Self-adjoint higher order differential operators with eigenvalue parameter dependent boundary conditions
- Manfred Möller^{1}Email author and
- Bertin Zinsou^{1}
https://doi.org/10.1186/s13661-015-0341-5
© Möller and Zinsou. 2015
Received: 22 January 2015
Accepted: 30 April 2015
Published: 16 May 2015
Abstract
Eigenvalue problems for even order regular quasi-differential equations with boundary conditions which depend linearly on the eigenvalue parameter λ can be represented by an operator polynomial \(L(\lambda)=\lambda^{2}M-i\lambda K-A\), where M is a self-adjoint operator. Necessary and sufficient conditions are given such that also K and A are self-adjoint.
Keywords
MSC
1 Introduction
In general, the spectrum of L is no longer real but still has some particularly nice properties if K, M, A are self-adjoint 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 self-adjoint 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 self-adjointness 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 self-adjointness for A in the case of formally symmetric even order quasi-differential 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 self-adjointness conditions makes use of quasi-derivatives, and we will henceforth mostly use quasi-derivatives \(y^{[j]}\) rather than derivatives \(y^{(j)}\). For the definition of the quasi-derivatives \(y^{[j]}\), we refer the reader to (2.2)-(2.5), see also Remark 3.2.
In this paper we consider 2nth order quasi-differential equations and derive necessary and sufficient conditions for 2n boundary conditions of the form (1.2) to generate self-adjoint 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 self-adjoint and for A to be symmetric. In Section 4 it is shown that A is self-adjoint if A is symmetric.
2 The eigenvalue problem
We first summarize some basic facts about quasi-differential equations for the convenience of the reader. For a more comprehensive discussion of quasi-differential 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.
Observe that the quasi-derivatives defined in (2.5) depend on G. However, since we are only going to deal with a single quasi-differential 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 \(w\in L^{1}(I)\).
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 \(\beta_{j}=\alpha_{j}\varepsilon_{j}\).
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 \(\{0\}\).
It is clear that M and K are bounded self-adjoint operators and that M is non-negative. The operator \(A(U)\) is not self-adjoint, in general, and we will give necessary and sufficient conditions for the operator \(A(U)\) to be self-adjoint.
3 Symmetry conditions for \(A(U)\)
We will denote the canonical inner product in \(L^{2}(I,w)\oplus\mathbb {C}^{k}\) by \(\langle\cdot,\cdot\rangle\).
Assumption 3.1
Remark 3.2
Proposition 3.3
Proof
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
Corollary 3.6
If \(A(U)\) is symmetric, then \(\operatorname {rank}W=2(2n-k)\) and \(W(N(U_{0}))=(N(U_{0}))^{\perp}\).
Proof
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
- 1.
For \(s\in\Theta_{1}\), \(p_{s}+q_{s}=2n-1\);
- 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
Corollary 3.9
Proof
We have seen in Proposition 3.8 that three sets of conditions have to be satisfied in order that the necessary condition \(\operatorname {rank}W = 2(2n-k)\) 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 self-adjoint 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 self-adjoint 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 Self-adjointness conditions for \(A(U)\)
From Corollary 3.9 we know that for self-adjointness 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 \(\varepsilon_{s}=(-1)^{q_{s}+n+1}\).
Theorem 4.2
Proof
Remark 4.3
Proposition 4.4
Assume that \(\operatorname {rank}W=2(2n-k)\). Then \(U_{1}D=V_{1}\) and \(V_{1}D=-U_{1}\).
Proof
Proposition 4.5
If \(A(U)\) is symmetric, then \(A(U)\) is self-adjoint.
Proof
Proposition 4.6
- (i)
\(p_{s}+p_{r}\ne2n-1\) for all \(r,s\in\Theta_{0}^{a}\),
- (ii)
\(p_{s}+p_{r}\ne2n-1\) for all \(r,s\in\Theta_{0}^{b}\).
Proof
Theorem 4.7
- 1.
\(\beta_{j}\in\mathbb{R}\) and \(p_{j}+q_{j}=2n-1\) for all \(j\in \Theta_{1}\);
- 2.
\(p_{s}+p_{r}\ne2n-1\) for all \(r,s\in\Theta_{0}^{a}\),
- 3.
\(p_{s}+p_{r}\ne2n-1\) for all \(r,s\in\Theta_{0}^{b}\).
Declarations
Acknowledgements
This research was partially supported by a grant from the NRF of South Africa, grant number 80956.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
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