Variational principles for spectral analysis of one Sturm-Liouville problem with transmission conditions
- Kadriye Aydemir^{1}Email author and
- Oktay Sh Mukhtarov^{2, 3}
https://doi.org/10.1186/s13662-016-0800-z
© Aydemir and Mukhtarov 2016
Received: 2 January 2016
Accepted: 2 March 2016
Published: 15 March 2016
Abstract
We study certain spectral aspects of the Sturm-Liouville problem with a finite number of interior singularities. First, for self-adjoint realization of the considered problem, we introduce a new inner product in the direct sum of the \(L_{2}\) spaces of functions defined on each of the separate intervals. Then we define some special solutions and construct the Green function in terms of them. Based on the Green function, we establish an eigenfunction expansion theorem. By applying the obtained results we extend and generalize such important spectral properties as the Parseval and Carleman equations, Rayleigh quotient, and Rayleigh-Ritz formula (minimization principle) for the considered problem.
Keywords
Sturm-Liouville problems boundary-transmission conditions transmission conditions expansions theorem Rayleigh-Ritz formula Parseval equality Carleman equation1 Introduction
2 Some preliminary results in according Hilbert space
- (i)
y and \(y'\) are absolutely continuous in each interval \(\Omega_{i}\) (\(i=1,2,\ldots,n+1\)) and has finite limits \(y(\xi_{0}+0)\), \(y'(\xi_{0}+0)\), \(y(\xi_{n+1}-0)\), \(y'(\xi _{n+1}-0)\), \(y(\xi_{k}\mp0)\), and \(y'(\xi_{k}\mp0)\) for \(k=1,2,\ldots,n\);
- (ii)
\(\mathcal{L}y(x) \in \mathcal{H}\), \(\mathcal{L}_{\alpha}y(x)=\mathcal{L}_{\beta}y(x)=\mathcal{L}_{2k-1}y(x)=\mathcal{L}_{2k} y(x)=0\), \(k=1,2,\ldots,n\). Then problem (1)-(5) is reduced to the operator equation \(\mathcal{A}y=\lambda y \) in the Hilbert space \(\mathcal{H}\).
Theorem 2.1
For all \(y, z \in D(\mathcal{A})\), we have the equality \(\langle\mathcal{A}y,z\rangle_{\mathcal{H}}=\langle y,\mathcal{A}z\rangle_{\mathcal{H}} \).
Proof
Lemma 2.2
The linear operator \(\mathcal{A}\) is densely defined in \(\mathcal{H}\).
Proof
It suffices to prove that if \(z \in\mathcal{H}\) is orthogonal to all \(y \in D(\mathcal{A})\), then \(z=0\). Suppose that \(\langle y,z\rangle_{\mathcal{H}}=0\) for all \(y \in D(\mathcal{A})\). Denote by \(\bigoplus_{i=1}^{n+1}C_{0}^{\infty}(\Omega_{i})\) the set of all infinitely differentiable functions in Ω vanishing on some neighborhoods of the points \(x=\xi_{k}\), \(k=0,1,2,\ldots,n+1\). Taking into account that \(C_{0}^{\infty}(\xi_{k},\xi_{k+1})\) is dense in \(L_{2}(\xi_{k},\xi_{k+1})\) (\(k=0,1,2,\ldots,n+1 \)), we have that the function \(z(x)\) vanishes on Ω. The proof is complete. □
Corollary 2.3
\(\mathcal{A}\) is symmetric linear operator in the Hilbert space \(\mathcal{H}\).
Corollary 2.4
Remark 2.5
In fact, as in our previous work [31], we can prove that the operator \(\mathcal{A}\) is self-adjoint in the Hilbert space \(\mathcal{H}\). Moreover, the resolvent operator \((A-\lambda I)^{-1}\) is compact in this space.
3 Eigenfunction expansion based on the Green function. Modified Parseval equality
Theorem 3.1
(Expansion theorem)
Proof
Theorem 3.2
(Modified Parseval equality)
Proof
4 Modified Carleman equality
5 The Rayleigh quotient and minimization principle for problem (1)-(5)
Lemma 5.1
(Rayleigh quotient)
Proof
Equation (41) is the Rayleigh quotient for considered problem (1)-(5).
Theorem 5.2
(Minimization principle)
Proof
Remark 5.3
In fact, it is proven that \(\lambda_{1}=\min R(y)\).
Corollary 5.4
Proof
Remark 5.5
By applying the Rayleigh-Ritz formula (43) it is difficult to explicitly compute the principal eigenvalues. But using the Rayleigh quotient (41) with appropriate test functions, we can obtain a good approximation for the eigenvalues. Moreover, from formula (50) it follows that \(\lambda_{k}\leq R(z_{k})\) for each test function \(z_{k}\in\Gamma_{k}\). Thus, we can also find an upper bound for the kth eigenvalue.
Declarations
Acknowledgements
The authors would like to thank the referees for their valuable comments.
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.
Authors’ Affiliations
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