Subordinacy Analysis and Absolutely Continuous Spectra for Sturm-Liouville Equations with Two Singular Endpoints

1998 ◽  
Vol 41 (1) ◽  
pp. 23-27
Author(s):  
Dominic P. Clemence
Keyword(s):  
Liouville Equation ◽  
Absolute Continuity ◽  
Absolutely Continuous ◽  
Particular Solution ◽  
Sturm Liouville ◽  
The One

AbstractThe Gilbert-Pearson characterization of the spectrum is established for a generalized Sturm-Liouville equation with two singular endpoints. It is also shown that strong absolute continuity for the one singular endpoint problem guarantees absolute continuity for the two singular endpoint problem. As a consequence, we obtain the result that strong nonsubordinacy, at one singular endpoint, of a particular solution guarantees the nonexistence of subordinate solutions at both singular endpoints.

On the Friedrichs extension of semi-bounded difference operators

1994 ◽  
Vol 116 (1) ◽  
pp. 167-177 ◽  
Author(s):  
M. Benammar ◽  
W. D. Evans
Keyword(s):  
Liouville Equation ◽  
Liouville Operator ◽  
Difference Operators ◽  
Dirichlet Integral ◽  
Friedrichs Extension ◽  
Sturm Liouville

In [5] Kalf obtained a characterization of the Friedrichs extension TF of a general semi-bounded Sturm–Liouville operator T, the only assumptions made on the coefficients being those necessary for T to be defined. The domain D(TF) of TF was described in terms of ‘weighted’ Dirichiet integrals involving the principal and non-principal solutions of an associated non-oscillatory Sturm–Liouville equation. Conditions which ensure that members of D(TF) have a finite Dirichlet integral were subsequently given by Rosenberger in [7].

Weighted Estimates for Solutions of the General Sturm-Liouville Equation and the Everitt-Giertz Problem. I

2014 ◽  
Vol 58 (1) ◽  
pp. 125-147 ◽  
Author(s):  
N. A. Chernyavskaya ◽  
L. A. Shuster
Keyword(s):  
Liouville Equation ◽  
Positive Constant ◽  
Absolute Constant ◽  
Absolutely Continuous ◽  
Weighted Estimates ◽  
Negative Function ◽  
Almost Everywhere ◽  
Absolute Positive Constant ◽  
Sturm Liouville ◽  
Image Position

AbstractConsider the equationwhereƒ∈Lp(ℝ),p∈ (1, ∞) andBy a solution of (*), we mean any functionyabsolutely continuous together with (ry′) and satisfying (*) almost everywhere on ℝ. In addition, we assume that (*) is correctly solvable in the spaceLp(ℝ), i.e.(1) for any function, there exists a unique solutiony∈Lp(ℝ) of (*);(2) there exists an absolute constantc1(p) > 0 such that the solutiony∈Lp(ℝ) of (*) satisfies the inequalityWe study the following problem on the strengthening estimate (**). Let a non-negative functionbe given. We have to find minimal additional restrictions forθunder which the following inequality holds:Here,yis a solution of (*) from the classLp(ℝ), andc2(p) is an absolute positive constant.

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.

On a variational characterization of the Fučík spectrum of the Laplacian and a superlinear Sturm–Liouville equation

2004 ◽  
Vol 134 (3) ◽  
pp. 557-577 ◽  
Author(s):  
Eugenio Massa
Keyword(s):  
Finite Number ◽  
Liouville Equation ◽  
Neumann Boundary ◽  
Linking Theorem ◽  
Variational Characterization ◽  
Fučík Spectrum ◽  
Fučik Spectrum ◽  
Sturm Liouville ◽  
Fucik Spectrum

In the first part of this paper, a variational characterization of parts of the Fučík spectrum for the Laplacian in a bounded domain Ω is given. The proof uses a linking theorem on sets obtained through a suitable deformation of subspaces of H1 (Ω). In the second part, a nonlinear Sturm–Liouville equation with Neumann boundary conditions on an interval is considered, where the nonlinearity intersects all but a finite number of eigenvalues. It is proved that, under certain conditions, this equation is solvable for arbitrary forcing terms. The proof uses a comparison of the minimax levels of the functional associated to this equation with suitable values related to the Fucík spectrum.

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.

scholarly journals On Functions of Bounded(p,k)-Variation

2012 ◽  
Vol 2012 ◽  
pp. 1-9 ◽  
Author(s):  
N. Merentes ◽  
S. Rivas ◽  
J. L. Sanchez
Keyword(s):  
Sobolev Space ◽  
Real Function ◽  
Compact Interval ◽  
Absolutely Continuous ◽  
Alternative Characterization ◽  
The One

We introduce and study the concept of(p,k)-variation (1<p<∞,k∈N)of a real function on a compact interval. In particular, we prove that a functionu:[a,b]→Rhas bounded(p,k)-variation if and only ifu(k-1)is absolutely continuous on[a,b]andu(k)belongs toLp[a,b]. Moreover, an explicit connection between the(p,k)-variation ofuand theLp-norm ofu(k)is given which is parallel to the classical Riesz formula characterizing functions in the spacesRVp[a,b]andAp[a,b]. This may also be considered as an alternative characterization of the one variable Sobolev spaceWpk[a,b].

scholarly journals Synthesis and Polymerization Kinetics of Rigid Tricyanate Ester

Polymers ◽  
2021 ◽  
Vol 13 (11) ◽  
pp. 1686
Author(s):  
Andrey Galukhin ◽  
Roman Nosov ◽  
Ilya Nikolaev ◽  
Elena Melnikova ◽  
Daut Islamov ◽  
...  
Keyword(s):  
Kinetic Analysis ◽  
Single Step ◽  
Polymerization Kinetics ◽  
Scanning Calorimetry ◽  
Modulated Differential Scanning Calorimetry ◽  
Diffusion Controlled ◽  
Polymerization Product ◽  
The One ◽  
Kinetics Of

A new rigid tricyanate ester consisting of seven conjugated aromatic units is synthesized, and its structure is confirmed by X-ray analysis. This ester undergoes thermally stimulated polymerization in a liquid state. Conventional and temperature-modulated differential scanning calorimetry techniques are employed to study the polymerization kinetics. A transition of polymerization from a kinetic- to a diffusion-controlled regime is detected. Kinetic analysis is performed by combining isoconversional and model-based computations. It demonstrates that polymerization in the kinetically controlled regime of the present monomer can be described as a quasi-single-step, auto-catalytic, process. The diffusion contribution is parameterized by the Fournier model. Kinetic analysis is complemented by characterization of thermal properties of the corresponding polymerization product by means of thermogravimetric and thermomechanical analyses. Overall, the obtained experimental results are consistent with our hypothesis about the relation between the rigidity and functionality of the cyanate ester monomer, on the one hand, and its reactivity and glass transition temperature of the corresponding polymer, on the other hand.

scholarly journals Congruence pairs of principal MS-algebras and perfect extensions

2020 ◽  
Vol 70 (6) ◽  
pp. 1275-1288
Author(s):  
Abd El-Mohsen Badawy ◽  
Miroslav Haviar ◽  
Miroslav Ploščica
Keyword(s):  
Boolean Algebra ◽  
Congruence Lattices ◽  
Descriptive Solution ◽  
The One ◽  
Special Case ◽  
Congruence Pair

AbstractThe notion of a congruence pair for principal MS-algebras, simpler than the one given by Beazer for K2-algebras [6], is introduced. It is proved that the congruences of the principal MS-algebras L correspond to the MS-congruence pairs on simpler substructures L°° and D(L) of L that were associated to L in [4].An analogy of a well-known Grätzer’s problem [11: Problem 57] formulated for distributive p-algebras, which asks for a characterization of the congruence lattices in terms of the congruence pairs, is presented here for the principal MS-algebras (Problem 1). Unlike a recent solution to such a problem for the principal p-algebras in [2], it is demonstrated here on the class of principal MS-algebras, that a possible solution to the problem, though not very descriptive, can be simple and elegant.As a step to a more descriptive solution of Problem 1, a special case is then considered when a principal MS-algebra L is a perfect extension of its greatest Stone subalgebra LS. It is shown that this is exactly when de Morgan subalgebra L°° of L is a perfect extension of the Boolean algebra B(L). Two examples illustrating when this special case happens and when it does not are presented.

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