Selfadjoint and m Sectorial Extensions of Sturm–Liouville Operators

The self-adjoint and m-sectorial extensions of coercive Sturm–Liouville operators are characterised, under minimal smoothness conditions on the coefficients of the differential expression.


Introduction
A non-negative, densely defined symmetric operator T acting in a Hilbert space H, has two distinguished self-adjoint extensions, namely the Friedrichs extension T F and the Krein extension T N ; for a strictly positive operator T , as will be the case in this paper, T N coincides with the von Neumann extension of T and is accordingly called the Krein-von Neumann extension.These extensions are extremal in the sense that all non-negative self-adjoint extensions T of T satisfy T N ≤ T ≤ T F in the form sense; recall that if T 1 , T 2 are non-negative self-adjoint operators with associated quadratic forms t 1 [•], t 2 [•] and form domains Q(T 1 ), Q(T 2 ) respectively, then T 1 ≤ T 2 in the form sense if Q(T 2 ) ⊂ Q(T 1 ) and 0 ≤ t 1 [u] ≤ t 2 [u] for all u ∈ Q(T 2 ).The abstract theory was initiated by Krein in [14,13] and developed further by Vishik in [18] and then by Birman in [6]; we shall refer to the work of the 3 authors in the cited papers as the Krein-Vishik-Birman theory.In the first, and main, part of this paper we determine all the positive self-adjoint extensions when T is the minimal operator generated by the general Sturm-Liouville expression on the interval [a, m), −∞ < a < m ≤ ∞, with the coefficients subject to the following minimal conditions: 1. k, p > 0 a.e. on [a, m); k, 1/p ∈ L 1 loc [a, m); 2. q ∈ L 1 loc [a, m) real-valued.We are assuming that a is a regular end-point of the interval, but m is allowed to be singular, i.e., at least one of the following is possible: The case of a a singular end-point can be treated similarly.
Let H denote the weighted space L 2 (a, m; k) with inner-product and norm The minimal operator T generated by τ in H is the closure of T ′ defined by where If m is regular, then T has domain 2) The adjoint T * of T is the maximal operator defined by In the main part of the paper, dealing with a symmetric T and its self-adjoint extensions, we shall be assuming that T is positive, i.e., T ≥ µ > 0, meaning and our objective is to contribute to the classical problem of characterising all the positive self-adjoint extensions of T .Our results extend previous ones in that we assume only the minimal requirements on the coefficients of τ and that T is positive.In order to achieve this degree of generality we use a theorem of Kalf in [11] in which he gives necessary and sufficient conditions for T to be bounded below, and a general characterisation of the Friedrichs extension of Sturm-Liouville operators in terms of weighted Dirichlet integrals.Kalf's result on the semi-boundedness of T is a consequence of a result of Rellich in [16] equating the lower semi-boundedness to the disconjugacy of the equation τ u = λu near m.Rellich proves that if is non-oscillatory near m for some λ ∈ R then T is bounded below, while conversely, if T is bounded below with lower bound α, then (1.4) is disconjugate for every λ < α, i.e. any solution u = 0 of (1.4) with λ ≤ α has at most one zero .Before quoting Kalf's result, we recall that if (1.4) is non-oscillatory at m, there exist linearly independent solutions f, g with the following properties: The functions f, g are called the principal and non-principal solutions of (1.4), respectively.Kalf's theorem implies the following when a is regular; see Corollary 2 in [17]: The operator T is bounded below if and only if there exist µ ∈ R and a function h ∈ AC loc [a, m) which is such that ph ′ ∈ AC loc [a, m), h > 0 in [s, m) for some s ∈ (a, m), The Friedrichs extension T F of T has domain It follows that the sesquilinear form associated with where q h = q − (ph ′ ) ′ h .We shall assume that µ > 0 so that T is positive.Kalf also allows for a to be singular.It is observed in [11] that the principal solution assumption on h can be replaced by a non-principal solution assumption, i.e., m s (1.7) Our principal tool is the result from the Krein-Vishik-Birman theory, that there is a one to one correspondence between the set of all positive self-adjoint extensions of T and the set of all pairs {N B , B}, where N B is a subspace of ker T * , and B is a positive self-adjoint operators in N B ; the Krein-von Neumann extension corresponds to B = 0 and the Friedrichs extension to B = ∞,(i.e., B acts (trivially) in a 0-dimensional vector space).The reader is referred to [9] for a comprehensive treatment, and in particular to section 2 of [10] in which an account is given of earlier works of the author; the latter are acknowledged in the addendum [2] to the survey article [1].If T is a positive self-adjoint extension of T , then T = T B for some B, where T B is associated with a form t B [•] which satisfies and b[•] is the form of B; we use the notation Q(A) to denote the form domain of a positive operator A, i.e., the domain of the associated form a see [6], [8], [9] and [1].The Friedrichs and Krein-von Neumann extensions have important roles in the application of the theory of symmetric operators.The Friedrichs extension has an acknowledged natural part to play in quantum mechanics, while in [10], Grubb describes an intimate connection between the eigenvalues of the Krein-von Neumann extension of a minimal elliptic differential operator of even order and those of a higher order problem concerning the buckling of a clamped plate.An abstract version of the latter connection is given in [5], establishing the Krein-von Neumann extension as a natural object in elasticity theory.
The last section of the paper is concerned with the more general problem of characterising all the coercive m-sectorial extensions of a coercive sectorial minimal operator T generated by the expression τ with a complex coefficient q.The same problem is discussed by Arlinskii in [3], Theorem 3.1, where previously determined abstract results are applied to a general second-order differential operator on [0, ∞) with L ∞ [0, ∞) coefficients.Theorems 2.1 and 2.2 below are analogous results for τ under the minimal conditions assumed here.In [8], Grubb developed methods of Birman and Vishik to characterise extensions of adjoint pairs of operators with bounded inverses and applied the results to elliptic partial differential operators.
An important tool in Kalf's paper, as well as in this paper, is the Jacobi factorisation identity . (1.10) 2 Self-adjoint extensions

The limit-point case at m
Since a is regular and T ≥ µ > 0, the deficiency index of (T − λ), and hence the dimension of ker(T * − λ), is constant for all λ ∈ R \ [µ, ∞).In the limit-point case of τ u = λu at m it therefore follows, in particular, that N := ker(T * ) is of dimension 1, and so, any η ∈ N can be written η = cψ, where ψ ∈ L 2 [a, m; k) is real, c ∈ C and τ ψ = 0. Our main result in the limit-point case is Theorem 2.1.Let τ be in the limit point case at m. Then there is a one-one correspondence between the positive self-adjoint extensions of T and the one-parameter family of operators {T l }, 0 ≤ l ≤ ∞, where T l is the restriction of T * to the domain The operator T 0 is the Krein-von Neumann extension and T ∞ is the Friedrichs extension.
Proof.If ψ(a) = 0, then ψ is an eigenfunction at zero of the self-adjoint extension T 0 of T determined by the Dirichlet boundary condition u(a) = 0. Thus N = ker T 0 and D(T ) ∔ N ⊆ D(T 0 ).Since D(T ) ∔ N is the domain of T N (see [1], Example 3.2.), it follows that T 0 = T N .However by [19, Theorem 5 ], T 0 is the Friedrichs extension of T and therefore has no null space.Consequently ψ(a) = 0 is not possible.Thus we may assume hereafter that ψ(a) = 0 and, without loss of generality, that Also, from (1.9) where By (1.10), It follows on integration by parts, and since θ(a) = ϕ(a) − ϕ(a)ψ(a) = 0, that For v ∈ D(T B ) ⊂ D(T * ) we have by (1.10) Thus + lv(a)ϕ(a) ψ 2 . (2.5) The value at m of the right-hand side of (2.5) is Suppose now that ϕ ∈ (T B ). Then the functions v, u, η, ϕ, θ, ξ are members of D(T * ) and thus since we have assumed the limit-point condition at m; see [7], Theorem III.10.13 or [15], Section 18.3.Hence the value of the right-hand side of (2.5) at m is Since u, θ ∈ D(T F ), we have But as u(a) = θ(a) = 0, Therefore, it follows that (2.7) The real constants l parameterise the operators T B : l = 0 corresponds to B = 0 and thus the Krein-von Neumann extension of T , while l = ∞ corresponds to B = ∞ and hence the Friedrichs extension.

2.2
The case of m regular or limit circle, and τ u = 0 nonoscillatory at m Let f, g be the principal and non-principal solutions respectively of τ u = 0. From Remark 3 in [11], and from Corollary 1 in [17] We still have that u(a) = 0 for u ∈ Q(T F ), and both where q g = q − (pg ′ ) ′ g > 0, and where We now have dimN = 2, and {f, g} is a fundamental system of solutions of τ u = 0.The self-adjoint operators B act in subspaces N B of N which therefore may be of dimension 1 or 2. In the case dimN B = 2 of the next theorem, {ψ 1 , ψ 2 } is a real orthonormal basis of N B .
Theorem 2.2.The positive self-adjoint extensions of T which correspond to operators B in the Krein-Vishik-Birman theory with dimN B = 1 form a one-parameter family T β of restrictions of T * with domains (2.12) where ψ is a real basis of N B with ψ(a) = 1.
The self-adjoint extensions corresponding to operators B with dimN B = 2 form a family T β , where now β is a matrix (b j,k ) j,k=1,2 of parameters, and the T β are restrictions of T * to domains where c 1 and c 2 are determined by the values of v at a and m by,

.14)
Proof.The case dimN B = 1 Let the real function ψ be a basis of N B , and we have that We now have from (1.8) that The argument following (2.3) can be repeated, with g replacing h and using the facts that now u g (m) = θ g (m) = 0, as well as u(a) = θ(a) = 0.The term corresponding to I 3 is now zero and the result is that since θ g (a) = θ g (m) = 0. We conclude from (2.16) and (2.17 The case dim(N B ) = 2 Let {ψ 1 , ψ 2 } be a real orthonormal basis for N. Then Furthermore, for some µ jk ∈ R, j, k = 1, 2, so that ψ j g (m) = µ j,2 , j = 1, 2. (2.20) From (2.15) and (2.17), we now have Since d 1 and d 2 are arbitrary, we have with no contribution from m as in the LP case.We could then repeat the above analysis with g replaced by f to get where ϕ f (m) = θ f (m)+c 1 .The equation (2.24) has to be satisfied for all θ ∈ Q(T F ).
3 Coercive sectorial operators 3.1 m-sectorial extensions and on the interval [a, m).Each of the expressions τ and τ + has a minimal and maximal operator associated with it, and the notation has to indicate this; the minimal operators T (τ ), T (τ + ) will be denoted by T, T + respectively.We shall assume that q 1 satisfies the minimal conditions 1. and 2. of the introduction and that for some ν > 0 and α ∈ (0, π/2).The minimal operators T, T + are then coercive and sectorial, i.e., the numerical range of T , Θ Furthermore, T, T + are J-symmetric with respect to the conjugation J : u → u, i.e., JT J ⊂ T * , J(T + )J ⊂ (T + ) * .
. We denote by T F and T N the Friedrichs and Krein-von Neumann extensions of T , and use a similar notation for T + .We recall that the Friedrichs extension T F of a sectorial operator T is an m-sectorial operator associated with the closure of the sesquilinear form t[u, v] = (T u, v), and is coercive if T is coercive.The form domain is , where T R F is the real part of T , i.e. the positive self-adjoint operator associated with the form u]; see [12] for details.In Arlinskii's construction of the m-sectorial extensions of a coercive sectorial operator T , the Krein-von Neumann extension T N has an important role.We refer to [4] for the definition of T N and a comprehensive treatment.From Theorem 3.6 in [4], where P is the projection of Q(T N ) onto Q(T ) with respect to the decomposition (3.3).Furthermore The identities (3.3) -(3.5) also have exact analogues for T + .From Edmunds/Evans Theorem III.10.7, and since def T and def T + are equal, being the dimensions of the kernels of T * , (T + ) * respectively, we have that and for all v ∈ D(T N ), Moreover, the form domain of T N is Hence, since u(a) = 0 for all u ∈ Q(T F ), we have from (3.3) the unique repre- for all ϕ ∈ Q(T N ), with We shall now follow Arlinskii's analysis in [3], Section 3.1,to characterise all coercive m-sectorial extension of T .Arlinskii considers a more general secondorder differential expression to generate his operator T , but with L ∞ coefficients, and not the minimal conditions we impose on ours.
Let X 0 denote Q(T ), with norm Arlynskii expresses T in divergence form; this requires T to be put in the form where q h = q − (ph ′ ) ′ h .The adjoint operators are given by Arlinskii's approach yields the following result; ψ is the solution of τ + ψ = 0 with ψ(a) = 1.establish a one to one correspondence between all coercive m-sectorial extensions T of T , excepting T F and T N , and the set of all pairs < w, y >, where w is a complex number with a positive real part, and y ∈ X 0 satisfies max{ℜ t R [(2y − ϕ, ϕ)], ϕ ∈ X 0 } < ℜw.
The associated closed form is given by The Friedrichs and Krein-von Neumann extensions are determined by the pairs < ∞, 0 > and < 0, 0 > respectively.
Let {ψ 1 , ψ 2 } be a basis for ker T * and let ψ = 2 j=1 c j ψ j be such that ψ(a) = 1.This determines a vector in ker T * , the complex number w in the open right half plane determines a one-dimensional coercive operator W (λψ) = λwψ .In general, parameter w can be a 2 × 2 sectorial/coercive matrix (as in Theorem 2.2) which determines a linear operator and in this case y is a linear operator.