STURM-LIOUVILLE OPERATORS WITH MATRIX DISTRIBUTIONAL COEFFICIENTS

Methods of Functionl Anlysis nd Topology Vol. 23 (2017), no. 1, pp. 51 59 STURM-LIOUVILLE OPERATORS WITH MATRIX DISTRIBUTIONAL COEFFICIENTS ALEXEI KONSTANTINOV AND OLEKSANDR KONSTANTINOV Abstrct. The pper dels with the singulr Sturm-Liouville expressions l(y) = (py ) +qy with the mtrix-vlued coefficients p, q such tht q = Q, p 1, p 1 Q, Qp 1, Qp 1 Q L 1, where the derivtive of the function Q is understood in the sense of distributions. Due to suitble regulriztion, the corresponding opertors re correctly defined s qusi-differentils. Their resolvent convergence is investigted nd ll self-djoint, mximl dissiptive, nd mximl ccumultive extensions re described in terms of homogeneous boundry conditions of the cnonicl form. 1. Introduction Mny problems of mthemticl physics led to study of Schrödinger-type opertors with strongly singulr (in prticulr distributionl) potentils, see the monogrphs [1, 2] nd the more recent ppers [5, 6, 18, 19] nd references therein. It should be noted tht the cse of very generl singulr Sturm-Liouville opertors defined in terms of pproprite qusi-derivtives hs been considered in [3] (see lso the book [7] nd erlier discussions of qusi-derivtives in [23, 26]). Higher-order qusi-differentil opertors with mtrixvlued vlued singulr coefficients were studied in [8, 9, 21, 25]. The pper [22] strted new pproch to study of one-dimensionl Schrödinger opertors with distributionl potentil coefficients in connection with such res s extension theory, resolvent convergence, spectrl theory nd inverse spectrl theory. An importnt development ws chieved in [11] (see lso [12, 14]), where it ws considered the cse of Sturm-Liouville opertors generted by the differentil expression (1) l(y) = (py ) (t)+q(t)y(t), t J with singulr distributionl coefficients on finite intervl J := (, b). Nmely it ws ssumed tht (2) q = Q, 1/p, Q/p, Q 2 /p L 1 (J,C), where the derivtive of Q is understood in the sense of distributions. A more generl clss of second order qusi-differentil opertors ws recently studied in [19]. In [12, 13] two-term singulr differentil opertors (3) l(y) = i m y (m) (t)+q(t)y(t), t J, m 2, with distributionl coefficient q were investigted. The cse of mtrix opertors of the form (3) ws considered in [17]. Let us lso mention [20] where the deficiency indices 2010 Mthemtics Subject Clssifiction. 34L40, 34B08, 47A10. Key words nd phrses. Sturm-Liouville problem, mtrix qusi-differentil opertor, singulr coefficients, resolvent pproximtion, self-djoint extension. 51

52 ALEXEI KONSTANTINOV AND OLEKSANDR KONSTANTINOV of mtrix Sturm-Liouville opertors with distributionl coefficients on hlf-line were studied. The purpose of the present pper is to extend the results of [11] to the mtrix Sturm- Liouville differentil expressions. In Section 2 we give regulriztion of the forml differentil expression (1) under mtrix nlogue of ssumptions (2). The question of norm resolvent convergence of such singulr mtrix Sturm-Liouville opertors is studied in Section 3. In Section 4 we consider the cse of the symmetric miniml opertor nd describe ll its self-djoint, mximl dissiptive, nd mximl ccumultive extensions. In ddition, we study in detils the cse of seprted boundry conditions. 2. Regulriztion of singulr expression For positive integer s, denote by M s C s s the vector spce of s s mtrices with complex coefficients. Let J := (, b) be finite intervl. Consider Lebesgue mesurble mtrix functions p, Q on J into M s such tht p is invertible lmost everywhere. In wht follows we shll lwys ssume tht (4) p 1, p 1 Q, Qp 1, Qp 1 Q L 1 (J,M s ). This condition should be considered s mtrix (noncommuttive) nlogue of the ssumption (2). In prticulr (4) is vlid under the (more restrictive) condition p 1 (t) (1+ Q(t) 2 )dt <, J which ws (loclly) ssumed in the bove-mentioned pper [20]. Consider the block Shin Zettl mtrix ( ) p (5) A := 1 Q p 1 Qp 1 Q Qp 1 L 1 (J,M 2s ) nd the corresponding qusi-derivtives D [0] y = y, D [1] y = py Qy, D [2] y = (D [1] y) +Qp 1 D [1] y +Qp 1 Qy. For q = Q the Sturm-Liouville expression (1) is defined by (6) l[y] := D [2] y. The qusi-differentil expression (6) gives rise to the mximl qusi-differentil opertor in the Hilbert spce L 2 (J,C s ) =: L 2 } L mx : y l[y], Dom(L mx ) := y L 2 y, D [1] y AC([,b],C s ),D [2] y L 2. The miniml qusi-differentil opertor is defined s restriction of the opertor L mx onto the set } Dom(L min ) := y Dom(L mx ) D [k] y() = D [k] y(b) = 0,k = 0,1. Note tht under the ssumption p 1,q L 1 (J,M s ), the opertors L mx,l min introduced bove coincide with the stndrd mximl nd miniml mtrix Sturm-Liouville opertors. The regulriztion of the formlly djoint differentil expression l + y := (p y ) (t)+q (t)y(t) cn be defined in n nlogous wy (here A = A T is the conjugte trnsposed mtrix to A). Let D k} (k = 0,1,2) be the Shin Zettl qusi-derivtives ssocited with l +. Denote by L + mx nd L + min the mximl nd the miniml opertors generted by this expression on the spce L 2. The following results re proved in [8] (see lso [21]) in the cse of generl qusi-differentil mtrix opertors.

STURM-LIOUVILLE OPERATORS 53 Lemm 1. (Green s formul). For ny y Dom(L mx ), z Dom(L + mx) there holds b ( ) D [2] y z y D 2} z dt = (D [1] y z y D 1} z) t=b t=. Lemm 2. For ny (α 0,α 1 ),(β 0,β 1 ) C 2s there exists function y Dom(L mx ) such tht D [k] y() = α k, D [k] y(b) = β k, k = 0,1. Theorem 1. The opertors L min, L + min, L mx, L + mx re closed nd densely defined on L 2 ([,b],c s ), nd stisfy L min = L + mx, L mx = L + min. In the cse of Hermitin mtrices p nd Q the opertor L min = L + min the deficiency indices (2s, 2s), nd L min = L mx, L mx = L min. is symmetric with 3. Convergence of resolvents Let l ε [y] = D [2] ε y, ε [0,ε 0 ], be qusi-differentil expressions with the coefficients p ε,q ε stisfying (4). These expressions generte the miniml opertors L ε min, Lε mx in L 2. Consider the qusi-differentil opertors L ε y = l ε [y], Dom(L ε ) = y Dom(L ε mx) α(ε)y ε ()+β(ε)y ε (b) = 0}. Here α(ε),β(ε) C 2s 2s re complex mtrices nd } Y ε () := y(),d ε [1] y(), Y ε (b) := } y(b),d ε [1] y(b). Clerly, L ε min L ε L ε mx, ε [0,ε 0 ]. Denote by ρ(l) the resolvent set of the opertor L. Recll tht L ε is sid to converge to L 0 in the norm resolvent sense, L ε R L0, if there is number µ ρ(l 0 ), such tht µ ρ(l ε ) for ll sufficiently smll ε, nd (7) (L ε µ) 1 (L 0 µ) 1 0, ε 0+. It should be noted tht if L ε R L0, then the condition (7) is fulfilled for ll µ ρ(l 0 ) (see [15]). Theorem 2. Suppose ρ(l 0 ) is not empty nd, for ε 0+, the following conditions hold: (1) p 1 ε p 1 0 1 0, (2) p 1 ε Q ε p 1 0 Q 0 1 0, (3) Q ε p 1 ε Q 0 p 1 0 1 0, (4) Q ε p 1 ε Q ε Q 0 p 1 0 Q 0 1 0, (5) α(ε) α(0), β(ε) β(0), where 1 is the norm in the spce L 1 (J,M s ). Then L ε R L0. Essentilly, the proof of Theorem 2 repets the rguments of [11] where the sclr cse s = 1 ws considered. Nevertheless the result seems to be new even in the cse of one-dimensionl Schrödinger opertors with distributionl mtrix-vlued potentils (p ε is the identity mtrix in C s ). Recll the following definition [16].

54 ALEXEI KONSTANTINOV AND OLEKSANDR KONSTANTINOV Definition 1. Denote by M m (J) =: M m, m N, the clss of mtrix-vlued functions R( ;ε) : [0,ε 0 ] L 1 (J,C m m ) prmetrized by ε such tht the solution of the Cuchy problem stisfies the limit condition where is the sup-norm. We need the following result [16]. Z (t;ε) = R(t;ε)Z(t;ε), Z(;ε) = I, lim Z( ;ε) I = 0, ε 0+ Theorem 3. Suppose tht the vector boundry-vlue problem (8) y (t;ε) = A(t;ε)y(t;ε)+f(t;ε), t J, ε [0,ε 0 ], (9) U ε y( ;ε) = 0, where the mtrix-vlued functions A(,ε) L 1 (J,C m m ), the vector-vlued functions f(,ε) L 1 (J,C m ), nd the liner continuous opertors stisfy the following conditions. U ε : C(J;C m ) C m, m N, 1) The homogeneous limit boundry-vlue problem (8),(9) with ε = 0 nd f( ;0) 0 hs only trivil solution; 2) A( ;ε) A( ;0) M m ; 3) U ε U 0 0, ε 0+. Then, for smll enough ε, there exist Green mtrices G(t,s;ε) for problems (8), (9) nd (10) G(, ;ε) G(, ;0) 0, ε 0+, where is the norm in the spce L (J J, C m m ). It follows from [24] tht conditions (1) (4) of Theorem 2 imply A( ;ε) A( ;0) M 2s, where the block Shin Zettl mtrix A( ;ε) is given by the formul ( ) p (11) A( ;ε) := 1 ε Q ε p 1 ε Q ε p 1 ε Q ε Q ε p 1. ε In prticulr A( ; 0) = A (see (5)). The following two lemms reduce Theorem 2 to Theorem 3. Lemm 3. The function y(t) is solution of the boundry-vlue problem (12) l ε [y](t) = f(t;ε) L 2, ε [0,ε 0 ], (13) α(ε)y ε ()+β(ε)y ε (b) = 0, if nd only if the vector-vlued function w(t) = (y(t),d [1] ε y(t)) is solution of the boundry-vlue problem (14) w (t) = A(t;ε)w(t)+ϕ(t;ε), (15) α(ε)w() + β(ε)w(b) = 0, where the mtrix-vlued function A( ;ε) is given by (11) nd ϕ( ;ε) := (0, f( ;ε)).

STURM-LIOUVILLE OPERATORS 55 Proof. Consider the system of equtions (D [0] ε y(t)) = p 1 ε (t)q ε (t)d ε [0] y(t)+p 1 ε (t)d ε [1] y(t), (D [1] ε y(t)) = Q ε (t)p 1 ε (t)q ε (t)d [0] ε y(t) Q ε (t)p 1 ε (t)d [1] ε y(t) f(t;ε). Let y( ) be solution of (12), then the definition of qusi-derivtive implies tht y( ) is solution of this system. On the other hnd, denoting w(t) = (D ε [0] y(t),d ε [1] y(t)) nd ϕ(t;ε) = (0, f(t;ε)), we rewrite this system in the form of eqution (14). Tking into ccount tht Y ε () = w(), Y ε (b) = w(b), one cn see tht the boundry conditions (13) re equivlent to the boundry conditions (15). Lemm 4. Let Green mtrix G(t,s,ε) = (g ij (t,s,ε)) 2 i,j=1 L (J J, C 2s 2s ) exist for the problem (14), (15) for smll enough ε. Then there exists Green function Γ(t, s; ε) for the semi-homogeneous boundry-vlue problem (12), (13) nd Γ(t,s;ε) = g 12 (t,s;ε).e. Proof. According to the definition of Green mtrix, unique solution of the problem (14), (15) cn be written in the form w ε (t) = b G(t,s;ε)ϕ(s;ε)ds, t J. Due to Lemm 3, the ltter equlity cn be rewritten in the form b D ε [0] y ε (t) = g 12 (t,s;ε)( f(s;ε))ds, b D ε [1] y ε (t) = g 22 (t,s;ε)( f(s;ε))ds, where y ε ( ) is unique solution of (12), (13). This implies the sttement of Lemm 4. Proof of Theorem 2. Consider the mtrices Q ε(t),µ = Q ε (t)+µti, p ε(t),µ = p ε (t) corresponding to the opertors L ε +µi. Clerly ssumption (4) nd conditions (1) (4) of Theorem 2 do not depend on µ nd we cn ssume without loss of generlity tht 0 ρ(l 0 ). It follows tht the homogeneous boundry-vlue problem l 0 [y](t) = 0, α(0)y 0 ()+β(0)y 0 (b) = 0 hs only trivil solution. Due to Lemm 3 the homogeneous boundry-vlue problem w (t) = A(t;0)w(t), α(0)w()+β(0)w(b) = 0 lso hs only trivil solution. By conditions (1) (4) of Theorem 2 we hve tht A( ;ε) A( ;0) M 2s, where A( ;ε) is given by formul (11). Thus the sttement of Theorem 2 implies tht the problem (14), (15) stisfies conditions of Theorem 3. It follows tht Green mtrices G(t,s;ε) of the problems (14), (15) exist. Tking into ccount Lemm 4 nd (10) we hve tht L 1 ε L 1 0 L 1 ε L 1 0 HS = Γ(, ;ε) Γ(, ;0) 2 (b ) Γ(, ;ε) Γ(, ;0) 0, ε 0+. Here HS is the Hilbert-Schmidt norm.

56 ALEXEI KONSTANTINOV AND OLEKSANDR KONSTANTINOV Remrk 1. It follows from the proof tht (L ε µ) 1 (L 0 µ) 1 in Hilbert-Schmidt norm for ll µ ρ(l 0 ). 4. Extensions of symmetric miniml opertor In wht follows we dditionlly suppose tht the mtrix functions p, Q nd, consequently, the distribution q = Q re Hermitin. By Theorem 1 the miniml opertor L min is symmetric nd one my consider problem of describing (in terms of homogeneous boundry conditions) ll self-djoint, mximl dissiptive, nd mximl ccumultive extensions of the opertor L min. Let us recll following definition. Definition 2. Let L be closed densely defined symmetric opertor on Hilbert spce H with equl (finite or infinite) deficient indices. A triplet (H,Γ 1,Γ 2 ), where H is n uxiliry Hilbert spce nd Γ 1, Γ 2 re liner mppings of Dom(L ) onto H, is clled boundry triplet of the symmetric opertor L, if (1) for ny f,g Dom(L ), (L f,g) H (f,l g) H = (Γ 1 f,γ 2 g) H (Γ 2 f,γ 1 g) H, (2) for ny f 1,f 2 H there is vector f Dom(L ) such tht Γ 1 f = f 1, Γ 2 f = f 2. The definition of boundry triplet implies tht f Dom(L) if nd only if Γ 1 f = Γ 2 f = 0. A boundry triplet exists for ny symmetric opertor with equl non-zero deficient indices (see [10] nd references therein). The following result is crucil for the rest of the pper. Lemm 5. A triplet (C 2s,Γ 1,Γ 2 ), where Γ 1,Γ 2 re the liner mppings ( ) Γ 1 y := D [1] y(), D [1] y(b), Γ 2 y := (y(),y(b)), from Dom(L mx ) onto C 2s is boundry triplet for the opertor L min. Proof. According to Theorem 1, L min = L mx. Due to Lemm 1, ( ) (L mx y,z) (y,l mx z) = y D [1] z D [1] b y z. But (Γ 1 y,γ 2 z) = D [1] y() z() D [1] y(b) z(b), (Γ 2 y,γ 1 z) = y() D [1] z() y(b) D [1] z(b). This mens tht condition 1) is fulfilled. Condition 2) is true due to Lemm 2. Let K be liner opertor on C 2s. Denote by L K the restriction of L mx onto the set of functions y Dom(L mx ) stisfying the homogeneous boundry condition in the cnonicl form (16) (K I)Γ 1 y +i(k +I)Γ 2 y = 0. Similrly, L K denotestherestrictionofl mx ontothesetofthefunctionsy Dom(L mx ) stisfying the boundry condition (17) (K I)Γ 1 y i(k +I)Γ 2 y = 0. Clerly, L K nd L K re the extensions of L for ny K. Recll tht densely defined liner opertor T on complex Hilbert spce H is clled dissiptive (resp. ccumultive) if I(Tx,x) H 0 (resp. 0), for ll x Dom(T)

STURM-LIOUVILLE OPERATORS 57 nd it is clled mximl dissiptive (resp. mximl ccumultive) if, in ddition, T hs no non-trivil dissiptive (resp. ccumultive) extensions in H. Every symmetric opertor is both dissiptive nd ccumultive, nd every self-djoint opertor is mximl dissiptive nd mximl ccumultive one. Lemm 5 together with results of [10, Ch. 3] leds to the following description of dissiptive, ccumultive, nd self-djoint extensions of L min. Theorem 4. Every L K with K being contrcting opertor in C 2s, is mximl dissiptive extension of L min. Similrly every L K with K being contrcting opertor in C 2s, is mximl ccumultive extension of the opertor L min. Conversely, for ny mximl dissiptive (respectively, mximl ccumultive) extension L of the opertor L min there exists contrcting opertor K such tht L = L K (respectively, L = L K ). The extensions L K nd L K re self-djoint if nd only if K is unitry opertor on C 2s. These correspondences between opertors K} nd the extensions L} re ll bijective. Remrk 2. It follows from Theorem 2 nd Theorem 4 tht the mpping K L K is not only bijective but lso continuous. More ccurtely, if contrcting opertors K n converge to n opertor K, then L Kn R LK. The converse is lso true, becuse the set of contrcting opertors in the spce C 2s is compct set. This mens tht the mpping K (L K λ) 1, Imλ < 0, is homeomorphism for ny fixed λ. Anlogous result is true for L K. Now we pss to description of seprted boundry conditions. Denote by f the germ of continuous function f t the point. Definition 3. The boundry conditions tht define the opertor L L mx re clled seprted if for rbitrry functions y Dom(L) nd ny g,h Dom(L mx ), such tht we hve g,h Dom(L). g = y, g b = 0, h = 0, h b = y b Theorem 5. Let K be liner opertor on C 2s. Boundry conditions (16), (17) defining L K nd L K respectively re seprted if nd only if K is block digonl, i.e., ( ) K 0 (18) K =, 0 K b where K,K b re rbitrry s s mtrices. Proof. We consider the opertors L K, the cse of L K cn be treted in similr wy. The ssumption y c = g c implies tht (19) y(c) = g(c), (D [1] y)(c) = (D [1] g)(c), c [,b]. Let K hve the form (18). Then (16) cn be written in the form of system, (K I)D [1] y()+i(k +I)y() = 0, (K b I)D [1] y(b)+i(k b +I)y(b) = 0. Clerly these conditions re seprted. Conversely, suppose tht boundry conditions (16) re seprted. The mtrix K C 2s 2s cn be written in the form ( ) K11 K K = 12. K 21 K 22 We need to prove tht K 12 = K 21 = 0. Let us rewrite (16) in the form of the system (K11 I)D [1] y() K 12 D [1] y(b)+i(k 11 +I)y()+iK 12 y(b) = 0, K 21 D [1] y() (K 22 I)D [1] y(b)+ik 21 y()+i(k 22 +I)y(b) = 0.

58 ALEXEI KONSTANTINOV AND OLEKSANDR KONSTANTINOV The fct tht the boundry conditions re seprted implies tht function g such tht g = y,g b = 0 lso stisfies this system. It follows from (19) tht for ny y Dom(L K ) ] K 11 [D [1] y()+iy() = D [1] y() iy(), ] K 21 [D [1] y()+iy() = 0. This mens tht for ny y Dom(L K ) (20) D [1] y()+iy() Ker(K 21 ). For ny z = (z 1,z 2 ) C 2s, consider the vectors i(k +I)z nd (K I)z. Due to Lemm 5 nd the definition of the boundry triplet, there exists function y z Dom(L mx ) such tht i(k +I)z = Γ1 y z, (21) (K I)z = Γ 2 y z. Clerly y z stisfies (16) nd y z Dom(L K ). Rewrite (21) in the form of the system i(k 11 +I)z 1 ik 12 z 2 = D [1] y z (), ik 21 z 1 i(k 22 +I)z 2 = D [1] y z (b), (K 11 I)z 1 +K 12 z 2 = y z (), K 21 z 1 +(K 22 I)z 2 = y z (b). The first nd the third equtions of the system bove imply tht for ny z 1 C s D [1] y z ()+iy z () = 2iz 1. Due to (20) we hve tht Ker(K 21 ) = C s nd therefore K 21 = 0. Similrly one cn prove tht K 12 = 0. Remrk 3. It follows from Lemm 5 nd Theorem 1 of [4] tht there is one-to-one correspondence between the generlized resolvents R λ of L min nd the boundry-vlue problems l[y] = λy +h, (K(λ) I)Γ 1 y +i(k(λ)+i)γ 2 y = 0. Here Imλ < 0, h L 2, nd K(λ) is n opertor-vlued function on the spce C 2s, regulr in the lower hlf-plne, such tht K(λ) 1. This correspondence is given by the identity R λ h = y, Imλ < 0. References 1. S. Albeverio, F. Gesztesy, R. Høegh-Krohn, nd H. Holden, Solvble models in quntum mechnics, Texts nd Monogrphs in Physics, Springer-Verlg, New York, 1988. 2. S. Albeverio nd P. Kursov, Singulr perturbtions of differentil opertors, London Mthemticl Society Lecture Note Series, vol. 271, Cmbridge University Press, Cmbridge, 2000. 3. C. Bennewitz nd W.N. Everitt, On second-order left-definite boundry vlue problems, Ordinry differentil equtions nd opertors (Dundee, 1982), Lecture Notes in Mth., vol. 1032, Springer, Berlin, 1983, pp. 31 67. 4. V.M. Bruk, A certin clss of boundry vlue problems with spectrl prmeter in the boundry condition, Mt. Sb. (N.S.) 100 (142) (1976), no. 2, 210 216. 5. J. Eckhrdt, F. Gesztesy, R. Nichols, A. Skhnovich, nd G. Teschl, Inverse spectrl problems for Schrödinger-type opertors with distributionl mtrix-vlued potentils, Differentil Integrl Equtions 28 (2015), no. 5-6, 505 522. 6. J. Eckhrdt, F. Gesztesy, R. Nichols, nd G. Teschl, Supersymmetry nd Schrödinger-type opertors with distributionl mtrix-vlued potentils, J. Spectr. Theory 4 (2014), no. 4, 715 768.

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