Canonical forms for boundary conditions of self-adjoint differential operators

Canonical forms of boundary conditions are important in the study of the eigenvalues of boundary conditions and their numerical computations. The known canonical forms for self-adjoint differential operators, with eigenvalue parameter dependent boundary conditions, are limited to 4-th order differential operators. We derive canonical forms for self-adjoint 2n-th order differential operators with eigenvalue parameter dependent boundary conditions. We compare the 4-th order canonical forms to the canonical forms derived in this article.


Introduction
Canonical forms of boundary conditions are important in the study of the eigenvalues of boundary conditions and their numerical computations [1].In [4], Hao, Sun and Zettl investigate canonical forms of self-adjoint boundary conditions for fourth order differential operators.They derive three mutually exclusive types of boundary conditions, which are separated, coupled and mixed boundary conditions.In [2] Bao, Hao, Sun and Zettl provide new canonical forms of self-adjoint boundary conditions for regular differential operators of order two and four.
In this paper, we extend the study conducted in [4] to 2n-th order differential operators.We start our investigation with sixth order differential operators with self-adjoint boundary conditions that we extend to 2n-th differential operators with self-adjoint boundary conditions and we show equivalence between the separated and coupled forms presented in [4] and those obtained during our investigation.
In Section 2, we introduce the self-adjoint sixth order differential operators with eigenvalue dependent boundary conditions under consideration.In Section 3, we present the types of boundary conditions for the self-adjoint sixth order differential operators.Next, using the CS-decomposition, we provide a classification of the different types of canonical forms for self-adjoint sixth order differential operators in Section 4 that we extend in Section 5 to canonical forms for self-adjoint 2n-th order differential operators.Finally, in Section 6 we show equivalences between the separated and coupled forms provided in [4] with those obtained in this paper.

Types of boundary conditions of sixth order differential operators
The following theorem gives conditions satisfied by the matrices A, B for the problems (1)-( 2) to be self-adjoint.Theorem 3.1.Assume that the matrices A, B ∈ M 6 (C) satisfy (4).Then Proof.See [10, Theorem 3].
Note that the boundary conditions (2) are invariant under left multiplication by a non singular matrix G ∈ M 6 (C) and if AC 6 A * = BC 6 B * , then Therefore, the boundary condition form (4) is invariant under elementary matrix row transformations of (A : B).
Next, we define the different types of boundary conditions based on Theorem 3.1.Definition 3.2.Let the hypotheses and notation of Theorem 3.1 hold.Then the boundary conditions (2), ( 4) are (1) separated if r = 0, (2) mixed if r = 1, 2, (3) coupled if r = 3. Remark 3.3.Note that the boundary conditions (2) are separated if each of the six boundary conditions involves only one endpoint, coupled if each of the six boundary conditions involves both endpoints, while they are mixed if there is at least one separated and one coupled boundary conditions.

Canonical forms for sixth order differential operators
Equation ( 4) can be written in the form (5) rank(A : where is a skew-Hermitian matrix with eigenvalues i and −i.Thus, each column vector x * j of (A : B) * may be written in the form (6) x where x * j,±i belongs to the eigenspace corresponding to the eigenvalue ±i.Condition (5) may now be written (7) x j,i x * k,i = x j,−i x * k,−i .Taking x j,i as the rows of X i and similarly for X −i , (6) may be summarized as (A : B) = X i + X −i and ( 7) as ( 8) where V is an arbitrary unitary matrix providing the diagonalization.From the ordering of eigenvectors (columns of V ) in ( 9) and the solution (6) in terms of eigenvectors, the matrix V may be chosen so that (A : B) has the form , where V is unitary and each y j,±i is an eigenvector corresponding to the eigenvalue ±i.Equation ( 8) yields and, since positive definite square roots are unique, the singular value decompositions yields the solution (6) and satisfies (7).Since rank(A : B) = 6, we have rank(Σ C ) = 6 and hence Σ C is invertible.By invariance of the boundary conditions under elementary row operations, we obtain the general form where V X and V Y are arbitrary unitary matrices.Here, the first 6 columns of V are eigenvectors corresponding to the eigenvalue i of C 6 ⊕ (−C 6 ), and the remaining 6 columns correspond to the eigenvalue −i.We write V as the block matrix Again, since the boundary conditions are invariant under row operations, we will assume Y is unitary.Choosing a particular V provides some additional insight.For the purpose of illustration, we also set W = I 6 in the following example. Let where Here rank(A) = rank(B) = 3. Choosing V as above, leads to a canonical form for separated boundary conditions in Lemma 4.1.Let Here we obtain coupled boundary conditions, leading to a canonical form in Lemma 4.1.From (13), we have Choosing appropriate W provides the remaining canonical forms.Thus It follows that and hence rank(A) = rank(I 3 ) + rank(W 3 ), rank(B) = rank(I 3 ) + rank(W 2 ).
Necessarily rank(W 3 ) = rank(W 2 ).The CS-decomposition, described in detail in [8] and [5, Theorem 2.7,1], provides a useful way to speak about rank.In particular, we obtain the CS-decomposition of W using [3, Corollary 3,1] for some unitary matrices U 1 , U 2 , V 1 and V 2 , and positive semi-definite diagonal matrices C and S satisfying C 2 + S 2 = I 3 .Hence, up to elementary row operations,  Let Z be the matrix There exist a 6 × 6 non singular matrix U , 3 × 3 unitary matrices U 1 , U 2 , V 1 and V 2 , and positive semi-definite diagonal matrices C and S with C 2 + S 2 = I 3 , such that and, the boundary conditions are (1) separated, if and only S = 0, (2) mixed, if and only if 0 < rank(S) < 3.
Remark 4.2.It may be assumed, in this representation of (A : B), that the diagonal entries of C are non-increasing and that the diagonal entries of S are non-decreasing.

Canonical forms for 2n-th order differential operators
We consider on the interval J = (a, b), −∞ ≤ a < b ≤ ∞, the 2n-th order differential equation with formally self-adjoint differential expression (with smooth coefficients) [7, Remark 3.2] where 1 p n exists on J, p j ∈ C j (J) and w ∈ L(J, R), w > 0 a.e. on J.If the coefficients are not smooth, we introduce the quasi-derivatives [7] y [1] = y ′ , y [2] = y ′′ , . . ., y The method in Section 4 generalizes in a straightforward way.Thus we obtain the following theorem.Let Z be the matrix Then there exists a 2n × 2n non singular matrix U and n × n unitary matrices V 1 , U * 1 , U * 2 and V 2 , and positive semi-definite diagonal matrices C and S with C 2 + S 2 = I n , such that and the boundary conditions are (1) separated, if and only if S = 0, (2) mixed, if and only 0 < rank(S) < n.

Revisiting canonical forms for fourth order differential operators
Hao, Sun and Zettl derived canonical forms for self-adjoint boundary conditions for differential equations of order four [4].In this section we will show some equivalences between the canonical forms in [4] and the forms presented in Lemma 4.1.
The following canonical forms are given [4].Then, the boundary conditions are (1) separated, if there exists 4 × 4 and 8 × 8 non singular matrices R and R ′ , respectively, such that ) mixed, if there exist 4 × 4 and 8 × 8 non singular matrix R and R ′ , respectively, such that for some r 1 , r  Let Z be the matrix Then there exists a 4 × 4 non singular matrix U and 2 × 2 unitary matrices (3) coupled, if and only if rank(S) = 2.
There are 36 canonical forms according to [4, Theorem 2], which yield (by elementary operations) the forms listed in Theorem 6.1.To show that Theorem 6.1 and Corollary 6.2 are equivalent, we need to show that U (and C, S, U 1 , U 2 , V 1 and V 2 ) and R and R ′ exist for each canonical form (i.e. for each type of boundary conditions) which gives equality of the forms.The forms given in both the theorem and the corollary ensure that AC 4 A * = BC 4 B * .An exhaustive comparison of all 36 forms is too lengthy and cumbersome to pursue here.We will show equivalence for two of the forms, which show clear connections between the two representations of boundary conditions.
First, we consider separated boundary conditions, i.e. S = 0: where the left hand side is obtained from the separated boundary conditions form of Corollary 6.2 and the right hand side from the separated boundary conditions form of Theorem 6.1.Thus for example, A 11 = r 1 a 21 a 21 r 2

4 where W 1 =
U 1 V 1 and W 2 = U 2 V 2are unitary.If rank S = 0, then W does not simplify in an obvious way.Thus we have the following Lemma.

2 and V 2 ,( 1 )
and positive semi-definite diagonal matrices C and S with C 2 +S 2 = I 2 , such that (A : B) = U C I 2 0 S −S 0 I 2 C separated, if and only if S = 0, (2) mixed, if and only if rank(S) = 1.