Avetik Pahlevanyan's Publications (Full profile)

Thesis

Pahlevanyan, A.A., On Some Problems of Spectral Theory, pp. 1-84, (2018)

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Paper

Pahlevanyan, A.A., Convergence of expansions for eigenfunctions and asymptotics of the spectral data of the Sturm-Liouville problem (in Russian), Izv. Nats. Akad. Nauk Armenii Mat., 52, (6), pp. 77-90, (2017)

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Abstract:
Uniform convergence of the expansion of an absolutely continuous function for eigenfunctions of the Sturm-Liouville problem $-y'' + q \left( x \right) y = \mu y,$ $y \left(0\right)=0,$ $y\left( \pi \right)\cos \beta + y'\left( \pi \right)\sin \beta = 0,$ $\beta \in \left( 0, \pi \right)$ with summable potential $q \in L_{\mathbb{R}}^1 \left[0, \pi \right]$ is proved. This result is used to obtain more precise asymptotic formulae for eigenvalues and norming constants of this problem.

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Convergence of expansions for eigenfunctions and asymptotics of the spectral data of the Sturm-Liouville problem.pdf
Harutyunyan, T.N., Pahlevanyan, A.A., On the norming constants of the Sturm-Liouville problem, Vestnik Kazan. Gos. Energet. Univ., (3(31)), pp. 7-26, (2016)

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Abstract:
We derive new asymptotic formulae for the norming constants of Sturm-Liouville problem with summable potentials, which generalize and make more precise previously known formulae. Moreover, our formulae take into account the smooth dependence of norming constants on boundary conditions. We also find some new properties of the remainder terms of asymptotics.

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On the norming constants of the Sturm-Liouville problem.pdf
Harutyunyan, T.N., Pahlevanyan, A.A. and Srapionyan, A.V., Riesz bases generated by the spectra of Sturm-Liouville problem, Electronic Journal of Differential Equations, 2013, (71), pp. 1-8, (2013)

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Abstract:
Let $\{\lambda _n^2\} _{n = 0}^\infty$ be the spectra of a Sturm-Liouville problem on $[0,\pi ]$. We investigate the question: Do the systems $\{ \cos(\lambda_nx)\} _{n = 0}^\infty$ or $\{ \sin(\lambda_n x)\} _{n = 0}^\infty$ form Riesz bases in ${L^2}[0,\pi ]$? The answer is almost always positive.

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Riesz bases generated by the spectra of Sturm-Liouville problem.pdf
Harutyunyan, T.N., Pahlevanyan, A.A. and Ashrafyan, Yu.A., On the "movement" of the zeros of eigenfunctions of the Sturm-Liouville problem, Proceedings of Artsakh State University, (1(27)), pp. 12-20, (2013)

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Abstract:
We study the dependence of the zeros of eigenfunctions of Sturm-Liouville problem on the parameters that define the boundary conditions. As a corollary, we obtain Sturm oscillation theorem, which states that the $n$-th eigenfunction has $n$ zeros.

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On the movement of the zeros of eigenfunctions of the Sturm-Liouville problem.pdf
Harutyunyan, T.N., Pahlevanyan, A.A. and Srapionyan, A.V., Riesz bases borned by the spectra of Sturm-Liouville problem (in Russian), Mathematics in Higher School, 8, (3), pp. 36-44, (2012)

Submitted Paper

Pahlevanyan, A.A., On the constructive solution of an inverse Sturm-Liouville problem, Electronic Journal of Differential Equations

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Abstract:
The necessary and sufficient conditions are found for the two sequences $\left\{\mu_n \right\}_{n=0}^{\infty}$ and $\left\{a_n \right\}_{n=0}^{\infty}$ to be the spectrum and the norming constants respectively, for a boundary value problem $-y'' + q \left( x \right) y = \mu y,$ $y \left(0\right)=0,$ $y\left( \pi \right)\cos \beta + y'\left( \pi \right)\sin \beta = 0,$ $\beta \in \left( 0, \pi \right),$ with $q \in L_{\mathbb{R}}^1 \left[0, \pi \right].$

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Thesis

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Submitted Paper