}. 5.13 Therefore, we hve to prove tht domt min +spn{v,w} = { f domt mx W b f,v = W b f,w = 0 } := D. 5.14 Indeed, the subspce on the left-hnd side is contined in D by Theorem 3.6 nd 5.11. In order to prove tht it is lso not lrger, consider the liner functionls F v, F w on domt mx defined by F v f = W b f,v nd F w f = W b f,w for f domt mx. 5.15 The intersection of the kernels of these functionls is precisely D. Furthermore, these functionls re linerly independent. Indeed, ssume c 1, c 2 C nd c 1 F v + c 2 F w = 0, then for ll f domt mx, 0 = c 1 F v f+c 2 F w f = c 1 W b f,v+c 2 W b f,w = W b f,c 1 v +c 2 w. 5.16 However, by Lemm 3.5 this yields Wf,c 1 v +c 2 w = Wf,c 1 v +c 2 wb = 0 5.17 for ll f domt mx nd consequently c 1 v +c 2 w domt min. Now since v, w re linerly independent modulo domt min we infer tht c 1 = c 2 = 0 nd Lemm 3.3 implies tht kerf v kerf w nd kerf w kerf v. 5.18 Hence, there exist f v, f w domt mx such tht Wf b v,v = Wf b w,w = 0, but for which Wf b v,w 0 nd Wf b w,v 0. Both f v nd f w do not lie in D nd re linerly independent; hence, D is t most two-dimensionl extension of domt min. In the cse when τ is in the l.c. cse t both endpoints, we my divide the selfdjoint restrictions of T mx into two clsses. Indeed, we sy some opertor S is self-djoint restriction of T mx with seprted boundry conditions if it is of the form Sf = τf, f doms = {g domt mx Wg,v = Wg,wb = 0}, 5.19 where v, w domt mx such tht Wv,v = Ww,wb = 0 but Wh,v 0 Wh,wbforsomeh domt mx. Conversely, echopertorofthisformis self-djoint restriction of T mx by Theorem 5.5 nd Lemm 3.5. The remining selfdjoint restrictions re clled self-djoint restrictions of T mx with coupled boundry conditions.

WEYL TITCHMARSH THEORY 17 6. Boundry Conditions In this section, let w 1, w 2 domt mx with Ww 1,w 2 = 1 nd Ww 1,w 1 = Ww 2,w 2 = 0, 6.1 if τ is in the l.c. cse t nd Ww 1,w 2 b = 1 nd Ww 1,w 1 b = Ww 2,w 2 b = 0, 6.2 if τ is in the l.c. cse t b. We will describe the self-djoint restrictions of T mx in terms of the liner functionls BC 1, BC 2, BC 1 b nd BC2 b on domt mx, defined by BC 1 f = Wf,w 2 nd BC 2 f = Ww 1,f for f domt mx, 6.3 if τ is in the l.c. cse t nd BC 1 bf = Wf,w 2 b nd BC 2 bf = Ww 1,fb for f domt mx, 6.4 if τ is in the l.c. cse t b. If τ is in the l.c. cse t some endpoint, functions with 6.1 resp., with 6.2 lwys exist. Indeed, one my tke them to coincide ner the endpoint with some rel solutions of τ zu = 0 with Wu 1,u 2 = 1 for some z R nd use Lemm 3.5. In the regulr cse these functionls my tke the form of point evlutions of the function nd its qusi-derivtive t the boundry point. Lemm 6.1. Suppose τ is regulr t. Then there re w 1, w 2 domt mx with 6.1 such tht the corresponding liner functionls BC 1 nd BC 2 stisfy BC 1 f = f nd BC 2 f = f [1] for f domt mx. 6.5 The nlogous result holds t the endpoint b. Proof. Tke w 1, w 2 domt mx to coincide ner with the rel solutions u 1, u 2 of τu = 0 with u 1 = u [1] 2 = 1 nd u[1] 1 = u 2 = 0. 6.6 Using the Plücker identity one esily obtins the equlity Wf,g = BC 1 fbc 2 g BC 2 fbc 1 g, f, g domt mx. 6.7 Then for ech v domt mx \domt min with Wv,v = 0 nd Wh,v 0 for some h domt mx, one my show tht there is ϕ [0,π such tht Wf,v = 0 iff BC 1 fcosϕ BC 2 fsinϕ = 0, f domt mx. 6.8 Conversely, if some ϕ [0,π is given, then there exists v domt mx, not belonging to domt min, with Wv,v = 0 nd Wh,v 0 for some h domt mx such tht Wf,v = 0 iff BC 1 fcosϕ BC 2 fsinϕ = 0, f domt mx. 6.9 Using this, Theorem 5.4 immeditely yields the following chrcteriztion of the self-djoint restrictions of T mx in terms of the boundry functionls.

18 J. ECKHARDT, F. GESZTESY, R. NICHOLS, AND G. TESCHL Theorem 6.2. Suppose τ is in the l.c. cse t nd in the l.p. cse t b. Then some opertor S is self-djoint restriction of T mx if nd only if there is some ϕ [0,π such tht Sf = τf, f doms = { g domt mx BC 1 gcosϕ BC 2 gsinϕ = 0 }. A similr result holds if τ is in the l.c. cse t b nd in the l.p. cse t. 6.10 Next we will give chrcteriztion of the self-djoint restrictions of T mx if τ is in the l.c. cse t both endpoints. Theorem 6.3. Suppose τ is in the l.c. cse t both endpoints. Then some opertor S is self-djoint restriction of T mx if nd only if there re mtrices B, B b C 2 2 with rnkb B b = 2 nd B JB = B b JBb 0 1 with J =, 6.11 1 0 such tht { Sf = τf, f doms = g domt mx B BC 1 g BC 1 BCg 2 = B b g b BCb 2g }. 6.12 Proof. If S is self-djoint restriction of T mx, there exist v, w domt mx, linerly independent modulo domt min, with such tht W b v,v = W b w,w = W b v,w = 0, 6.13 doms = { f domt mx W b f,v = W b f,w = 0 }. 6.14 Let B, B b C 2 2 be defined by BC 2 B = v BCv 1 BCw 2 BCw 1 Then simple computtion shows tht nd B b = BC 2 b v BCb 1v BCb 2w BC1 b w. 6.15 B JB = B b JB b iff W b v,v = W b w,w = W b v,w = 0. 6.16 In order to prove rnkb B b = 2, let c 1, c 2 C nd BCv 2 BCw 2 BCc 2 1 v +c 2 w 0 = c 1 BCv 1 BCb 2v +c BCw 1 2 BC 2 BCb 1v b w = BCc 1 1 v +c 2 w BC 2 BCb 1w b c 1v +c 2 w. 6.17 BCb 1c 1v +c 2 w Hence, the function c 1 v + c 2 w lies in the kernel of BC, 1 BC, 2 BCb 1 nd BC2 b, nd therefore, Wc 1 v + c 2 w,f = 0 nd Wc 1 v + c 2 w,fb = 0 for ech f domt mx. This mens tht c 1 v+c 2 w domt min nd hence c 1 = c 2 = 0, since v, w re linerly independent modulo domt min. This proves tht B B b hs rnk two. Furthermore, clcultion yields tht for f domt mx W b f,v = W b f,w = 0 iff B BC 1 f BC 2 f which proves tht S is given s in the clim. = B b BC 1 b f BC 2 b f, 6.18

WEYL TITCHMARSH THEORY 19 Conversely, let B, B b C 2 2 with the climed properties be given. Then there re v, w domt mx such tht BC 2 B = v BCv 1 BC 2 BCw 2 BCw 1 nd B b = b v BCb 1v BCb 2w BC1 b w. 6.19 In order to prove tht v nd w re linerly independent modulo domt min, let c 1, c 2 C nd c 1 v +c 2 w domt min, then BCc 2 1 v +c 2 w BCv 2 BCw 2 0 = BCc 1 1 v +c 2 w BCb 2c 1v +c 2 w = c BCv 1 1 BC 2 BCb 1c b v +c BCw 1 2 BC 2 1v +c 2 w BCb 1v b w. 6.20 BCb 1w Now the rows of B B b re linerly independent, hence c 1 = c 2 = 0. Since gin B JB = B b JB b iff W b v,v = W b w,w = W b v,w = 0, 6.21 the functions v, w stisfy the ssumptions of Theorem 5.5. As bove, one infers once gin tht for f domt mx, BC 1 B f BC 1 BCf 2 = B b f b BCb 2f iff Wf,w b = Wf,w b = 0. 6.22 Hence, S is self-djoint restriction of T mx by Theorem 5.5. As in the preceding section, if τ is in the l.c. cse t both endpoints, we my divide the self-djoint restrictions of T mx into two clsses. Theorem 6.4. Suppose τ is in the l.c. cse t both endpoints. Then some opertor S is self-djoint restriction of T mx with seprted boundry conditions if nd only if there re ϕ, ϕ b [0,π such tht Sf = τf, 6.23 { } f doms = g domt mx BCgcosϕ 1 BCgsinϕ 2 = 0, BCb 1gcosϕ b BCb 2gsinϕ. b = 0 Furthermore, S is self-djoint restriction of T mx with coupled boundry conditions if nd only if there re φ [0,π nd R R 2 2 with detr = 1 i.e., R SL 2 R such tht Sf = τf, f doms = { g domt mx BC 1 b g BCb 2g = e iφ R } BC 1 g BCg 2. 6.24 Proof. Using 6.8 nd 6.9 one esily sees tht the self-djoint restrictions of T mx with seprted boundry conditions re precisely the ones given in 6.23. Hence, we only hve to prove the second clim. Let S be self-djoint restriction of T mx with coupled boundry conditions nd B, B b C 2 2 mtrices s in Theorem 6.3. Then by 6.11 either both of them hve rnk one or both hve rnk two. In the first cse we hve B z = c zw nd B b z = c b zw b 6.25

20 J. ECKHARDT, F. GESZTESY, R. NICHOLS, AND G. TESCHL for some c, c b, w, w b C 2 \{0,0}. Since the vectors w nd w b re linerly independent recll tht rnkb B b = 2 one infers tht BC 1 B f BC 1 BCf 2 = B b f BC 1 b BCb 2f iff B f BC 1 BCf 2 = B b f b BCb 2f = 0. 6.26 In prticulr, B JB = B b JB b iff B JB = B b JB b = 0. 6.27 Nowletv domt mx withbc 2 v = c 1 ndbc 1 v = c 2. Asimpleclcultion yields 0 = B JB = Ww 1,w 2 BC 1 vbc 2 v BC 2 vbc 1 vw w = Ww 1,w 2 Wv,vw w. 6.28 Hence, Wv,v = 0 nd since BCv,BC 1 v 2 = c 2,c 1 0, v domt min. Furthermore, for ech f domt mx, BC 1 B f BCf 2 = BCfBC 1 v BC 2 fbc 2 vw 1 = Wf,vw. 6.29 Similrly one obtins function f domt mx \domt min with Ww,wb = 0 nd BC 1 B b f b BCb 2f = Wf,wbw b, f domt mx. 6.30 However, this shows tht S is self-djoint restriction with seprted boundry conditions. Hence, both mtrices, B nd B b, hve rnk two. If we set B = B 1 b B, then B = JB 1 J nd therefore, detb = 1; hence, detb = e 2iφ for some φ [0,π. If we set R = e iφ B, one infers from the identities b11 b B = 12 = JB 1 J = e 2iφ 0 1 b22 b 21 0 1 b 21 b 22 1 0 b 12 b 11 1 0 6.31 = e 2iφ b11 b 12, b 21 b 22 tht R R 2 2 with detr = 1. Now becuse for ech f domt mx BC 1 B f BC 1 BCf 2 = B b f BC 1 b BCb 2f iff b f BCb 2f = e iφ BC 1 R f BCf 2, 6.32 S hs the climed representtion. Conversely, if S is of the form 6.24, then Theorem 6.3 shows tht it is selfdjoint restriction of T mx. Now if S were self-djoint restriction with seprted boundry conditions, there would exist n f doms\domt min, vnishing in some vicinity of. By the boundry condition we would lso hve BCb 1f = BCb 2f = 0, tht is, f domt min. Hence, S cnnot be self-djoint restriction with seprted boundry conditions. We note tht the seprted self-djoint extensions described in 6.23 re lwys rel tht is, commute with the ntiunitry opertor of complex conjugtion, resp., the nturl conjugtion in L 2,b;rxdx. The coupled boundry conditions in 6.24 re rel if nd only if φ = 0 see lso [160, Sect. 4.2].

WEYL TITCHMARSH THEORY 21 7. The Spectrum nd the Resolvent In this section we will compute the resolvent R z = S zi r 1 of self-djoint restriction S of T mx. First we del with the cse when both endpoints re in the l.c. cse. Theorem 7.1. Suppose τ is in the l.c. cse t both endpoints nd S is self-djoint restriction of T mx. Then for ech z ρs, the resolvent R z is n integrl opertor R z gx = G z x,ygyrydy, x,b, g L 2,b;rxdx, 7.1 with squre integrble kernel G z, tht is, R z is Hilbert-Schmidt opertor, R z B 2 L 2,b;rxdx. For ny two given linerly independent solutions u 1, u 2 of τ zu = 0, there re coefficients m ± ij z C, i, j {1,2}, such tht the kernel is given by { 2 i,j=1 G z x,y = m+ ij zu ixu j y, y,x], 2 i,j=1 m ij zu ixu j y, y [x,b.. 7.2 Proof. Let u 1, u 2 be two linerly independent solutions of τ zu = 0 with Wu 1,u 2 = 1. If g L 2 c,b;rxdx, then R z g is solution of τ zf = g which lies in doms. Hence, from Lemm 2.4 we get for suitble constnts c 1, c 2 C R z gx = u 1 x c 1 + x u 2 tgtrtdt +u 2 x c 2 x u 1 tgtrtdt, 7.3 for x,b. Furthermore, since R z g stisfies the boundry conditions, we obtin BC 1 B R z g BC 1 BCR 2 = B b R z g z g b BCb 2R, 7.4 zg for some suitble mtrices B, B b C 2 2 s in Theorem 6.3. Now since g hs compct support, we infer tht BC 1 R z g c1 BC BCR 2 = u 1 1 +c 2 BCu 1 2 BC 1 z g c 1 BCu 2 1 +c 2 BCu 2 = u 1 BCu 1 2 c1 2 BCu 2 1 BCu 2 2 c 2 c1 = M α, 7.5 c 2 s well s BC 1 b R z g c 1 + b BCb 2R = u 2tgtrtdt BCb 1u 1 zg c 1 + b u 2tgtrtdt BCb 2u 1 c 2 b + u 1tgtrtdt BCb 1u 2 c 2 b u 1tgtrtdt BCb 2u 2 BC 1 = b u 1 BCb 1u 2 c 1 + b BCb 2u 1 BCb 2u u 2tgtrtdt 2 c 2 u 1tgtrtdt b c1 = M β +M u 2tgtrtdt c β 2 u. 7.6 1tgtrtdt

22 J. ECKHARDT, F. GESZTESY, R. NICHOLS, AND G. TESCHL Consequently, b c1 B M α B b M β = B c b M u 2tgtrtdt β 2 u. 7.7 1tgtrtdt Now if B M α B b M β were not invertible, we would hve d1 C 2 d1 d1 \{0,0} with B M α = B d b M β, 7.8 2 d 2 d 2 nd the function d 1 u 1 + d 2 u 2 would be solution of τ zu = 0 stisfying the boundry conditions of S, nd consequently would be n eigenvector with eigenvlue z. However, this would contrdict z ρs, nd it follows tht B M α B b M β must be invertible. Since c1 = B c M α B b M β 1 B b M u 2tgtrtdt β 2 u 1tgtrtdt the constnts c 1 nd c 2 my be written s liner combintions of u 2 tgtrtdt nd, 7.9 u 1 tgtrtdt, 7.10 where the coefficients re independent of g. Using eqution 7.3 one verifies tht R z g hs n integrl-representtion with function G z s climed. The function G z is squre-integrble, since the solutions u 1 nd u 2 lie in L 2,b;rxdx by ssumption. Finlly, since the opertor K z defined K z gx = G z x,ygyrydy, x,b, g L 2,b;rxdx, 7.11 on L 2,b;rxdx, nd the resolvent R z re bounded, the clim follows since they coincide on dense subspce. Since the resolvent R z is compct, in fct, Hilbert Schmidt, this implies discreteness of the spectrum. Corollry 7.2. Suppose τ is in the l.c. cse t both endpoints nd S is self-djoint restriction of T mx. Then S hs purely discrete spectrum, tht is, σs = σ d S. Moreover, 1 < nd dimkers λ 2, λ σs. 7.12 1+λ2 λ σs If S is self-djoint restriction of T mx with seprted boundry conditions or if t lest one endpoint is in the l.c. cse, then the resolvent hs simpler form. Theorem 7.3. Suppose S is self-djoint restriction of T mx with seprted boundry conditions if τ is in the l.c. t both endpoints nd z ρs. Furthermore, let u nd u b be nontrivil solutions of τ zu = 0, such tht { stisfies the boundry condition t if τ is in the l.c. cse t, u lies in L 2 7.13,b;rxdx ner if τ is in the l.p. cse t, nd u b { stisfies the boundry condition t b if τ is in the l.c. cse t b, lies in L 2,b;rxdx ner b if τ is in the l.p. cse t b. 7.14

WEYL TITCHMARSH THEORY 23 Then the resolvent R z is given by where R z gx = G z x,ygyrydy, x,b, g L 2,b;rxdx, 7.15 G z x,y = { 1 u yu b x, y,x], Wu b,u u xu b y, y [x,b. 7.16 Proof. The functions u, u b re linerly independent; otherwise, they would be eigenvectors of S with eigenvlue z. Hence, they form fundmentl system of τ zu = 0. Now for ech f L 2,b;rxdx we define function f g by f g x = Wu b,u 1 u b x x u tgtrtdt+u x x u b tgtrtdt x,b. 7.17 If f L 2 c,b;rxdx, then f g is solution of τ zf = g by Lemm 2.4. Moreover, f g is sclr multiple of u ner nd sclr multiple of u b ner b. Hence, the function f g stisfies the boundry conditions of S nd therefore, R z g = f g. Now if g L 2,b;rxdx is rbitrry nd g n L 2 c,b;rxdx is sequence with g n g s n, we obtin R z g n R z g since the resolvent is bounded. Furthermore, f gn converges pointwise to f g, hence R z g = f g. If τ is in the l.p. cse t some endpoint, then Corollry 4.3 shows tht there is lwys, unique up to sclr multiples, nontrivil solution of τ zu = 0, lying in L 2,b;rxdx ner this endpoint. Also if τ is in the l.c. cse t some endpoint, there exists, unique up to sclr multiples, nontrivil solution of τ zu = 0, stisfying the boundry condition t this endpoint. Hence, functions u nd u b, s in Theorem 7.3 lwys exist. Corollry 7.4. If S is self-djoint restriction of T mx with seprted boundry conditions if τ is in the l.c. t both endpoints, then ll eigenvlues of S re simple. Proof. Suppose λ R is n eigenvlue nd u i doms with τu i = λu i for i = 1,2, tht is, they re solutions of τ λu = 0. If τ is in the l.p. cse t some endpoint, then clerly the Wronskin Wu 1,u 2 vnishes. Otherwise, since both functions stisfy the sme boundry conditions this follows using the Plücker identity. Since the deficiency index of T min is finite, the essentil spectrum of self-djoint reliztions is independent of the boundry conditions, tht is, ll self-djoint restrictions of T mx hve the sme essentil spectrum cf., e.g., [156, Theorem 8.18] We conclude this section by proving tht the essentil spectrum of the self-djoint restrictions of T mx is determined by the behvior of the coefficients in some rbitrrily smll neighborhood of the endpoints. In order to stte this result we need some nottion. Fix some c,b nd denote by τ,c resp., by τ c,b the differentil expression on, c resp., on c, b corresponding to our coefficients restricted to,c resp., to c,b. Furthermore, let S,c resp., S c,b be some self-djoint extension of τ,c resp., of τ c,b. Theorem 7.5. For ech c,b we hve σ e S = σ e S,c σe Sc,b. 7.18,