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Selfadjoint higher order differential operators with eigenvalue parameter dependent boundary conditions
Boundary Value Problems volume 2015, Article number: 79 (2015)
Abstract
Eigenvalue problems for even order regular quasidifferential equations with boundary conditions which depend linearly on the eigenvalue parameter λ can be represented by an operator polynomial \(L(\lambda)=\lambda^{2}Mi\lambda KA\), where M is a selfadjoint operator. Necessary and sufficient conditions are given such that also K and A are selfadjoint.
1 Introduction
In order to solve linear partial differential equations of the form
where \(\mathcal{A}\) is a linear differential operator with respect to the variable x on an interval I, the separation of variables method \(u(x,t)=y(x)e^{i\omega t}\) leads to
For tindependent boundary conditions \(Bu=0\), setting \(\lambda =\omega^{2}\), the operator theoretic realization leads to an eigenvalue problem for an operator A in the Lebesgue space \(L^{2}(I)\) with domain
Such problems are well studied, and of particular importance is the case that A is selfadjoint. Many applications in physics and engineering can be represented by such selfadjoint operators.
However, problems like the Regge problem and the vibrating beam problem have boundary conditions with partial first order derivatives with respect to t or whose mathematical model leads to an eigenvalue problem with the eigenvalue parameter \(\lambda=\omega\) occurring linearly in the boundary conditions. Such problems have an operator representation of the form
in a Hilbert space \(H=L^{2}(I)\oplus\mathbb{C}^{k}\), where k is the number of eigenvalue dependent boundary conditions.
In general, the spectrum of L is no longer real but still has some particularly nice properties if K, M, A are selfadjoint with \(M\ge0\) and \(K\ge0\), the resolvent set of L is nonempty, and L has a compact resolvent: it is symmetric with respect to the imaginary axis and eigenvalues with negative imaginary parts must lie on the imaginary axis. In this situation, the operators M and K are quite simple bounded selfadjoint operators. However, the operator A is determined by three ingredients: the differential equation \(\mathcal{A}\), the parameter independent boundary conditions as homogeneous boundary conditions for A, and the parameter dependent boundary conditions as an inhomogeneous part of A. Hence one cannot make use of the criteria for selfadjointness in the case of parameter independent boundary conditions. Rather, the parameter dependent case is a proper extension of the parameter independent case.
For parameter independent boundary conditions, i.e., \(k=0\), characterizations of selfadjointness for A in the case of formally symmetric even order quasidifferential expressions are known both for the regular and the singular cases, see [1] and in particular [1], Theorem 6 for the regular case. The simplest formulation of these selfadjointness conditions makes use of quasiderivatives, and we will henceforth mostly use quasiderivatives \(y^{[j]}\) rather than derivatives \(y^{(j)}\). For the definition of the quasiderivatives \(y^{[j]}\), we refer the reader to (2.2)(2.5), see also Remark 3.2.
Some special cases of selfadjoint boundary conditions for regular 2nth order differential equations with \(k>0\) are known. In [2], the second order problem related to the Regge problem was investigated, whereas the fourth order differential equation \(y^{(4)}(gy')'\) related to a vibrating beam was dealt with in [3], where the boundary conditions are of the form
with exactly one boundary condition depending on λ. A classification of all selfadjoint boundary conditions of the form (1.2) was obtained in [4]. A corresponding result for sixth order differential equations was given in [5].
In this paper we consider 2nth order quasidifferential equations and derive necessary and sufficient conditions for 2n boundary conditions of the form (1.2) to generate selfadjoint operators K and A.
In Section 2 we give a precise definition of the boundary value problem and the quadratic operator pencil L associated with it. In Section 3 we derive necessary and sufficient conditions for K to be selfadjoint and for A to be symmetric. In Section 4 it is shown that A is selfadjoint if A is symmetric.
2 The eigenvalue problem
We first summarize some basic facts about quasidifferential equations for the convenience of the reader. For a more comprehensive discussion of quasidifferential equations, the reader is referred to [6] and to [7] in the scalar case and to [8, 9] for the general case with matrix coefficients.
Let \(I=(a, b)\) be an interval with \(\infty< a <b <\infty\), and let m be a positive integer. For a given set S, \(M_{m}(S)\) denotes the set of \(m\times m\) matrices with entries from S. Let
where \(L^{1}(I)\) denotes the complexvalued Lebesgue integrable functions on I.
For \(G\in Z_{m}(I)\), define
and
Inductively, for \(r=1,\dots,m\), we define
where \(g_{m,m+1}:=1\) and where \(AC(I)\) denotes the set of complexvalued functions which are absolutely continuous on I. Finally we set
The expression \(\mathcal{A}=\mathcal{A}_{G}\) is called the quasidifferential expression associated with G, and the function \(y^{[r]}\), \(0\le r\le m\), is called the rth quasiderivative of y. We also write \(\mathscr{D}(\mathcal{A})\) for \(Q_{m}\).
Observe that the quasiderivatives defined in (2.5) depend on G. However, since we are only going to deal with a single quasidifferential equation, we will not indicate this dependence explicitly.
In the remainder of the paper, we assume that \(m=2n\) is an even positive integer, that \(G=(g_{r,s})_{r,s=1}^{2n}\in Z_{2n}(I)\), and that \(w:I\to\mathbb{R}\) is positive a.e. and satisfies \(w\in L^{1}(I)\).
Together with (2.6) we consider the boundary conditions \(B_{j}(\lambda)y=0\), \(j=1,\dots,2n\), taken at the endpoint a for \(j=1,\dots,n\) and at the endpoint b for \(j=n+1,\dots, 2n\). We assume for simplicity that
where \(a_{j}=a \) for \(j=1,\dots,n\), \(a_{j}=b\) for \(j=n+1,\dots,2n\), \(\beta _{j}\in\mathbb{C}\) and \(0\le p_{j},q_{j}\le2n1\). Of course, the numbers \(q_{j}\) are ambiguous and irrelevant in case \(\beta_{j}=0\).
The differential expression (2.6) and the boundary conditions (2.7) define the eigenvalue problem
We put
and
Assumption 2.1
We assume that the numbers \(p_{1},\dots,p_{n},q_{j}\) for \(j\in\Theta_{1}^{a}\) are distinct and that the numbers \(p_{n+1}, \dots ,p_{2n},q_{j}\) for \(j\in\Theta_{1}^{b}\) are distinct.
Assumption 2.1 means that for any pair \((r, a_{j})\) the term \(y^{[r]}(a_{j})\) occurs at most once in the boundary conditions (2.7).
For \(j\in\Theta_{1}\), we choose \(\alpha_{j}\in\mathbb{R}\) and \(\varepsilon_{j}\in\mathbb{C}\) such that \(\beta_{j}=\alpha_{j}\varepsilon_{j}\).
For \(y\in\mathscr{D}(\mathcal{A})\), we define \(Y_{R}= \bigl ({\scriptsize\begin{matrix} Y(a)\cr Y(b)\end{matrix}} \bigr ) \) with \(Y= (y^{[0]},y^{[1]},\dots,y^{[2n1]} )^{\mathsf{T}}\). We denote the collection of the 2n boundary conditions (2.9) by U and define the following matrices related to U:
Remark 2.2
In case that \(\Theta_{r}=\emptyset\) for \(r=0\) or \(r=1\), the corresponding matrix \(U_{r}\) will be identified with the ‘zero’ operator from \(\mathbb{C}^{2n}\) into \(\{0\}\).
The weighted Lebesgue space \(L^{2}(I,w)\) is the Hilbert space of all equivalence classes of complexvalued measurable functions f such that \((f,f)_{w}:=\int_{I}w(x)f(x)^{2}\,dx<\infty\). For convenience we define the operator \(\mathcal{A}_{\max}\) on \(L^{2}(I,w)\) by
We will associate the quadratic operator pencil
in the space \(L^{2}(I,w)\oplus\mathbb{C}^{k}\) with problem (2.8), (2.9), where
The operator \(A(U)\) in \(L^{2}(I,w)\oplus\mathbb{C}^{k}\) is defined by
It is easy to see that a function \(y\in\mathscr{D}(\mathcal{A}_{\max})\) satisfies \(\mathcal{A}y=\lambda^{2}wy\) and \(B_{j}(\lambda)y=0\) for \(j=1,\dots,2n\) if and only if there is \(c\in\mathbb{C}^{k}\) such that \((y,c)^{\mathsf{T}}\in \mathscr{D}(A(U))\) such that \(L(\lambda)(y,c)^{\mathsf{T}}=0\). In this case c is uniquely determined by y. Indeed, if \(y\in\mathscr{D}(\mathcal{A}_{\max})\) with \(\mathcal {A}y=\lambda^{2}wy\) and \(B_{j}(\lambda)y=0\) for \(j=1,\dots,2n\), then \(U_{0}Y_{R}=0\) shows that \((y,V_{1}Y_{R})^{\mathsf{T}} \in\mathscr{D}(A(U))\) and
Clearly, the first component is 0, and so is the second component since
Hence the operator pencil L is an operator realization of the eigenvalue problem (2.8), (2.9).
It is clear that M and K are bounded selfadjoint operators and that M is nonnegative. The operator \(A(U)\) is not selfadjoint, in general, and we will give necessary and sufficient conditions for the operator \(A(U)\) to be selfadjoint.
3 Symmetry conditions for \(A(U)\)
We will denote the canonical inner product in \(L^{2}(I,w)\oplus\mathbb {C}^{k}\) by \(\langle\cdot,\cdot\rangle\).
The Lagrange form of \(A(U)\) is defined by
The operator \(A(U)\) is symmetric if and only if its Lagrange form is identically zero. For this it is necessary that \(\mathcal{A}\) is formally symmetric, and for the remainder of this paper we make therefore the following assumption.
Assumption 3.1
We assume that
where
and δ is the Kronecker delta.
It is easy to verify that Assumption 3.1 holds if and only if
Remark 3.2
Classical formally selfadjoint differential expressions are of the form
with \(g_{j} \in C^{j}[0,a]\) for \(j=0,\dots,n\) and invertible \(g_{n}\). It is easy to verify that this is a quasidifferential equation with quasiderivatives
The corresponding matrix \(G=(g_{r,s})_{r,s=1}^{2n}\) has the entries \(g_{r,r+1}=1\) for \(r=1,\dots,n1\) and \(r=n+1,\dots, 2n1\), \(g_{n,n+1}=g_{n}^{1}\), \(g_{r,2nr+1}=g_{2nr}\) for \(r=n+1,\dots,2n\), while all other entries are zero. It is easy to see that Assumption 3.1 holds in this case if and only if \(g_{j}=\overline{g_{j}}\) for \(j=0,\dots,n\), so that the formal selfadjointness condition reduces to the wellknown condition that all \(g_{j}\), \(j=0,\dots,n\), are realvalued functions.
From [10], Lemma 3.3 we know that the Lagrange identity
holds, where
Proposition 3.3
The Lagrange form \(F_{U}\) of \(A(U)\) has the representation
where
Proof
Let \(\tilde{y},\tilde{z}\in\mathscr{D}(A(U))\). Then
and an application of the Lagrange identity (3.3) completes the proof of the lemma. □
By definition, an operator in a Hilbert space is symmetric if and only if its Lagrange form is identically zero. Hence we have the following.
Corollary 3.4
The differential operator \(A(U)\) is symmetric if and only if \(Z_{R}^{*}WY_{R}=0\) for all \(\widetilde{y},\widetilde{z}\in\mathscr{D}(A(U))\).
The nullspace and range of a matrix M are denoted by \(N(M)\) and \(R(M)\), respectively.
Proposition 3.5
The differential operator \(A(U)\) is symmetric if and only if \(W(N(U_{0}))\subset(N(U_{0}))^{\perp}\).
Proof
From [10], Corollary 5.5 we know that
Hence \(\{Y_{R}:\widetilde{y}\in\mathscr{D}(A(U)) \}=N(U_{0})\). An application of Proposition 3.4 completes the proof. □
Corollary 3.6
If \(A(U)\) is symmetric, then \(\operatorname {rank}W=2(2nk)\) and \(W(N(U_{0}))=(N(U_{0}))^{\perp}\).
Proof
Since \(\dim(N(U_{0}))^{\perp}=\operatorname {rank}U_{0}=2nk\), we have
Hence \(\operatorname {rank}W\le2(2nk)\). Since \(V_{1}^{*}U_{1}U_{1}^{*}V_{1}\) has 2k nonzero entries and D is invertible, \(\operatorname {rank}W\ge2(2n k )\) and \(\operatorname {rank}W=2(2n k )\) follows. In this case, all the inequalities of (3.7) are equalities and \(\dim W(N(U_{0}))=\dim(N(U_{0}))^{\perp}\) holds. Thus it follows from Proposition 3.5 that \(W(N(U_{0}))=(N(U_{0}))^{\perp}\). □
In view of Corollary 3.6, we may assume that \(\operatorname {rank}W=2(2n k )\) when investigating the symmetry of \(A(U)\). Since \((N(U_{0}))^{\perp }=R(U_{0}^{*})\), see [11], Theorem IV.5.13, Proposition 3.5 and Corollary 3.6 lead to the following.
Corollary 3.7
Let \(\operatorname {rank}W=2(2n k )\). Then the differential operator \(A(U)\) is symmetric if and only if \(W(N(U_{0}))=R(U_{0}^{*})\).
We now give an explicit description for the condition \(\operatorname {rank}W=2(2n k )\).
Proposition 3.8
\(\operatorname {rank}W=2(2n k )\) if and only if the following conditions hold:

1.
For \(s\in\Theta_{1}\), \(p_{s}+q_{s}=2n1\);

2.
For \(s\in\Theta_{1}^{(a)}\), \(\varepsilon_{s}=(1)^{q_{s}+n}\);

3.
For \(s \in\Theta_{1}^{(b)}\), \(\varepsilon_{s}=(1)^{q_{s}+n+1}\).
Proof
Note that
where
Since D has exactly one nonzero entry in each row and column and \(V_{1}^{*}V_{0}V_{0}^{*}V_{1}\) has exactly 2k nonzero entries, it follows that \(\operatorname {rank}W=2(2n k )\) if and only if each nonzero entry of \(V_{2}\) cancels a nonzero entry of \((1)^{n1}C\) and each nonzero entry of \(V_{3}\) cancels a nonzero entry of \((1)^{n}C\). Since the nonzero entries of C are in rows i and columns j such that \(i+j=2n+1\), we obtain that \(\operatorname {rank}W=2(2nk)\) if and only if conditions 1, 2, and 3 are satisfied. □
Corollary 3.9
The boundary eigenvalue problem (2.8), (2.9) has an operator pencil representation (2.12) with selfadjoint operator K and symmetric operator \(A(U)\) if and only if

1.
\(\beta_{j}\in\mathbb{R}\) and \(p_{j}+q_{j}=2n1\) for all \(j\in \Theta_{1}\);

2.
\(W(N(U_{0}))=R(U_{0}^{*})\).
Proof
We have seen in Proposition 3.8 that three sets of conditions have to be satisfied in order that the necessary condition \(\operatorname {rank}W = 2(2nk)\) for symmetry of \(A(U)\) holds. Conditions 2 and 3 can always be satisfied if we put \(\alpha_{j}=\beta_{j} (1)^{q_{s}+n}\) for \(j\in\Theta_{1}^{a}\) and \(\alpha _{j}=\beta_{j} (1)^{q_{s}+n+1}\) for \(j\in\Theta_{1}^{b}\), and for K to be selfadjoint it is therefore necessary and sufficient that \(\beta_{j}\) are real. The remaining conditions now follow easily from Proposition 3.8 and Corollary 3.7. □
We could now give explicit conditions for symmetry of \(A(U)\) in terms of the boundary conditions (2.7). However, we will see in the next section that \(A(U)\) is selfadjoint if and only if it is symmetric. In order to avoid duplication we will therefore postpone deriving these explicit conditions to the next section.
4 Selfadjointness conditions for \(A(U)\)
From Corollary 3.9 we know that for selfadjointness of K and \(A(U)\) the condition \(\beta_{j}\in\mathbb{R}\) for all \(j\in\Theta_{1}\) is necessary. Hence we require without loss of generality that the numbers \(\varepsilon_{s}\) for \(s\in\Theta_{1}\) are chosen as in Proposition 3.8, conditions 2 and 3.
Assumption 4.1
For \(s\in\Theta_{1}^{(a)}\), let \(\varepsilon_{s}=(1)^{q_{s}+n}\), and for \(s \in\Theta_{1}^{(b)}\), let \(\varepsilon_{s}=(1)^{q_{s}+n+1}\).
For convenience, we set
The range \(R(U_{r}^{*})\) of \(U_{r}^{*}\) for \(r=0,1\) is the span of all standard unit vectors \(e_{\tilde{p}_{j}}\) in \(\mathbb{C}^{4n}\) with \(j\in\Theta_{r}\), and \(R(V_{1}^{*})\) is the span of all standard unit vectors \(e_{\tilde{q}_{j}}\) in \(\mathbb{C}^{4n}\) with \(j\in\Theta_{1}\). Hence it follows from Assumptions 2.1 and 4.1 that
Theorem 4.2
The operator \(A(U)\) is densely defined, the domain \(\mathscr{D}((A(U))^{*})\) of its adjoint \((A(U))^{*}\) is the set of all \(\widetilde{z}= \bigl ({\scriptsize\begin{matrix} z\cr d \end{matrix}} \bigr ) \) in \(L^{2}(I,w)\oplus\mathbb{C}^{k }\) such that there is \(c\in\mathbb{C}^{k}\) such that \(z\in\mathscr {D}(\mathcal{A}_{\max})\) and
For \(\widetilde{z}= \bigl ({\scriptsize\begin{matrix} z\cr d \end{matrix}} \bigr ) \in\mathscr{D}((A(U)^{*}))\), the vectors d and c are uniquely determined by z, namely, \(d=U_{1}D^{*}Z_{R}\) and \(c=V_{1}D^{*}Z_{R}\), and
Proof
By definition of the adjoint (possibly as a linear relation), \(\widetilde{z}= \bigl ({\scriptsize\begin{matrix}z\cr d \end{matrix}} \bigr ) \in L^{2}(I,w)\oplus\mathbb{C}^{k }\) belongs to \(\mathscr{D}((A(U))^{*})\) if and only if there is \(\widetilde{u}= \bigl ({\scriptsize\begin{matrix}u\cr c \end{matrix}} \bigr ) \in L^{2}(I,w)\oplus\mathbb{C}^{k }\) such that for all \(\widetilde{y}= \bigl ({\scriptsize\begin{matrix} y\cr V_{1}Y_{R} \end{matrix}} \bigr ) \in\mathscr{D}(A(U))\) the identity
holds.
Hence let \(\widetilde{z},\widetilde{u}\in L^{2}(I,w)\oplus\mathbb{C}^{k }\) such that (4.5) holds for all \(\widetilde{y}\in\mathscr{D}(A(U))\). If y has compact support in I, then (4.5) reduces to
This, the formal symmetry Assumption 3.1 and [10], Theorem 4.2 show that \(z\in\mathscr {D}(\mathcal{A}_{\max})\) and \(\mathcal{A}_{\max}z=u\). We can now conclude that (4.5) holds if and only if
In view of the Lagrange identity (3.3), the above is equivalent to
Since the range of all \(Y_{R}\) with \(y\in\mathscr{D}(A(U))\) is \(N(U_{0})\), it follows that (4.5) is equivalent to \(z\in\mathscr{D}(\mathcal{A}_{\max})\), \(u=\mathcal{A}_{\max}z\) and
Applying \(U_{1}\) and \(V_{1}\), respectively, to (4.6) and observing (4.1) and (4.2) it follows that d and c are uniquely given by \(d=U_{1}D^{*}Z_{R}\) and \(c=V_{1}D^{*}Z_{R}\). From the uniqueness of u and c we see that \((A(U))^{*}\) is not only a linear relation but a linear operator, so that \(A(U)\) is densely defined. □
Remark 4.3
The matrix D is invertible and
see [8], (2.7).
Proposition 4.4
Assume that \(\operatorname {rank}W=2(2nk)\). Then \(U_{1}D=V_{1}\) and \(V_{1}D=U_{1}\).
Proof
By definition of \(U_{1}\) and D we can write
where \(U_{1}^{\alpha}=(\delta_{j,p_{i}+1})_{i\in\Theta_{1}^{\alpha},j=1,\dots ,2n}\) for \(\alpha=a,b\). In view of Proposition 3.8 we conclude that
Hence \(U_{1}D=V_{1}\), and (4.7) gives \(V_{1}D=U_{1}D^{2}=U_{1}\). □
Proposition 4.5
If \(A(U)\) is symmetric, then \(A(U)\) is selfadjoint.
Proof
We have to show that \(\mathscr{D}((A(U))^{*})\subset\mathscr{D}(A(U))\). By Theorem 4.2, \(\mathscr{D}((A(U))^{*})\) is the set of all \(\bigl ({\scriptsize\begin{matrix} z\cr V_{1}Z_{R} \end{matrix}} \bigr ) \) such that \(z\in\mathscr{D}(\mathcal{A}_{\max})\) and \(D^{*}Z_{R}+U_{1}^{*}dV_{1}c\in R(U_{0}^{*})\). But Theorem 4.2, Proposition 4.4 and (4.7) imply
so that \(\mathscr{D}((A(U))^{*})\subset\mathscr{D}(A(U))\) if and only if \(W^{1}(R(U_{0}^{*}))\subset N(U_{0})\).
We know that \(\operatorname {rank}U_{0}=2nk\) and \(\dim N(U_{0})=4n\operatorname {rank}U_{0}=2n+k\), whereas \(\dim N(W)=4n\operatorname {rank}W=2k\) by Corollary 3.6. Altogether, we conclude
But from Corollary 3.7 we conclude that \(N(U_{0})\subset W^{1}(R(U_{0}^{*}))\), and it follows that \(N(U_{0})= W^{1}(R(U_{0}^{*}))\). □
Proposition 4.6
Assume \(\operatorname {rank}W=2(2nk)\). Then \(W(N(U_{0}))=R(U_{0}^{*})\) if and only if

(i)
\(p_{s}+p_{r}\ne2n1\) for all \(r,s\in\Theta_{0}^{a}\),

(ii)
\(p_{s}+p_{r}\ne2n1\) for all \(r,s\in\Theta_{0}^{b}\).
Proof
Defining for \(c=a,b\),
where \(V_{2}\) and \(V_{3}\) are as in (3.7), it follows that
and
in view of (3.5) and (3.8). Therefore \(W(N(U_{0}))=R(U_{0}^{*})\) if and only if \(W_{c}(N_{c})=M_{c}\) for \(c=a,b\). Now let \(c\in\{a,b\}\). From Proposition 3.8 and its proof we find for \(j\in\{1,\dots ,2n\}\) that
Observing condition 1 in Proposition 3.8 it follows that
Hence \(W_{c}(N_{c})=M_{c}\) holds if and only if the sets
are complementary subsets of \(\{1,\dots,2n\}\). But by Assumption 2.1 and condition 1 in Proposition 3.8 the listed elements in \(\Psi_{0}^{c}\) as well as in \(\Psi_{1}^{c}\) are mutually distinct, so that the sets \(\Psi_{0}^{c}\) and \(\Psi_{1}^{c}\) are complementary if and only if they are disjoint. It is clear that this latter property holds if and only if \(2np_{j}\notin\Psi_{0}^{c}\) for all \(j\in\Theta_{0}^{c}\). This completes the proof of the proposition. □
Theorem 4.7
The boundary eigenvalue problem (2.8), (2.9) has an operator pencil representation (2.12) with selfadjoint operators K and \(A(U)\) if and only if

1.
\(\beta_{j}\in\mathbb{R}\) and \(p_{j}+q_{j}=2n1\) for all \(j\in \Theta_{1}\);

2.
\(p_{s}+p_{r}\ne2n1\) for all \(r,s\in\Theta_{0}^{a}\),

3.
\(p_{s}+p_{r}\ne2n1\) for all \(r,s\in\Theta_{0}^{b}\).
Proof
This theorem is an immediate consequence of Corollary 3.9 and Propositions 4.5 and 4.6. □
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This research was partially supported by a grant from the NRF of South Africa, grant number 80956.
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Möller, M., Zinsou, B. Selfadjoint higher order differential operators with eigenvalue parameter dependent boundary conditions. Bound Value Probl 2015, 79 (2015). https://doi.org/10.1186/s1366101503415
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DOI: https://doi.org/10.1186/s1366101503415