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
1 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
References
- Hinton, D: Spectral Theory and Computational Methods of Sturm-Liouville Problems. CRC Press, Boca Raton (1997) Google Scholar
- Levitan, BM: The Eigenfunction Expansion for the Second Order Differential Operator (1950) Google Scholar
- Marchenko, VA: Sturm-Liouville Operator and Application. Birkhäuser, Basel (1986) View ArticleGoogle Scholar
- Naimark, MA: The study of eigenfunction expansion of non-selfadjoint differential of the second order on the half line. Tr. Mosk. Mat. Obŝ. 3, 181-270 (1954) MathSciNetGoogle Scholar
- Rotenberg, M: Advances in Atomic and Molecular Physics, vol. 6 (1970) Google Scholar
- Sherstyuk, AI: Problems of Theoretical Physics, vol. 3. Leningrad. Gos. University, Leningrad (1988) Google Scholar
- Tupitsyn, II, Volotka, AV, Glazov, DA, Shabaev, VM: Magnetic-dipole transition probabilities in B-like and Be-like ions. Phys. Rev. A 72, 062503 (2005) View ArticleGoogle Scholar
- Germinet, F, Klein, A: Bootstrap multiscale analysis and localization in random media. Commun. Math. Phys. 222, 415-448 (2001) MathSciNetView ArticleMATHGoogle Scholar
- Stollmann, P: Caught by Disorder, Bound States in Random Media. Progress in Math. Phys., vol. 20. Birkhäuser, Boston (2001) View ArticleMATHGoogle Scholar
- Stollmann, P: Localization for acoustic waves in random perturbations of periodic media. Isr. J. Math. 107, 125-139 (1998) MathSciNetView ArticleMATHGoogle Scholar
- Mamedov, KR, Cetinkaya, FA: Eigenparameter dependent inverse boundary value problem for a class of Sturm-Liouville operator. Bound. Value Probl. 2014, Article ID 194 (2014) MathSciNetView ArticleMATHGoogle Scholar
- Titchmarsh, EC: Eigenfunctions Expansion Associated with Second Order Differential Equations, 2nd edn. Oxford University Press, London (1962) MATHGoogle Scholar
- Likov, AV, Mikhalilov, YA: The Theory of Heat and Mass Transfer, Qosenergaizdat (1963) (in Russian) Google Scholar
- Titeux, I, Yakubov, Y: Completeness of root functions for thermal conduction in a strip with piecewise continuous coefficients. Math. Models Methods Appl. Sci. 7(7), 1035-1050 (1997) MathSciNetView ArticleMATHGoogle Scholar
- Mukhtarov, OS, Kadakal, M, Muhtarov, FS: On discontinuous Sturm-Liouville problems with transmission conditions. J. Math. Kyoto Univ. 44(4), 779-798 (2004) MathSciNetMATHGoogle Scholar
- Mukhtarov, OS, Aydemir, K: Eigenfunction expansion for Sturm-Liouville problems with transmission conditions at one interior point. Acta Math. Sci. 35(3), 639-649 (2015) MathSciNetView ArticleMATHGoogle Scholar
- Aliyev, ZS: Some global results for nonlinear fourth order eigenvalue problems. Cent. Eur. J. Math. 12(12), 1811-1828 (2014) MathSciNetMATHGoogle Scholar
- Aliyev, ZS, Kerimov, NB: Spectral properties of the differential operators of the fourth-order with eigenvalue parameter dependent boundary condition. Int. J. Math. Math. Sci. 2012, Article ID 456517 (2012) MathSciNetView ArticleMATHGoogle Scholar
- Locker, J: Spectral Theory of Non-selfadjoint Two-Point Differential Operators. Am. Math. Soc., Providence (2000) MATHGoogle Scholar
- Mennicken, R, Möller, M: Non-self-Adjoint Boundary Eingenvalue Problems. North-Holland Mathematics Studies, vol. 192. North-Holland, Amsterdam (2003) MATHGoogle Scholar
- Tretter, C: On λ-Nonlinear Boundary Eigenvalue Problems. Mathematical Research, vol. 71. Akademie Verlag, Berlin (1993) Google Scholar
- Yakubov, S: Completeness of Root Functions of Regular Differential Operators. Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 71. Longman, Harlow (1994); Copublished in the United States with Wiley, New York MATHGoogle Scholar
- Allahverdiev, BP, Bairamov, E, Ugurlu, E: Eigenparameter dependent Sturm-Liouville problems in boundary conditions with transmission conditions. J. Math. Anal. Appl. 401(1), 388-396 (2013) MathSciNetView ArticleMATHGoogle Scholar
- Amirov, RK, Ozkan, AS: Discontinuous Sturm-Liouville problems with eigenvalue dependent boundary condition. Math. Phys. Anal. Geom. 17, 483-491 (2014). doi:10.1007/s11040-014-9166-1 MathSciNetView ArticleMATHGoogle Scholar
- Aydemir, K, Mukhtarov, OS: Second-order differential operators with interior singularity. Adv. Differ. Equ. 2015, Article ID 26 (2015). doi:10.1186/s13662-015-0360-7 MathSciNetView ArticleGoogle Scholar
- Aydemir, K, Mukhtarov, OS: Green’s function method for self-adjoint realization of boundary-value problems with interior singularities. Abstr. Appl. Anal. 2013, Article ID 503267 (2013). doi:10.1155/2013/503267 MathSciNetView ArticleMATHGoogle Scholar
- Bairamov, E, Ugurlu, E: The determinants of dissipative Sturm-Liouville operators with transmission conditions. Math. Comput. Model. 53(5/6), 805-813 (2011) MathSciNetView ArticleMATHGoogle Scholar
- Kandemir, M: Irregular boundary value problems for elliptic differential-operator equations with discontinuous coefficients and transmission conditions. Kuwait J. Sci. Eng. 39(1A), 71-97 (2010) MathSciNetGoogle Scholar
- Kadakal, M, Mukhtarov, OS: Sturm-Liouville problems with discontinuities at two points. Comput. Math. Appl. 54, 1367-1379 (2007) MathSciNetView ArticleMATHGoogle Scholar
- Mukhtarov, OS, Yakubov, S: Problems for differential equations with transmission conditions. Appl. Anal. 81, 1033-1064 (2002) MathSciNetView ArticleMATHGoogle Scholar
- Mukhtarov, OS, Aydemir, K: New type Sturm-Liouville problems in associated Hilbert spaces. J. Funct. Spaces Appl. 2014, Article ID 606815 (2014) MathSciNetMATHGoogle Scholar
- Hıra, F, Altınışık, N: Sampling theorems for Sturm-Liouville problem with moving discontinuity points. Bound. Value Probl. 2015, Article ID 237 (2015) View ArticleMATHGoogle Scholar
- Petrovsky, IG: Lectures on Partial Differential Equations, 1st English edn. Interscience Publishers, New York (1954) (translated from Russian by A. Shenitzer) MATHGoogle Scholar