On two-point boundary value problems for the Sturm-Liouville operator
aa r X i v : . [ m a t h . SP ] D ec On two-point boundary value problems forthe Sturm-Liouville operatorAlexander Makin
Abstract
In this paper, we study spectral problems for the Sturm-Liouville operator with ar-bitrary complexvalued potential q ( x ) and two-point boundary conditions. All types ofmentioned boundary conditions are considered. We ivestigate in detail the completenessproperty and the basis property of the root function system.
1. Introduction.
The spectral theory of two-point differentialoperators was begun by Birkhoff in his two papers [1, 2] of 1908 wherehe introduced regular boundary conditions for the first time. It wascontinued by Tamarkin [3, 4] and Stone [5, 6]. Afterwards their in-vestigations were developed in many directions. There is an enormousliterature related to the spectral theory outlined above, and we referto [7-18] and their extensive reference lists for this activity.The present communication is a brief survey of results in the spectraltheory of the Sturm-Liouville equation u ′′ − q ( x ) u + λu = 0 (1)with two-point boundary conditions B i ( u ) = a i u ′ (0) + a i u ′ ( π ) + a i u (0) + a i u ( π ) = 0 , (2)where the B i ( u ) ( i = 1 ,
2) are linearly independent forms with arbitrarycomplex-valued coefficients and q ( x ) is an arbitrary complex-valuedfunction of class L (0 , π ).Our main focus is on the non-self-adjoint case, and, in particular, thecase when boundary conditions are degenerate. We will study the com-pleteness property and the basis property of the root function system f operator (1), (2). The convergence of spectral expansions is investi-gated only in classical sense, i.e. the question about the summabilityof divergent series by a generalized method is not considered.We consider the operator Lu = u ′′ − q ( x ) u as a linear operatoron L (0 , π ) with the domain D ( L ) = { u ∈ L (0 , π ) | u ( x ) , u ′ ( x ) areabsolutely continuous on [0 , π ], u ′′ − q ( u ) u ∈ L (0 , π ), B i ( u ) = 0( i = 1 , } .By an eigenfunction of the operator L corresponding to an eigen-value λ ∈ C we mean any function u ( x ) ∈ D ( L ) ( u ( x )
0) whichsatisfies the equation L u + λ u = 0almost everywhere on [0 , π ].By an associated function of the operator L of order p ( p =1 , , . . . ) corresponding to the same eigenvalue λ and the eigenfunc-tion u ( x ) we mean any function p u ( x ) ∈ D ( L ) which satisfies theequation L p u + λ p u = p − u almost everywhere on [0 , π ]. One can also say that an eigenfunction u ( x ) is an associated function of zero order. The set of all eigen-and associated functions (or root functions) corresponding to the sameeigenvalue λ together with the function u ( x ) ≡ L be countable andall root linear manifolds be root subspaces. Let us choose a basis ineach root subspace. Any system { u n ( x ) } obtained as the union ofchosen bases of all the root subspaces is called a system of eigen- andassociated functions (or root function system) of the operator L .The main purpose of this lecture is to study the basis property ofthe root function system of the operator L . Before starting our in- estigation we must verify completeness of the root function system in L (0 , π ).It is convenient to write conditions (2) in the matrix form A = (cid:18) a a a a a a a a (cid:19) and denote the matrix composed of the ith and jth columns of A (1 ≤ i < j ≤
4) by A ( ij ); we set A ij = detA ( ij ).Denote by c ( x, µ ) , s ( x, µ ) ( λ = µ ) the fundamental system of solu-tions to equation (1) with the initial conditions c (0 , µ ) = s ′ (0 , µ ) = 1, c ′ (0 , µ ) = s (0 , µ ) = 0. The eigenvalues of problem (1), (2) are theroots of the characteristic determinant∆( µ ) = (cid:12)(cid:12)(cid:12)(cid:12) B ( c ( x, µ )) B ( s ( x, µ )) B ( c ( x, µ )) B ( s ( x, µ )) (cid:12)(cid:12)(cid:12)(cid:12) . Simple calculations show that∆( µ ) = − A − A + A s ( π, µ ) − A s ′ ( π, µ ) − A c ( π, µ ) − A c ′ ( π, µ ) . It is easily seen that if q ( x ) ≡ ( µ ) of the corresponding problem (1), (2) has the form∆ ( µ ) = − A − A + A sin πµµ − ( A + A ) cos πµ + A µ sin πµ. Boundary conditions (2) are called nondegenerate if they satisfy oneof the following relations:1) A = 0 , A = 0 , A + A = 0 , A = 0 , A + A = 0 , A = 0 . Evidently, boundary conditions (2) are nondegenerate iff ∆ ( µ ) = const .Notice, that for any nondegenerate boundary conditions an asymp-totic representation for the characteristic determinant ∆( µ ) as | µ | →∞ one can find in [10]. heorem 1 ([10]). For any nondegenerate conditions the spec-trum of problem (1), (2) consists of a countable set { λ n } of eigen-values with only one limit point ∞ , and the dimensions of thecorresponding root subspaces are bounded by one constant. Thesystem { u n ( x ) } of eigen- and associated functions is complete andminimal in L (0 , π ) ; hence, it has a biorthogonally dual system { v n ( x ) } . For convenience, we introduce numbers µ n , where µ n is the squareroot of λ n with nonnegative real part.It is known that nondegenerate conditions can be divided into threeclasses:1) strengthened regular conditions;2) regular but not strengthened regular conditions;3) irregular conditions.The definitions are given in [8]. These three cases should be consid-ered separately.
2. Strengthened regular conditions.
Let boundary condi-tions (2) belong to class 1). According to [8], this is equivalent to thefulfillment one of the following conditions: A = 0; A = 0 , A + A = 0 , A + A = ∓ ( A + A ); A = 0 , A + A = 0 , A + A = 0 , A = A , A = 0 . It is well known that, all but finitely many eigenvalues λ n are simple(in other words, they are asymptotically simple), and the number ofassociated functions is finite. Moreover, the λ n are separated in thesense that there exists a constant c > λ k and λ m , we have | µ k − µ m | ≥ c . (3) Theorem 2.
The system of root functions { u n ( x ) } forms a Rieszbasis in L (0 , π ) . This statement was proved in [21], [22] and [9, Chapter XIX]. lass 1) contains many types of boundary conditions, for exam-ple, the Dirichlet boundary conditions u (0) = u ( π ) = 0, the New-mann boundary conditions u ′ (0) = u ′ ( π ) = 0, the Dirichlet-Newmannboundary conditions u (0) = u ′ ( π ) = 0 and others.
3. Regular but not strengthened regular conditions.
Let boundary conditions belong to class 2). According to [8], this isequivalent to the fulfillment of the conditions A = 0 , A + A = 0 , A + A = ( − θ +1 ( A + A ) , (4)where θ = 0 ,
1. It is well known [10] that the eigenvalues of problem(1), (2) form two series: λ = µ , λ n,j = (2 n + o (1)) (5)(if θ = 0) and λ n,j = (2 n − o (1)) (6)(if θ = 1). Here, in both cases, j = 1 , n = 1 , , . . . . We denote µ n,j = p λ n,j = 2 n − θ + o (1). It follows from [8] that asymptoticformulas (6) and (7) can be refined. Specifically, µ n,j = 2 n − θ + O ( n − / ) . Obviously, | µ n, − µ n, | = O ( n − / ); i.e. µ n, and µ n, become infinitelyclose to each other as n → ∞ . If µ n, = µ n, for all n , except,possibly, a finite set, then the spectrum of problem (1), (2) is called asymptotically multiple . If the set of multiple eigenvalues is finite,then the spectrum of problem (1), (2) is called asymptotically simple .There exist numerous examples when the number of multiple eigen-values is finite or infinite, and the total number of associated functionsis finite or infinite also. We see that separation condition (3) neverholds. Depending on the particular form of the boundary conditionsand the potential q ( x ) the system of root functions may have or maynot have the basis property [17], [22], [23], and even for fixed bound-ary conditions, this property may appear or disappear under arbitrary mall variations of the coefficient q ( x ) in the corresponding metric [24].Thus, the considered case is much more complicated than the previousone, so we will study it in detail.For any problem (1), (2) let Q denote the set of potentials q ( x ) fromthe class L (0 , π ) such that the system of root functions forms a Rieszbasis in L (0 , π ), ¯ Q = L (0 , π ) \ Q .To analyze this class of problems, it is reasonable [12] to divideconditions (2) satisfying (4) into three types:I) A = A , A = 0;II) A = A , A = 0;III) A = A The eigenvalue problem for operator (1) with boundary conditions oftype I, II, or III, is called the problem of type I, II, or III, respectively.At first we consider the problems of type I. It was shown in [12]that any boundary conditions of type I are equivalent to the boundaryconditions specified by the matrix A = (cid:18) − θ +1 − θ +1 (cid:19) , i.e., to periodic or antiperiodic boundary conditions. These boundaryconditions are selfadjoint. Theorem 3 ([25]).
The sets Q and ¯ Q are everywhere dense in L (0 , π ) . Recently, (see [26-37] and their extensive reference lists) by a numberof authors, a very nice theory of the problems of type I was built.Let us consider the problems of type II. It was also established in [12]that any boundary conditions of type II are equivalent to the boundaryconditions specified by the matrix A = (cid:18) − a − (cid:19) or A = (cid:18) a (cid:19) , here a = 0 in both cases. If a is a real number and q ( x ) is a realfunction, then the corresponding boundary value problem is selfadjoint. Theorem 4 ([38]). If A = A and A = 0 , then the sys-tem { u n ( x ) } forms a Riesz basis in L (0 , π ) , and the spectrum isasymptotically simple. Denote by { v n ( x ) } the biorthogonally dualsystem. The key point in the proof of Theorem 4 is obtaining theestimate max ( x,ξ ) ∈ [0 ,π ] × [0 ,π ] | u n ( x ) v n ( ξ ) | ≤ C, (7)which is valid for any number n . It follows from (7) and [39] that thesystem { u n ( x ) } forms a Riesz basis in L (0 , π ).A comprehensive description of boundary conditions of type III wasgiven in [12]. In particular, it is known that all of them are non-self-adjoint. Theorem 5 ([38]). If A = A , then the system of root func-tions { u n ( x ) } of problem (1), (2) is a Riesz basis in L (0 , π ) if andonly if the spectrum is asymptotically multiple . Thus, we have established that for problems of type III the ques-tion about the basis property for the system of eigen- and associatedfunctions is reduced to the question about asymptotic multiplicity ofthe spectrum. The presence of this property depends essentially on theparticular form of the boundary conditions and the function q ( x ). Theorem 6 ([40, 41]). If A = A , then, for any function q ( x ) ∈ L (0 , π ) and any ε > , there exists a function ˜ q ( x ) ∈ L (0 , π ) such that || q ( x ) − ˜ q ( x ) || L (0 ,π ) < ε and problem (1), (2)with the potential ˜ q ( x ) has an asymptotically multiple spectrum. For A = A and A = 0, a similar proposition was deduced in[42].Theorems 3, 4, 6 and the results of [43] imply that the whole classof regular but not strengthened regular boundary conditions splits intotwo subclasses (a) and (b). Subclass (a) coincides with the second ype of boundary conditions and is characterized by the fact that thesystem of root functions of problem (1), (2) with boundary conditionsfrom this subclass forms a Riesz basis in L (0 , π ) for any potential q ( x ) ∈ L (0 , π ); i.e. Q = L (0 , π ), ¯ Q = ∅ . We will see belowthat boundary conditions from the subclass (a) are the only boundaryconditions (in addition to strengthened regular ones) that ensure theRiesz basis property of the system of root functions for any potential q ( x ) ∈ L (0 , π ).Subclass (b) contains the remaining regular but not strengthenedregular boundary conditions. An entirely different situation takes placein this case. For any problem with boundary conditions from thissubclass, the sets Q and ¯ Q are dense everywhere in L (0 , π ).
4. Irregular conditions.
Let boundary conditions (2) belong toclass 3). According to [8, 12], this is equivalent to the fulfillment oneof the following conditions: A = 0 , A + A = 0 , A + A = 0 , A = A , A = 0; A = 0 , A + A = 0 , A + A = 0 , A = 0 . According to [12], any boundary conditions of the considered classare equivalent to the boundary conditions determined by the matrix A = (cid:18) ± b ∓ (cid:19) , where b = 0 , or A = (cid:18) b b − b (cid:19) , where b = ± , b = 0 , or A = (cid:18) a
00 0 0 1 (cid:19) , where a = 0 . In case 3), as well as in case 1), all but finitely many eigenvalues λ n are simple, the number of associated functions is finite, and separation ondition (3) holds. However, the system { u n ( x ) } never forms even ausual basis in L (0 , π ), because || u n || L (0 ,π ) || v n || L (0 ,π ) → ∞ as n →∞ . Here { v n ( x ) } is the biorthogonally dual system.This case wasinvestigated in [5], [6], [44].
5. Degenerate conditions.
Let boundary conditions (2) bedegenerate. According to [10, 12], this is equivalent to the fulfillmentof the following conditions: A = 0 , A + A = 0 , A = 0 . According to [12], any boundary conditions of the considered classare equivalent to the boundary conditions determined by the matrix A = (cid:18) d − d (cid:19) , or A = (cid:18) (cid:19) . If in the first case d = 0 then for any potential q ( x ) we have the initialvalue problem (the Cauchy problem) which has no eigenvalues. Thesame situation takes place in the second case.Further we will consider the first case if d = 0. Then the boundaryconditions can be written in more visual form u ′ (0) + du ′ ( π ) = 0 , u (0) − du ( π ) = 0 . (8) By P W σ we denote the class of entire functions f ( z ) of exponentialtype ≤ σ such that || f ( z ) || L ( R ) < ∞ , and by P W − σ we denote the setof odd functions in P W σ .By performing simple manipulations, we obtain the relation∆( µ ) = d − d + c ( π, µ ) − s ′ ( π, µ ) = d − d + R π K ( t ) sin µtµ dt == d − d + f ( µ ) µ , where K ( t ) ∈ L (0 , π ). If q ( x ) ∈ L (0 , π ) then K ( t ) ∈ L (0 , π ) and f ( µ ) ∈ P W − π . Notice, that simple calculations show that if d = ± nd q ( x ) ≡ λ ∈ C is an eigenvalue of infinite multiplicity.This abnormal example constructed by Stone illustrates the difficultyof investigation of problems with boundary conditions of the consideredclass.It is well known that the characteristic determinant ∆( µ ) of problem(1), (8) is an entire function of the parameter µ , consequently, for theoperator (1), (8) we have only the following possibilities:1) the spectrum is absent;2) the spectrum is a finite nonempty set;3) the spectrum is a countable set without finite limit points;4) the spectrum fills the entire complex plane.One can prove [45] that case 2) is impossible. It is known that c ( π, µ ) − s ′ ( π, µ ) ≡ Q ( x ) = q ( x ) − q ( π − x ) = 0 (9)almost everywhere on the segment [0 , π ]. Evidently, lim µ →∞ ( c ( π, µ ) − s ′ ( π, µ )) = 0. Hence it follows that if c ( π, µ ) − s ′ ( π, µ ) ≡ C then C = 0. We see that case 1) takes place if and only if condition (9)holds and d = ±
1, and case 4) takes place if and only if condition (9)holds and d = ±
1. If condition (9) does not hold we have case 3).
Completeness of the root function system of problem (1), (8) wasinvestigated in [46-47]. The main result of the mentioned papers is:
Theorem 7 ([47]). If q ( x ) ∈ C k [0 , π ] for some k = 0 , . . . and q ( k ) (0) = ( − k q ( k ) ( π ) , then the system of root functions is completein the space L p (0 , π ) if ≤ p < ∞ . It follows from [47] that depending on the potential q ( x ) the systemof root functions may have or may not have the completeness property,moreover, this property may appear or disappear under arbitrary smallvariations of the coefficient q ( x ) in the corresponding metric even forfixed boundary conditions. f the conditions d = 1 and q ( x ) ∈ C [0 , π ] hold necessary andsufficient conditions of the completeness of root function system ofproblem (1), (8) were found in [48]. Theorem 8 ([49]).
If for a number ρ > h → R ππ − h Q ( x ) dxh ρ = ν, and ν = 0 , then the root function system of problem (1), (8) iscomplete in the space L p (0 , π ) if ≤ p < ∞ . Since for a wide class of potentials q ( x ) the root function system ofproblem (1), (8) is complete in L (0 , π ) one can set a question whetherthe mentioned system forms a basis. Recently, it was proved in [50] that the root function system neverforms an unconditional basis in L (0 , π ) if multiplicities of the eigen-values are uniformly bounded by some constant. Moreover, under thecondition mentioned above it was established there that if the eigen-and associated function system of general ordinary differential opera-tor with two-point boundary conditions forms an unconditional basisthen the boundary conditions are regular. Article [50] was publishedin 2006. At that time it was unknown whether there exists a potential q ( x ) providing unbounded growth of multiplicities of the eigenvalues. However, in 2010 in [45] an example of a potential q ( x ) for whichthe characteristic determinant has the roots of arbitrary high multi-plicity was constructed. Hence, the corresponding root function sys-tem { u n ( x ) } contains associated functions of arbitrary high order. Itmeans, that paper [50] does not give the definitive solution of basisproperty problem. Below we will show a method to construct a poten-tial q ( x ) providing unbounded growth of multiplicities of eigenvalues. Theorem 9 ([51]).
Suppose that a function v ( µ ) can be repre- ented in the form v ( µ ) = γ + f ( µ ) µ , (10) where γ is some complex number, the function f ( µ ) ∈ P W − π satis-fies the condition Z ∞−∞ | µ m f ( µ ) | dµ < ∞ , where m is a nonnegative integer number. Then there exists afunction q ( x ) ∈ W m (0 , π ) such that the characteristic determinant ∆( µ ) of problem (1), (8), where either d = ( γ + p γ + 4) / or d = ( γ − p γ + 4) / and with the potential q ( x ) is identicallyequal to the function v ( µ ) . Therefore, Theorem 9 reduces the problem on the structure of thespectrum of problem (1), (8) with degenerate boundary conditions tothe problem on the expansion of a function of the form (10) into acanonical product.
Example 1.
Let us define a sequence { a k } ( k = 1 , , . . . ) in this way: a = 1 , a = 3 , a = 5, a k +1 = a k + 2 p , if 2 p < k < p +1 ( p = 1 , , . . . ),and a k +1 = a k + ( a k − a k − ) + 2, if k = 2 p ( p = 2 , , . . . ). Set F ( µ ) = ∞ Y k =1 (cid:18) − µ a k (cid:19) a k +1 − a k − δ k , where δ k = 0, if k = 2 p , and δ k = 1, if k = 2 p , ( p = 2 , , . . . ). Theorem 10 ([52]).
For any real x the following inequality holds | F ( x ) | ≤ C ( | x | + 1) M , where M is a sufficiently large number. Denote f ( µ ) = µ ∞ Y k = M +1 (cid:18) − µ a k (cid:19) a k +1 − a k − δ k . t is easily shown that f ( µ ) ∈ P W − π . This, together with Theorem 9implies that there exists a potential q ( x ) ∈ L (0 , π ), such that for thecharacteristic determinant ∆ ( µ ) of problem u ′′ − q ( x ) u + λu = 0 , u ′ (0) + du ′ ( π ) = 0 , u (0) − du ( π ) = 0 (11)( d = ±
1) we have the equality∆ ( µ ) = f ( µ ) /µ. It follows from the definition of sequence { a k } that multiplicities of ze-ros a k of constructed above function f ( µ ) monotonically not decreaseand tend to infinity as k → ∞ . Therefore, the eigenvalues λ n = µ n of problem (11) have the desired property: their multiplicities m ( λ n )tend to infinity and the corresponding root function system containsassociated functions of arbitrary high order, i.e. the dimensions of rootsubspaces infinitely grow. Moreover, the following inequality takesplace c ln µ n ≤ m ( λ n ) ≤ c ln µ n . Theorem 11 ([52]).
The root function system { u n ( x ) } of prob-lem (11) is complete in L (0 , π ) .Example 2. Denote ˜ a k = a k − α k + iβ k , ( k = 1 , , . . . ) where α k = a k − p a k − β k , β k = ( a k − a k − ) /
10) ( a = 0). Denote h k = a k +1 − a k − δ k , F ( µ ) = ∞ Y k =1 (cid:18) − µ ˜ a k (cid:19) [ h k / (cid:18) − µ ¯˜ a k (cid:19) [ h k / (cid:18) − µ ˜ a k (cid:19) h k − h k / . Theorem 12 ([53]).
For any real x the following inequality holds | F ( x ) | ≤ C ( | x | + 1) M , where M is a sufficiently large number. Denote f ( µ ) = µ ∞ Y k = M +1 (cid:18) − µ ˜ a k (cid:19) [ h k / (cid:18) − µ ¯˜ a k (cid:19) [ h k / (cid:18) − µ ˜ a k (cid:19) h k − h k / . t is easily shown that f ( µ ) ∈ P W − π . This, together with Theorem9 implies that there exists a potential q ( x ) ∈ L (0 , π ), such that forthe characteristic determinant ∆ ( µ ) of problem u ′′ − q ( x ) u + λu = 0 , u ′ (0) + du ′ ( π ) = 0 , u (0) − du ( π ) = 0 (12)( d = ±
1) we have the equality∆ ( µ ) = f ( µ ) /µ. It follows from the definition of sequence { ˜ a k } that multiplicities ofzeros ˜ a k , ¯˜ a k of constructed above function f ( µ ) monotonically notdecrease and tend to infinity as k → ∞ . Therefore, the eigenval-ues ˜ λ n = ˜ µ n of problem (12) have two properties: their multiplicities m (˜ λ n ) tend to infinity, hence, the corresponding root function systemcontains associated functions of arbitrary high order, and | I m ˜ µ n | → ∞ as n → ∞ . Moreover, the following two inequalities hold c ln | ˜ µ n | ≤ m ( λ n ) ≤ c ln | ˜ µ n | , c | I m ˜ µ n | ≤ m (˜ λ n ) ≤ c ln | ˜ µ n | . Theorem 13 ([53]).
The root function system { u n ( x ) } of prob-lem (12) is complete in L (0 , π ) . For any problem (1), (8) let Ω denote the set of potentials q ( x )from the class L (0 ,
1) such that the system of root functions containsassociated functions of arbitrary high order, ¯Ω = L (0 , π ) \ Ω. Theorem 14 ([54]).
The sets Ω and ¯Ω are everywhere dense in L (0 , π ) . Since for a wide class of potentials q ( x ) the root function system ofproblem (1), (8) is complete in L (0 , π ) one can set a question whetherthe mentioned system forms a basis. Let λ n = µ n ( Reµ n ≥ , n = 1 , , . . . ) be the eigenvalues of problem(1), (8) numbered neglecting their multiplicities in nondecreasing orderof absolute value. By m ( λ n ) we denote the multiplicity of an eigenvalue λ n . heorem 15 ([55-57]). If lim n →∞ m ( λ n ) p | µ n | = 0 , (13) then the system of eigenfunctions and associated functions of prob-lem (1), (8) is not a basis in L (0 , π ) . Clearly, since Theorem 15 contains supplementary condition (13), itdoes not give the definitive solution of the basis property problem. Ifthis condition does not hold then the mentioned problem has not beensolved.
Acknowledgement
This work was supported by the Russian Foundation for Basic Re-search, project No. 13-01-00241.
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On a two-point boundary value problem for theSturm-Liouville Operator with nonclassical spectral asymptotics. Dif-fer. Equations, 49, No. 5, 536-544 (2013); translation from Differ.Uravn., 49, No. 5, 564-572 (Russian) (2013).[53] A.S. Makin. Problem with Nonclassical Eigenvalue Asymp-totics. Differ. Equations, 51, No. 3, 318-324 (2015); translation fromDiffer. Uravn., 51, No. 3, 317-322 (Russian) (2015).[54] A.S. Makin. On a new class of boundary value problems forthe Sturm-Liouville operator, Differ. Equations, V. 49, No. 2, 262-266(2013); translation from Differ. Uravn., 49, No. 2, 260-264 (Russian)(2013).[55] A.S. Makin. On two-point boundary value problems for the theSturm-Liouville operator. International Conference ”Modern methodsof theory of boundary value problems”. Voronezh, Russia. May 3-9,2015. Abstracts of talks, p. 138-140.[56] Alexander Makin. Spectral Analysis for the Sturm-LiouvilleOperator with Degenerate Boundary Conditions. International Confer- nce on Differential and Difference Equations and Applications. Amadora,Portugal, May 18-22, 2015. Abstracts of talks, p. 93-94.[57] A.S. Makin. On spectral expansions for the the Sturm-Liouvilleoperator with two-point boundary conditions. International Confer-ence ”Function Spaces and Function Approximation Theory”. Moscow,Russia. May 25-29, 2015. Abstracts of talks, p. 177.E-mail: [email protected] on Differential and Difference Equations and Applications. Amadora,Portugal, May 18-22, 2015. Abstracts of talks, p. 93-94.[57] A.S. Makin. On spectral expansions for the the Sturm-Liouvilleoperator with two-point boundary conditions. International Confer-ence ”Function Spaces and Function Approximation Theory”. Moscow,Russia. May 25-29, 2015. Abstracts of talks, p. 177.E-mail: [email protected]