REGULARIZATION OF SINGULAR STURM-LIOUVILLE EQUATIONS

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1 REGULARIZATION OF SINGULAR STURM-LIOUVILLE EQUATIONS ANDRII GORIUNOV, VLADIMIR MIKHAILETS Abstrct. The pper dels with the singulr Sturm-Liouville expressions l(y) = (py ) + qy with the coefficients q = Q, 1/p, Q/p, Q 2 /p L 1, where the derivtive of the function Q is understood in the sense of distributions. Due to new regulriztion, the corresponding opertors re correctly defined s qusi-differentils. Their resolvent pproximtion is investigted nd ll self-djoint nd mximl dissiptive extensions nd generlized resolvents re described in terms of homogeneous boundry conditions of the cnonicl form. 1. Introduction This pper studies opertors generted by the differentil expressions (1) l(y) = (py ) (t) + q(t)y(t), t J on finite intervl J := (, b). If the coefficients in (1) re rel-vlued nd (2) q C(J ), 0 < p C 1 (J ), then the eqution l(y) = f is differentil Sturm-Liouville eqution tht hs been investigted quite comprehensively. A modern exposition of the clssicl Sturm-Liouville theory my be found in mny studies. Principl sttements of this theory remin true under the weker ssumptions (3) q, 1/p L 1 (J, C) =: L 1, see [1] nd references therein. This is chieved through regulriztion of the expression l(y) pplying Shin-Zettl qusi-derivtives. They were introduced in [2] nd lter generlized in [3], see lso [4]. A further essentil development of tht pproch ws chieved in the pper [5]. It ws proved there tht if p(t) 1, then the condition on q my be significntly wekened. Nmely, it is sufficient to suppose tht (4) p(t) 1, q = Q, Q L 2 (J, C) =: L 2, where the derivtive of the function Q is understood in the sense of distributions. Note tht the one-dimension Schrödinger opertors with potentils tht re Rdon mesures were introduced nd investigted long before tht by physicists pplying opertor theory methods (see [6] nd references therein) Mthemtics Subject Clssifiction. Primry 34L40; Secondry 34B08, 47A10. Key words nd phrses. Sturm-Liouville problem, qusi-differentil expression, singulr coefficients, resolvent pproximtion, self-djoint extension, generlized resolvent. 1

2 2 ANDRII GORIUNOV, VLADIMIR MIKHAILETS The min gol of this pper is to define nd investigte Sturm-Liouville opertors on finite intervl J under the ssumptions more generl thn those in (3) nd (4), (5) q = Q, 1/p, Q/p, Q 2 /p L 1. To chieve this gol, in Section 2, we propose new regulriztion of the forml differentil expression (1) under ssumptions (5) by mens of Shin-Zettl qusi-derivtives. We lso define the corresponding mximl nd miniml opertors on the Hilbert spce L 2. If conditions (3) hold, then these opertors coincide with the clssicl ones nd, under ssumptions (4), they re identicl to the opertors introduced in [5]. Section 3 shows tht, in the cse of two-point boundry conditions, resolvents of the constructed opertors my be pproximted in the sense of the norm with resolvents of other Sturm-Liouville opertors; for instnce, ones tht hve more regulr coefficients. In Section 4, the miniml opertor is supposed to be symmetric nd ll its self-djoint extensions re described in terms of the homogeneous boundry conditions of the cnonicl form. In ddition, in Section 5, ll mximl dissiptive extensions nd generlized resolvents of the miniml symmetric opertor re described in the sme form. Extensions in Sections 4 nd 5 re described by pplying the boundry triplet theory (see [7] nd references therein). They re prmetrized by certin clsses of opertors on C 2, nd this prmetriztion is bijective nd continuous. Also, seprted boundry conditions re singled out. Note tht in the cse where p(t) 1, the results of Sections 3 nd 4 improve the corresponding results of [5] where stronger conditions re required for the pproximtion, nd self-djoint extensions re described on the bsis of the Glzmn-Krein-Nimrk theory. Results of Section 5 del with the questions not considered in [5]. 2. Regulriztion of singulr expression Consider the forml differentil expression (1), ssuming tht conditions (5) hold. We introduce the qusi-derivtives D [0] y = y, D [1] y = py Qy, D [2] y = (D [1] y) + Q p D[1] y + Q2 p y. Then expression (1) is defined to be the qusi-differentil expression l[y] := D [2] y. Definition 1. A solution of the Cuchy problem for the resolvent eqution (6) l[y] λy = f L 2, y(c) = α 1, (D [1] y)(c) = α 2, where c J nd α 1, α 2 re rbitrry complex numbers, is defined to be the first component of the solution of the Cuchy problem for the correspondent system of the first order differentil equtions (7) w (t) = A λ (t)w(t) + ϕ(t), w(c) = (α 1, α 2 ),

3 REGULARIZATION OF SINGULAR STURM-LIOUVILLE EQUATIONS 3 where w(t) = (y(t), D [1] y(t)), the mtrix-vlued function is ( ) nd ϕ(t) := (0, f(t)). A λ (t) := Q p 1 p Q2 p λ Q p L 2 2 1, Lemm 1. Problem (6), with ssumptions (5), hs only unique solution defined on J. Proof of Lemm 1. Problem (7) with A λ ( ) L hs only unique solution for ny c J nd (α 1, α 2 ) C 2 due to Theorem [1]. This implies the sttement of Lemm 1 by Definition 1. The qusi-differentil expression l[y] gives rise to the mximl qusi-differentil opertor L mx : y l[y], Dom(L mx ) := { y L 2 y, D [1] y AC(J, C), D [2] y L 2 } on the Hilbert spce L 2 (see [3, 4]). 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 }. Remrk 1. One cn esily see tht if Q is replced with Q := Q + c, c C, then the opertors L mx, L min do not chnge. If the coefficients in (1) stisfy (3), then the opertors L mx, L min introduced bove coincide with the usul mximl nd miniml Sturm-Liouville opertors [1]. Consider the expression l + (y) = (py ) (t) + q(t)y(t), formlly djoint to (1), where the br denotes complex conjugtion. Denote by L + mx nd L + min the mximl nd the miniml opertors generted by this expression on the spce L 2. Then results of this section, together with results of [4] for generl qusi-differentil expressions, yield following theorem. Theorem 1. The opertors L min, L + min, L mx, L + mx re closed nd densely defined on the spce L 2, L min = L + mx, L mx = L + min. In the cse where p nd Q re rel-vlued, the opertor L min = L + min is symmetric with the deficiency index (2, 2), nd L min = L mx, L mx = L min. 3. Approximtion of resolvent Consider the clss of qusi-differentil expressions l ε [y] = D [2] ε y with the coefficients p ε, q ε = Q ε, ε [0, ε 0 ]. On the Hilbert spce L 2 with norm 2, these expressions generte the opertors L ε min, L ε mx for every ε. Let α(ε), β(ε) C 2 2 be mtrices nd consider the vectors Y ε () := { y(), D [1] ε y() }, Y ε (b) := { y(b), D [1] ε y(b) } C 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}.

4 4 ANDRII GORIUNOV, VLADIMIR MIKHAILETS It is evident tht L ε min L ε L ε mx, ε [0, ε 0 ]. We denote by ρ(l) the resolvent set of the opertor L. Recll tht the opertors L ε converge to the opertor L 0 in the sense of the norm resolvent convergence, L R ε L 0, if the there is number µ C such tht µ ρ(l 0 ) nd µ ρ(l ε ) (for ll sufficiently smll ε), nd (L ε µ) 1 (L 0 µ) 1 0, ε 0 +. This definition does not depend on the choice of the point µ ρ(l 0 ) [8]. In the cse where the mtrices α(ε), β(ε) do not depend on ε nd p ε (t) 1, it is shown in [5] tht if Q ε Q for ε 0+ nd the resolvent set of the opertor L 0 is not empty, then R L 0. The following theorem generlizes this result. L ε Theorem 2. Suppose ρ(l 0 ) is not empty nd, for ε 0+, the following conditions hold: (1) 1/p ε 1/p 0 1 0, (2) Q ε /p ε Q 0 /p 0 1 0, (3) Q 2 ε/p ε Q 2 0/p 0 1 0, (4) α(ε) α(0), β(ε) β(0), where 1 is the norm in the spce L 1 (J, C). Then L ε R L 0. Remrk 2. In the cse where p ε (t) 1, condition (1) is utomticlly fulfilled nd conditions (2) nd (3) re weker thn the ssumption tht Q ε Q To prove Theorem 2 we will need some uxiliry results. We strt by introducing the following definition ([9, 10]). Definition 2. Denote by M m (J ) =: M m, m N, the clss of mtrix-vlued functions R( ; ε) : [0, ε 0 ] L m m 1 prmetrized by ε such tht the solution of the Cuchy problem stisfies the limit condition where C is the sup-norm. Z (t; ε) = R(t; ε)z(t; ε), Z(; ε) = I m, lim Z( ; ε) I m C = 0, ε 0+ In pper [10], the following generl result is estblished: 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 m m 1, the vector-vlued functions f(, ε) L m 1, nd the liner continuous opertors U ε : C(J ; C m ) C m, m N,

5 stisfy the following conditions. REGULARIZATION OF SINGULAR STURM-LIOUVILLE EQUATIONS 5 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, on the squre J J, (10) G(, ; ε) G(, ; 0) 0, ε 0+, where is the norm in the spce L. Remrk 3. Condition 3) in Theorem 3 cnnot be replced with the weker condition on the opertor U ε to strongly converge, U ε s U 0 [10]. However, one cn esily see tht, for the two-point boundry opertors U ε y := B 1 (ε)y() + B 2 (ε)y(b), B k (ε) C m m, k = 1, 2, both the strong convergence nd the norm convergence conditions re equivlent to B k (ε) B k (0) 0, ε 0+, k = 1, 2. There re different sufficient conditions for the mtrix-vlued function R( ; ε) to belong to M m. In prticulr, the results of [11] give tht conditions (1), (2), (3) of Theorem 2 imply A( ; ε) A( ; 0) M 2, where the mtrix-vlued function A( ; ε) is given by the formul ( ) Qε /p (11) A( ; ε) := ε 1/p ε Q 2 L ε/p ε Q ε /p ε Before proving Theorem 2, we will need the following two lemms require to reduce Theorem 2 to Theorem 3. Lemm 2. 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 boundryvlue 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)). Proof of Lemm 2. Consider the system of equtions (D ε [0] y(t)) = Q ε(t) p ε (t) D[0] ε y(t) + 1 p ε (t) D[1] ε y(t), (D [1] ε y(t)) = Q2 ε(t) p ε (t) D[0] ε y(t) Q ε(t) p ε (t) D[1] ε y(t) f(t; ε).

6 6 ANDRII GORIUNOV, VLADIMIR MIKHAILETS 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). Due to Lemm 2, tht sttement tht (U) the homogeneous boundry-vlue problem l 0 [y](t) = 0, α(0)y 0 () + β(0)y 0 (b) = 0, hs only trivil solution implies tht the homogeneous boundry-vlue problem hs only trivil solution. Lemm 3. Let Green mtrix w (t) = A(t; 0)w(t), α(0)w() + β(0)w(b) = 0 G(t, s, ε) = (g ij (t, s)) 2 i,j=1 L 2 2 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) Proof of Lemm 3. According to the definition of Green mtrix, unique solution of problem (14), (15) cn be written in the form w ε (t) = b.e. G(t, s; ε)ϕ(s; ε)ds, t J. Due to Lemm 2, 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 the problem (12), (13). This implies the sttement of Lemm 3. Proof of Theorem 2. Note tht, due to the equlity (Q ε + µ) 2 /p ε (Q 0 + µ) 2 /p 0 = (Q 2 ε/p ε Q 2 0/p 0 ) + 2µ(Q ε /p ε Q 0 /p 0 ) + µ 2 (1/p ε 1/p 0 ), where µ C, conditions (1) (3) of Theorem 2 imply tht we cn ssume without loss of generlity tht 0 ρ(l 0 ). We need to show tht sup L 1 ε f L 1 0 f 0, ε 0+. f 2 =1 The eqution L 1 ε f = y ε is equivlent to L ε y ε = f, i. e., y ε is solution of problem (12), (13). Also the sttement (U) is verified due to 0 ρ(l 0 ). From the conditions 1) 3) of Theorem 2, it follows tht A( ; ε) A( ; 0) M 2, where A( ; ε) is given by formul (11). Thus sttement of Theorem 2 implies tht the problem (14), (15) stisfies conditions of Theorem 3. This mens

7 REGULARIZATION OF SINGULAR STURM-LIOUVILLE EQUATIONS 7 tht Green mtrices G(t, s; ε) of the problems (14), (15) exist nd the limit reltion (10) is stisfied. Tking into ccount Lemm 3 this yields limit equlity Then L 1 ε which proves Theorem 2. L 1 0 = sup f 2 =1 Γ(, ; ε) Γ(, ; 0) 0, ε 0 +. b (b ) 1/2 [Γ(t, s; ε) Γ(t, s; 0)] f(s) ds 2 sup f 2 =1 b Γ(t, s; ε) Γ(t, s; 0) f(s) ds C (b ) Γ(, ; ε) Γ(, ; 0) 0, ε 0+, For the cse p ε (t) 1, sttement stronger thn Theorem 2 ws proved in [12]. 4. Self-djoint boundry conditions In wht follows we will require the functions p, Q nd, consequently, the distribution q = Q to be rel-vlued. In this cse, the expression l[y] is formlly self-djoint [4] nd, ccording to Theorem 1, the miniml opertor L min is symmetric. So one my pose problem of describing (in terms of homogeneous boundry conditions) ll extensions of the opertor L min tht re self-djoint in the spce L 2. To give n nswer to this question, we will pply the concept of the boundry triplet. Let us recll following definition. Definition 3. Let L be closed densely defined symmetric opertor in Hilbert spce H with equl (finite or infinite) deficient indices. The triplet (H, Γ 1, Γ 2 ), where H is n uxiliry Hilbert spce nd Γ 1, Γ 2 re the 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 [7] nd references therein). It is not unique. The following result is crucil for the rest of the pper. Bsic Lemm. Triplet (C 2, Γ 1, Γ 2 ), where Γ 1, Γ 2 re the liner mppings (16) Γ 1 y := ( D [1] y(), D [1] y(b) ), Γ 2 y := (y(), y(b)), from Dom(L mx ) onto C 2 is boundry triplet for the opertor L min. For convenience, we introduce the following nottion. Definition 4. Denote by L K the restriction of the opertor L mx onto the set of functions y(t) Dom(L mx ) stisfying the homogeneous boundry condition in the cnonicl form (17) (K I) Γ 1 y + i (K + I) Γ 2 y = 0, where K is ny bounded opertor on the spce C 2.

8 8 ANDRII GORIUNOV, VLADIMIR MIKHAILETS Bsic Lemm together with results of [7, Ch. 3] gives the following description of ll selfdjoint extensions of L min. Theorem 4. Every L K, with K being unitry opertor on the spce C 2, is self-djoint extension of the opertor L min. Conversely, for ny self-djoint extension L of the opertor L min there is unitry opertor K such tht L = L K. This correspondence between unitry opertors {K} nd self-djoint extensions { L} is bijective. We strt proof of the Bsic Lemm with the following two lemms tht re specil cses of the corresponding results for generl qusi-differentil expressions (see [4]). Lemm 4. Suppose y, z Dom(L mx ). Then b ( ) D [2] y z y D [2] z dt = ( ) D [0] y D [1] z + D [1] y D [0] b z. Lemm 5. Suppose tht {α 0, α 1 }, {β 0, β 1 } re rbitrry sets of complex numbers. Then there is function y Dom(L mx ) such tht D [k] y() = α k, D [k] y(b) = β k, k = 0, 1. Proof of the Bsic Lemm. To prove the Bsic Lemm, we need to prove tht the triplet (C 2, Γ 1, Γ 2 ) stisfies conditions 1) nd 2) in the definition of the boundry triplet for the opertor L min. According to Theorem 1, L min = L mx. Due to Lemm 4, ( ) (L mx y, z) (y, L mx z) = D [0] y D [1] z D [1] y D [0] b z. But (Γ 1 y, Γ 2 z) = D [1] y() D [0] z() D [1] y(b) D [0] z(b), (Γ 2 y, Γ 1 z) = D [0] y() D [1] z() D [0] y(b) D [1] z(b). This mens tht condition 1) is fulfilled. Condition 2) is true due to Lemm 5. Proof of Theorem 4. The clim in Theorem 4 follows from the Bsic Lemm nd Theorem 1.6 Ch. 3 [7] for the boundry triplet of n bstrct symmetric opertor. Remrk 4. Theorem 2, together with Theorem 4, implies tht the mpping K L K is not only bijective but lso continuous. More ccurtely, if unitry opertors K n converge to n opertor K, then (LK λ) 1 (L Kn λ) 1 0, n, Im λ 0. The converse is lso true, becuse the set of unitry opertors in the spce C 2 is compct set. This mens tht the mpping is homeomorphism for ny fixed λ C \ R. K (L K λ) 1, Im λ 0, Now we pss to description of seprted self-djoint boundry conditions for expression (1). Denote by f the germ of continuous function f t the point.

9 REGULARIZATION OF SINGULAR STURM-LIOUVILLE EQUATIONS 9 Definition 5. The boundry conditions tht define the opertor L L mx re clled seprted if for rbitrry functions y Dom(L) nd g, h Dom(L mx ), g, h Dom(L) if g = y, g b = 0, h = 0, h b = y b. Theorem 5. Self-djoint boundry conditions (17) re seprted if nd only if the mtrix K is of the form (18), where K, K b C nd K = K b = 1. A proof of Theorem 5 is bsed on the following lemm. Lemm 6. Boundry conditions of the form (17), with K being ny C 2 2 -mtrix re seprted if nd only if ( ) K 0 (18) K =, 0 K b where K, K b C. Proof of Lemm 6. It is evident tht y c = g c implies (19) y(c) = g(c), (D [1] y)(c) = (D [1] g)(c), c [, b]. Let the mtrix K hve the form (18) in boundry condition (17). Then conditions (17) cn be written in the form of system, { (K 1)D [1] y() + i(k + 1)y() = 0, (K b 1)D [1] y(b) + i(k b + 1)y(b) = 0. It is evident tht these boundry conditions re seprted. Inversely, suppose tht the boundry conditions (17) re seprted. The mtrix K C 2 2 cn be written in the form ( ) K11 K K = 12. K 21 K 22 We need to prove K 12 = K 21 = 0. Let us rewrite the boundry conditions (17) in the form of the system { (K11 1)D [1] y() K 12 D [1] y(b) + i(k )y() + ik 12 y(b) = 0, K 21 D [1] y() (K 22 1)D [1] y(b) + ik 21 y() + i(k )y(b) = 0. The fct tht the boundry conditions re seprted implies tht function g such tht g = y, g b = 0 lso stisfies this system. Due to equlities (19) this gives { K11 [ D [1] y() + iy() ] = D [1] y() iy(), K 21 [ D [1] y() + iy() ] = 0 for ny y Dom(L K ). This mens tht either K 21 = 0 or D [1] y() + iy() = 0 for ny y Dom(L K ). Suppose K Let us return to the boundry conditions (17). For ny F = (F 1, F 2 ) C 2, consider the vectors i (K + I) F nd (K I) F. Due to the Bsic Lemm nd the definition of the boundry triplet, there exists function y F Dom(L mx ) such tht { i (K + I) F = Γ1 y F, (20) (K I) F = Γ 2 y F.

10 10 ANDRII GORIUNOV, VLADIMIR MIKHAILETS A simple clcultion shows tht y F stisfies the boundry conditions (17) nd, therefore, y F Dom(L K ). We cn rewrite (20) in the form of the system i(k )F 1 ik 12 F 2 = D [1] y F (), ik 21 F 1 i(k )F 2 = D [1] y F (b), (K 11 1)F 1 + K 12 F 2 = y F (), K 21 F 1 + (K 22 1)F 2 = y F (b). The first nd the third equtions of the system bove imply tht 0 = D [1] y F () + iy F () = 2iF 1 for ny F 1 C. We rrived t contrdiction, therefore, K 21 = 0. Similrly one my prove K 12 = 0. Proof of Theorem 5. Due to Lemm 6, we only need to remrk tht mtrix of the form (18) is unitry if nd only if K = K b = Non-self-djoint boundry conditions nd generlized resolvents Recll the following definition. Definition 6. A densely defined liner opertor L on complex Hilbert spce H is clled dissiptive if Im (Lf, f) H 0, f Dom(L) nd it is clled mximl dissiptive if, besides this, L hs no nontrivil dissiptive extensions on the spce H. For instnce, every symmetric opertor is dissiptive nd every self-djoint opertor is mximl dissiptive one. Thus, if the miniml opertor L min is symmetric, then one cn stte the problem of describing its mximl dissiptive extensions. According to Phillips Theorem [7, 13], every mximl dissiptive extension of symmetric opertor is restriction of its djoint opertor. Therefore, every mximl dissiptive extension of the opertor L min is restriction of opertor L mx. Prmetric bijective description of the clss of mximl dissiptive extensions of the symmetric qusi-differentil opertor L min is given by the following theorem. Theorem 6. Every L K, with K being contrcting opertor on the spce C 2, is mximl dissiptive extension of the opertor L min. Conversely, for ny mximl dissiptive extension L of the opertor L min there exists contrcting opertor K such tht L = L K. This correspondence between contrcting opertors {K} nd the mximl dissiptive extensions { L} is bijective. Proof of Theorem 6. Theorem 6 is direct consequence of Bsic Lemm nd Theorem 1.6 Ch. 3 [7] for the boundry triplet of n bstrct symmetric opertor. Remrk 5. The mpping K (L K λ) 1, Im λ < 0, for ny fixed λ is homeomorphism (see Remrk 4). Theorem 7. Dissiptive boundry conditions (17) re seprted if nd only if the mtrix K is of the form (18), where K 1, K b 1. Proof of Theorem 7. As in the proof of Theorem 5, due to Lemm 6, we only need to remrk tht the mtrix K of the form (18) is contrcting opertor on C 2 if nd only if K 1, K b 1.

11 REGULARIZATION OF SINGULAR STURM-LIOUVILLE EQUATIONS 11 Recll the following definition. Definition 7. A generlized resolvent of closed symmetric opertor L is the opertor-vlued function R λ of the complex prmeter λ C\R which cn be represented in the form R λ f = P + ( L + λi +) 1 f, f H, where L + is self-djoint extension of the opertor L, generlly, on the spce H + which is wider thn H, I + is the identity opertor on H +, nd P + is the orthogonl projection opertor from H + onto H. The opertor-vlued function R λ opertor L if nd only if (R λ f, g) H = (Im λ 0) is generlized resolvent of symmetric + d (F µ f, g), f, g H, µ λ where F µ is the generlized spectrl function of the opertor L. In other words, the opertorvlued function F µ should hve following properties [14]: 1 0. For µ 2 > µ 1, the difference F µ2 F µ1 is bounded non-negtive opertor, 2 0. F µ+ = F µ for ny rel µ, 3 0. for ny x H, lim F µx H = 0, µ lim F µx x H = 0. µ + A prmetric inner description of ll generlized resolvents of the opertor L min is given by the following theorem. Theorem 8. There is one-to-one correspondence between the generlized resolvents of the opertor 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, where λ C, Im λ < 0, h(x) L 2, nd K(λ) is n opertor-vlued function into the spce C 2, regulr in the lower hlf-plne, such tht K(λ) 1. This correspondence is given by the identity R λ h = y, Im λ < 0. Proof of Theorem 8. Due to Bsic Lemm Theorem, 8 is consequence of Theorem 1 of the pper [15]. For generl qusi-differentil opertors of even nd odd orders, respectively, the ssertions of Theorems 4, 6 nd 8 re nnounced without proofs in [16, 17]. References [1] A. Zettl, Sturm-Liouville Theory, Mth. Surveys Monogr., Amer. Mth. Soc., Providence, RI, [2] D. Shin, Qusi-differentil opertors in Hilbert spce, Mt. Sb. (N.S.) 13(55) (1943), no. 1, (Russin) [3] A. Zettl, Formlly self-djoint qusi-differentil opertors, Rocky Mountin J. Mth. 5 (1975), no. 3, [4] W. N. Everitt, L. Mrkus, Boundry Vlue Problems nd Symplectic Algebr for Ordinry Differentil nd Qusi-differentil Opertors, Mth. Surveys Monogr., Amer. Mth. Soc., Providence, RI, [5] A. M. Svchuk, A. A. Shklikov, Sturm-Liouville opertors with singulr potentils, Mth. Notes 66 (1999), no. 5 6,

12 12 ANDRII GORIUNOV, VLADIMIR MIKHAILETS [6] S. Albeverio, F. Gestezy, R. Hoegh-Krohn, H. Holden, Solvble Models in Quntum Mechnics, Texts nd Monogrphs in Physics, Springer-Verlg, New York, [7] V. I. Gorbchuk nd M. L. Gorbchuk, Boundry Vlue Problems for Opertor Differentil Equtions, Kluwer Acdemic Publishers, Dordrecht Boston London, (Russin edition: Nukov Dumk, Kiev, 1984) [8] T. Kto, Perturbtion Theory for Liner Opertors, Reprint of the 1980 ed., Clssics in Mthemtics, Springer-Verlg, Berlin, [9] V. A. Mikhilets, N. V. Rev, Generliztions of Kigurdze s theorem on the well-posedness of liner boundry-vlue problems, Dopov. Nts. Akd. Nuk Ukr. (2008), no. 9, (Russin) [10] V. A. Mikhilets, N. V. Rev, Continuous dependence on prmeter of solutions of generl boundry-vlue problems, Proc. Inst. Mt. Nts. Akd. Nuk Ukr. 5 (2008), no. 1, (Russin) [11] J. D. Tmrkin, A lemm of the theory of liner differentil systems, Bull. Amer. Mth. Soc. 36 (1930), no. 2, [12] A. S. Goriunov, V. A. Mikhilets, Resolvent convergence of Sturm-Liouville opertors with singulr potentils, Mth. Notes 87 (2010), no. 2, ArXiv: [mth.fa] 23 Jn [13] R. S. Phillips, Dissiptive opertors nd hyperbolic systems of prtil differentil equtions, Trns. Amer. Mth. Soc. 90 (1959), [14] N. I. Akhiezer, I. M. Glzmn, Theory of Liner Opertors in Hilbert Spce, vol. II, Monogrphs nd Studies in Mthemtics, vol. 10, Pitmn Advnced Publishing Progrm, Pitmn, Boston Msschusetts London, [15] 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(6), (Russin) [16] A. S. Goriunov, V. A. Mikhilets, On extensions of symmetric qusi-differentil opertors of even order, Dopov. Nts. Akd. Nuk Ukr. (2009), no. 4, (Russin) [17] A. S. Goriunov, V. A. Mikhilets, On extensions of symmetric qusi-differentil opertors of odd order, Dopov. Nts. Akd. Nuk Ukr. (2009), no. 9, (Russin) Institute of Mthemtics of Ntionl Acdemy of Sciences of Ukrine, Tereshchenkivsk str., 3, Kyiv-4, Ukrine, E-mil ddress, Andrii Goriunov: goriunov@imth.kiev.u E-mil ddress, Vldimir Mikhilets: mikhilets@imth.kiev.u