A new approach to asymptotic formulas for eigenfunctions of discontinuous non-selfadjoint Sturm–Liouville operators

2020 ◽  
Vol 11 (4) ◽  
pp. 1805-1820
Author(s):  
Seyfollah Mosazadeh
Keyword(s):  
Asymptotic Formulas ◽  
New Approach ◽  
Sturm Liouville ◽  
Liouville Operators

Asymptotic formulas for Dirichlet boundary value problems

2005 ◽  
Vol 42 (2) ◽  
pp. 153-171 ◽  
Author(s):  
Bülent Yilmaz ◽  
O. A. Veliev
Keyword(s):  
Boundary Conditions ◽  
Main Part ◽  
Dirichlet Boundary ◽  
Arbitrary Order ◽  
Dirichlet Boundary Conditions ◽  
Asymptotic Formulas ◽  
Summable Function ◽  
Special Cases ◽  
Sturm Liouville ◽  
Liouville Operators

In this article we obtain asymptotic formulas of arbitrary order for eigenfunctions and eigenvalues of the nonselfadjoint Sturm-Liouville operators with Dirichlet boundary conditions, when the potential is a summable function. Then using these we compute the main part of the eigenvalues in special cases.

scholarly journals Asymptotic Behavior of Solutions of the Dirac System with an Integrable Potential

2021 ◽  
Vol 93 (5) ◽  
Author(s):  
Łukasz Rzepnicki
Keyword(s):  
Asymptotic Behavior ◽  
Spectral Parameter ◽  
Asymptotic Formulas ◽  
Asymptotic Behavior Of Solutions ◽  
Dirac System ◽  
Eigenvalues And Eigenfunctions ◽  
Integrable Potential ◽  
Sturm Liouville ◽  
Complex Valued ◽  
Liouville Operators

AbstractWe consider the Dirac system on the interval [0, 1] with a spectral parameter $$\mu \in {\mathbb {C}}$$ μ ∈ C and a complex-valued potential with entries from $$L_p[0,1]$$ L p [ 0 , 1 ] , where $$1\le p$$ 1 ≤ p . We study the asymptotic behavior of its solutions in a strip $$|\mathrm{Im}\,\mu |\le d$$ | Im μ | ≤ d for $$\mu \rightarrow \infty $$ μ → ∞ . These results allow us to obtain sharp asymptotic formulas for eigenvalues and eigenfunctions of Sturm–Liouville operators associated with the aforementioned Dirac system.

scholarly journals Spectral analysis of the matrix Sturm–Liouville operator

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Natalia P. Bondarenko
Keyword(s):  
Liouville Operator ◽  
Finite Interval ◽  
Spectral Characteristics ◽  
Asymptotic Formulas ◽  
Inverse Spectral Theory ◽  
Different Types ◽  
The Matrix ◽  
Sturm Liouville ◽  
Shaped Graph ◽  
Liouville Operators

AbstractThe self-adjoint matrix Sturm–Liouville operator on a finite interval with a boundary condition in general form is studied. We obtain asymptotic formulas for the eigenvalues and the weight matrices of the considered operator. These spectral characteristics play an important role in the inverse spectral theory. Our technique is based on an analysis of analytic functions and on the contour integration in the complex plane of the spectral parameter. In addition, we adapt the obtained asymptotic formulas to the Sturm–Liouville operators on a star-shaped graph with two different types of matching conditions.

scholarly journals Asymptotic analysis of non-self-adjoint Hill operators

2013 ◽  
Vol 11 (12) ◽  
Author(s):  
Oktay Veliev
Keyword(s):  
Sufficient Conditions ◽  
Periodic Boundary Conditions ◽  
Asymptotic Formulas ◽  
Spectral Operator ◽  
Hill Operator ◽  
Eigenvalues And Eigenfunctions ◽  
Spectral Singularities ◽  
Sturm Liouville ◽  
The Hill ◽  
Liouville Operators

AbstractWe obtain uniform asymptotic formulas for the eigenvalues and eigenfunctions of the Sturm-Liouville operators L t (q) with a potential q ∈ L 1[0,1] and t-periodic boundary conditions, t ∈ (−π, π]. Using these formulas, we find sufficient conditions on the potential q such that the number of spectral singularities in the spectrum of the Hill operator L(q) in L 2(−∞,∞) is finite. Then we prove that the operator L(q) has no spectral singularities at infinity and it is an asymptotically spectral operator provided that the potential q satisfies sufficient conditions.

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