Multi-interval Sturm-Liouville boundary-value problems with distributional potentials
aa r X i v : . [ m a t h . SP ] A ug MULTI-INTERVAL STURM–LIOUVILLE BOUNDARY-VALUE PROBLEMSWITH DISTRIBUTIONAL POTENTIALS
ANDRII GORIUNOVA
BSTRACT . We study the multi-interval boundary-value Sturm-Liouville problems with distributional po-tentials. For the corresponding symmetric operators boundary triplets are found and the constructive descrip-tions of all self-adjoint, maximal dissipative and maximal accumulative extensions and generalized resolventsin terms of homogeneous boundary conditions are given. It is shown that all real maximal dissipative andmaximal accumulative extensions are self-adjoint and all such extensions are described.
In recent years the interest in multi-interval differential and quasi-differential operators has increased(see [1, 2, 3, 4]). The main attention is paid to the case where a (quasi-)differential expression is formallyself-adjoint. From the operator-theoretic point of view this corresponds to the situation where we inves-tigate extensions of a symmetric (quasi-)differential operator with equal deficiency indices in the directsum of Hilbert spaces on the basis of Glazman-Krein-Naimark theory [5, 6, 7, 8]. In the present paper wedevelop another approach to such problems based on the concept of boundary triplets [9, 10].Let m ∈ N , a = a < a < · · · < a m = b be a partition of a finite interval [ a , b ] into m parts and on everyinterval ( a i − , a i ) , i ∈ { , . . . , m } let the formal Sturm-Liouville expression(1) l i ( y ) = − ( p i ( t ) y ′ ) ′ + q i ( t ) y be given. Here, the measurable finite functions p i and Q i are such that(2) 1 / p i , Q i / p i , Q i / p i ∈ L ([ a i − , a i ] , R ) , the potentials q i = Q ′ i and the derivative is understood in the sense of distributions.For m = m ∈ N .We introduce the quasi-derivatives D [ ] i y = y , D [ ] i y = p i y ′ − Q i y , D [ ] i y = ( D [ ] i y ) ′ + Q i p i D [ ] i y + Q i p i y . on every interval ( a i − , a i ) , as in [11].Then the maximal and minimal operators L i , : y → l i [ y ] , Dom ( L i , ) : = n y ∈ L (cid:12)(cid:12)(cid:12) y , D [ ] i y ∈ AC ([ a i − , a i ] , C ) , D [ ] i y ∈ L o , L i , : y → l i [ y ] , Dom ( L i , ) : = n y ∈ Dom ( L i , ) (cid:12)(cid:12)(cid:12) D [ k ] i y ( a i − ) = D [ k ] i y ( a i ) = , k = , o are defined in the spaces L (( a i − , a i ) , C ) . According to [11] the operators L i , , L i , are closed and denselydefined in L ([ a i − , a i ] , C ) . The operator L i , is symmetric with the deficiency indices ( , ) and L ∗ i , = L i , , L ∗ i , = L i , . Recall that a boundary triplet of a closed densely defined symmetric operator T with equal (finite orinfinite) deficiency indices is called a triplet ( H , G , G ) where H is an auxiliary Hilbert space and G , G are the linear maps from Dom ( T ∗ ) to H such that Mathematics Subject Classification.
Key words and phrases.
Sturm-Liouville operator; multi-interval boundary value problems; distributional coefficients; self-adjoint extension; maximal dissipative extension; generalized resolvent. (1) for any f , g ∈ Dom ( T ∗ ) there holds ( T ∗ f , g ) H − ( f , T ∗ g ) H = ( G f , G g ) H − ( G f , G g ) H ;(2) for any g , g ∈ H there is a vector f ∈ Dom ( T ∗ ) such that G f = g and G f = g .It is proved in [11] that for every i = , . . . , m the triplet ( C , G , i , G , i ) , where G , i , G , i are linear maps G , i y : = (cid:16) D [ ] i y ( a i − +) , − D [ ] i y ( a i − ) (cid:17) , G , i y : = ( y ( a i − +) , y ( a i − )) , from Dom ( L i , ) to C is a boundary triplet for the operator L i , .We consider the space L ([ a , b ] , C ) as a direct sum ⊕ mi = L ([ a i − , a i ] , C ) which consists of vector func-tions f = ⊕ mi = f i such that f i ∈ L ([ a i − , a i ] , C ) . In this space we consider operators L max = ⊕ mi = L i , and L min = ⊕ mi = L i , .Then the operators L max , L min are closed and densely defined in L ([ a , b ] , C ) . The operator L min issymmetric with the deficiency indices ( m , m ) and L ∗ min = L max , L ∗ max = L min . Note that the minimal operator L min may be not semi-bounded even in the case of a single-intervalboundary-value problem since the function p may reverse sign. Theorem 1.
The triplet ( C m , G , G ) where G , G are linear maps G y : = ( G , y , G , y , . . . , G , m y ) , G y : = ( G , y , G , y , . . . , G , m y ) from Dom ( L max ) onto C m is a boundary triplet for L min . Denote by L K the restriction of L max onto the set of functions y ( t ) ∈ Dom ( L max ) satisfying the homo-geneous boundary condition ( K − I ) G y + i ( K + I ) G y = . Similarly, denote by L K the restriction of L max onto the set of functions y ( t ) ∈ Dom ( L max ) satisfyingthe homogeneous boundary condition ( K − I ) G y − i ( K + I ) G y = . Here K is a bounded operator in C m .The constructive description of the various classes of extensions of the operator L min is given by thefollowing theorem. Theorem 2.
Every L K with K being a contracting operator in C m is a maximal dissipative extension ofL min . Similarly every L K with K being a contracting operator in C m is a maximal accumulative extensionof the operator L min .Conversely, for any maximal dissipative (respectively, maximal accumulative) extension e L of the oper-ator L min there exists the unique contracting operator K such that e L = L K (respectively, e L = L K ).The extensions L K and L K are self-adjoint if and only if K is a unitary operator on C m . Recall that a linear operator T acting in L ([ a , b ] , C ) is called real if:(1) for every function f from Dom ( T ) the complex conjugate function f also lies in Dom ( T ) ;(2) the operator T maps complex conjugate functions into complex conjugate functions, that is T ( f ) = T ( f ) .One can see that the maximal and minimal operators are real. Theorem 3.
All real maximal dissipative and maximal accumulative extensions of the minimal operatorL min are self-adjoint. The self-adjoint extension L K or L K is real if and only if the unitary matrix K issymmetric. Let us recall that a generalized resolvent of a closed symmetric operator T in a Hilbert space H is anoperator-valued function l R l defined on C \ R , which can be represented as R l f = P + (cid:0) T + − l I + (cid:1) − f , f ∈ H , ULTI-INTERVAL STURM–LIOUVILLE BOUNDARY-VALUE PROBLEMS WITH DISTRIBUTIONAL POTENTIALS 3 where T + is a self-adjoint extension of T which acts in a certain Hilbert space H + ⊃ H , I + is theidentity operator on H + , and P + is the orthogonal projection operator from H + onto H . It is knownthat an operator-valued function R l is a generalized resolvent of a symmetric operator T if and only if itcan be represented as ( R l f , g ) H = Z + ¥ − ¥ d (cid:0) F m f , g (cid:1) m − l , f , g ∈ H , where F m is a generalized spectral function of the operator T , i. e. m F m is an operator-valued function F m defined on R and taking values in the space of continuous linear operators in H with the followingproperties:(1) for m > m the difference F m − F m is a bounded non-negative operator;(2) F m + = F m for any real m ;(3) for any x ∈ H there holdslim m →− ¥ || F m x || H = , lim m → + ¥ || F m x − x || H = . The following theorem provides a description of all generalized resolvents of the operator L min . Theorem 4. ) Every generalized resolvent R l of the operator L min in the half-plane Im l < acts by therule R l h = y, where y is a solution of the boundary-value probleml ( y ) = l y + h , ( K ( l ) − I ) G f + i ( K ( l ) + I ) G f = . Here h ( x ) ∈ L ([ a , b ] , C ) and K ( l ) is a m × m matrix-valued function which is holomorphic in the lowerhalf-plane and satisfies || K ( l ) || ≤ . ) In the half-plane Im l > every generalized resolvent of L min acts by the rule R l h = y where y is asolution of the boundary-value problem l ( y ) = l y + h , ( K ( l ) − I ) G f − i ( K ( l ) + I ) G f = . Here h ( x ) ∈ L ([ a , b ] , C ) and K ( l ) is a m × m matrix-valued function which is holomorphic in the lowerhalf-plane and satisfies || K ( l ) || ≤ .The parametrization of the generalized resolvents by the matrix-valued functions K is bijective. R EFERENCES [1] W. N. Everitt, A. Zettl,
Sturm-Liouville differential operators in direct sum spaces , Rocky Mountain J. Math. , 3 (1986),497–516.[2] W. N. Everitt, A. Zettl, Quasi-differential operators generated by a countable number of expressions on the real line ,Proc. London Math. Soc. , 3 (1992), 524–544.[3] M. S. Sokolov, An abstract approach to some spectral problems of direct sum differential operators , Electron. J. Differ-ential Equations , 75 (2003), 1–6.[4] M. S. Sokolov,
Representation results for operators generated by a quasi-differential multi-interval system in a Hilbertdirect sum space , Rocky Mountain J. Math , 2 (2006), 721–739.[5] A. Zettl, Formally self-adjoint quasi-differential operators , Rocky Mountain J. Math , 3 (1975), 453–474.[6] W. N. Everitt, L. Markus, Boundary Value Problems and Symplectic Algebra for Ordinary Differential and Quasi–differential Operators , AMS, Providence, 1999.[7] A. Zettl,
Sturm-Liouville Theory , AMS, Providence, 2005.[8] M. A. Naimark,
Linear differential operators , Part 2, F. Ungar, New York, 1968.[9] V. I. Gorbachuk, M. L. Gorbachuk,
Boundary Value Problems for Operator Differential Equations , Kluwer, Dordrecht,1991.[10] A. N. Kochubei
Symmetric operators and nonclassical spectral problems (Russian), Mat. Zametki , 3 (1979), 425–434.[11] A. S. Goriunov, V. A. Mikhailets Regularization of singular Sturm-Liouville equations , Meth. Funct. Anal. Topol. , 2(2010), 120–130.I NSTITUTE OF M ATHEMATICS OF N ATIONAL A CADEMY OF S CIENCES OF U KRAINE , K
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