scholarly journals A New Type of Sturm-Liouville Equation in the Non-Newtonian Calculus

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Sertac Goktas
Keyword(s):  
Liouville Equation ◽  
Spectral Properties ◽  
Liouville Problem ◽  
Mathematical Physics ◽  
Asymptotic Estimates ◽  
One Dimensional ◽  
Multiplicative Calculus ◽  
New Type ◽  
Sturm Liouville ◽  
The One

In mathematical physics (such as the one-dimensional time-independent Schrödinger equation), Sturm-Liouville problems occur very frequently. We construct, with a different perspective, a Sturm-Liouville problem in multiplicative calculus by some algebraic structures. Then, some asymptotic estimates for eigenfunctions of the multiplicative Sturm-Liouville problem are obtained by some techniques. Finally, some basic spectral properties of this multiplicative problem are examined in detail.

On the Canonical Solution of the Sturm–Liouville Problem with Singularity and Turning Point of Even Order

2011 ◽  
Vol 54 (3) ◽  
pp. 506-518 ◽  
Author(s):  
A. Neamaty ◽  
S. Mosazadeh
Keyword(s):  
Liouville Equation ◽  
Turning Point ◽  
Infinite Product ◽  
Liouville Problem ◽  
Asymptotic Estimates ◽  
Product Representation ◽  
Representation Of Solutions ◽  
Even Order ◽  
Sturm Liouville ◽  
Infinite Product Representation

AbstractIn this paper, we are going to investigate the canonical property of solutions of systems of differential equations having a singularity and turning point of even order. First, by a replacement, we transform the system to the Sturm–Liouville equation with turning point. Using of the asymptotic estimates provided by Eberhard, Freiling, and Schneider for a special fundamental system of solutions of the Sturm–Liouville equation, we study the infinite product representation of solutions of the systems. Then we transform the Sturm–Liouville equation with turning point to the equation with singularity, then we study the asymptotic behavior of its solutions. Such representations are relevant to the inverse spectral problem.

A Sturm-Liouville Equation: The Time-Independent One-Dimensional Schrödinger Equation

2017 ◽  
Author(s):  
John A. Adam
Keyword(s):  
Schrödinger Equation ◽  
Liouville Equation ◽  
Schrodinger Equation ◽  
Bound States ◽  
Classical Case ◽  
Liouville Theory ◽  
Weyl’S Theorem ◽  
One Dimensional ◽  
Sturm Liouville ◽  
The One

This chapter examines the mathematical properties of the time-independent one-dimensional Schrödinger equation as they relate to Sturm-Liouville problems. The regular Sturm-Liouville theory was generalized in 1908 by the German mathematician Hermann Weyl on a finite closed interval to second-order differential operators with singularities at the endpoints of the interval. Unlike the classical case, the spectrum may contain both a countable set of eigenvalues and a continuous part. The chapter first considers the one-dimensional Schrödinger equation in the standard dimensionless form (with independent variable x) and various relevant theorems, along with the proofs, before discussing bound states, taking into account bound-state theorems and complex eigenvalues. It also describes Weyl's theorem, given the Sturm-Liouville equation, and looks at two cases: the limit point and limit circle. Four examples are presented: an “eigensimple” equation, Bessel's equation of order ? greater than or equal to 0, Hermite's equation, and Legendre's equation.

Perturbation theory for a Sturm-Liouville problem with an interior singularity

1987 ◽  
Vol 414 (1846) ◽  
pp. 255-269 ◽  
Keyword(s):  
Perturbation Theory ◽  
Differential Operators ◽  
Liouville Problem ◽  
One Dimensional ◽  
Resolvent Convergence ◽  
Norm Resolvent Convergence ◽  
Sturm Liouville ◽  
Quantitative Properties ◽  
The One ◽  
Interior Singularity

The perturbation theory of operators and forms is used to construct Sturm-Liouville differential operators for potentials with I/ x , Pf (1/ x ) and, for ϵ →0+, 1/( x -i ϵ ) interior singularities. The norm resolvent convergence of approximating sequences of operators with smooth potentials is established and various qualitative and quantitative properties of their spectra are obtained. For an infinite interval, a comparison is made with the one-dimensional Coulomb problem possessing a potential 1/| x |.

scholarly journals On partial fractional Sturm–Liouville equation and inclusion

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Zohreh Zeinalabedini Charandabi ◽  
Hakimeh Mohammadi ◽  
Shahram Rezapour ◽  
Hashem Parvaneh Masiha
Keyword(s):  
Differential Equation ◽  
Liouville Equation ◽  
Existence Of Solutions ◽  
Liouville Problem ◽  
Existence Of A Solution ◽  
Contractive Mappings ◽  
Sturm Liouville

AbstractThe Sturm–Liouville differential equation is one of interesting problems which has been studied by researchers during recent decades. We study the existence of a solution for partial fractional Sturm–Liouville equation by using the α-ψ-contractive mappings. Also, we give an illustrative example. By using the α-ψ-multifunctions, we prove the existence of solutions for inclusion version of the partial fractional Sturm–Liouville problem. Finally by providing another example and some figures, we try to illustrate the related inclusion result.

Sturm-Liouville operators with an indefinite weight function

1977 ◽  
Vol 78 (1-2) ◽  
pp. 161-191 ◽  
Author(s):  
K. Daho ◽  
H. Langer
Keyword(s):  
Weight Function ◽  
Liouville Equation ◽  
Spectral Properties ◽  
Scalar Product ◽  
Main Tool ◽  
Indefinite Weight ◽  
Indefinite Weight Function ◽  
Sturm Liouville ◽  
Liouville Operators

SynopsisSpectral properties of the singular Sturm-Liouville equation –(p−1y′)′ + qy = λry with an indefinite weight function r are studied in . The main tool is the theory of definitisable operators in spaces with an indefinite scalar product.

scholarly journals On the spectral properties of the Schrödinger operator with a periodic PT-symmetric potential

2017 ◽  
Vol 14 (05) ◽  
pp. 1750065 ◽  
Author(s):  
Oktay Veliev
Keyword(s):  
Schrödinger Operator ◽  
Spectral Properties ◽  
Schrodinger Operator ◽  
Symmetric Potential ◽  
One Dimensional ◽  
Symmetric Complex ◽  
The One ◽  
Complex Valued

In this paper, we investigate the spectrum and spectrality of the one-dimensional Schrödinger operator with a periodic PT-symmetric complex-valued potential.

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