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].

scholarly journals An inverse problem for the non-self-adjoint matrix Sturm-Liouville operator

2018 ◽  
Vol 50 (1) ◽  
pp. 71-102 ◽  
Author(s):  
Natalia Pavlovna Bondarenko
Keyword(s):  
Inverse Problem ◽  
Liouville Operator ◽  
Finite Interval ◽  
Spectral Characteristics ◽  
Sufficient Conditions ◽  
Adjoint Matrix ◽  
Method Of Spectral Mappings ◽  
Sturm Liouville ◽  
Necessary And Sufficient

The inverse problem of spectral analysis for the non-self-adjoint matrix Sturm-Liouville operator on a finite interval is investigated. We study properties of the spectral characteristics for the considered operator, and provide necessary and sufficient conditions for the solvability of the inverse problem. Our approach is based on the constructive solution of the inverse problem by the method of spectral mappings. The characterization of the spectral data in the self-adjoint case is given as a corollary of the main result.

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 Schrödinger's factorization method for Sturm-Liouville operators

1978 ◽  
Vol 80 (1-2) ◽  
pp. 67-84 ◽  
Author(s):  
U.-W. Schmincke
Keyword(s):  
Discrete Spectrum ◽  
Liouville Operator ◽  
Differential Operators ◽  
Factorization Method ◽  
Friedrichs Extension ◽  
First Order ◽  
Sturm Liouville ◽  
Image Position ◽  
Special Case ◽  
Liouville Operators

SynopsisWe consider the Friedrichs extension A of a minimal Sturm-Liouville operator L0 and show that A admits a Schrödinger factorization, i.e. that one can find first order differential operators Bk with where the μk are suitable numbers which optimally chosen are just the lower eigenvalues of A (if any exist). With the help of this theorem we derive for the special case L0u = −u″ + q(x)u with q(x) → 0 (|x| → ∞) the inequalityσd(A) being the discrete spectrum of A. This inequality is seen to be sharp to some extent.

scholarly journals On the multiplicity of the eigenvalues of the vectorial Sturm-Liouville equation

2011 ◽  
Vol 42 (3) ◽  
pp. 265-274
Author(s):  
Chien-Wen Lin
Keyword(s):  
Potential Function ◽  
Liouville Equation ◽  
Liouville Operator ◽  
Symmetric Matrix ◽  
Algebraic Multiplicity ◽  
Dirichlet Eigenvalue ◽  
Geometric Multiplicity ◽  
Real Symmetric Matrix ◽  
Sturm Liouville

Let $ Q(x) $ be a continuous $ m\times m $ real symmetric matrix-valued function defined on $ [0,1] $, and denote the Sturm-Liouville operator $ -\frac{d^2}{dx^2}+Q(x) $ as $ L_Q $ with $ Q(x)$ as its potential function. In this paper we prove that for each Dirichlet eigenvalue $ \lambda_* $ of $L_Q$, the geometric multiplicity of $ \lambda_* $ is equal to its algebraic multiplicity. Applying this result, we get a necessary and sufficiently condition such that each Dirichlet eigenvalue of $ L_Q $ is of multiplicity $ m $.

Inverse nodal problems for the Sturm–Liouville operator with nonlocal integral conditions

2017 ◽  
Vol 25 (6) ◽  
Author(s):  
Yi-Teng Hu ◽  
Chuan-Fu Yang ◽  
Xiao-Chuan Xu
Keyword(s):  
Potential Function ◽  
Liouville Equation ◽  
Liouville Operator ◽  
Dense Subset ◽  
Integral Conditions ◽  
End Points ◽  
Nodal Points ◽  
Inverse Nodal Problems ◽  
Sturm Liouville ◽  
Nonlocal Integral Conditions

AbstractIn this work, we consider inverse nodal problems of the Sturm–Liouville equation with nonlocal integral conditions at two end-points. We prove that a dense subset of nodal points uniquely determine the potential function of the Sturm–Liouville equation up to a constant.

Characterization of the scattering data for the Sturm-Liouville operator

2013 ◽  
Vol 37 (17) ◽  
pp. 2626-2637
Author(s):  
A.A. Nabiev ◽  
S. Saltan ◽  
M. Gürdal
Keyword(s):  
Liouville Operator ◽  
Scattering Data ◽  
Sturm Liouville

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.

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