scholarly journals On Povzner–Wienholtz-type self-adjointness results for matrix-valued Sturm–Liouville operators

2003 ◽  
Vol 133 (4) ◽  
pp. 747-758 ◽  
Author(s):  
Steve Clark ◽  
Fritz Gesztesy
Keyword(s):  
Half Line ◽  
Sturm Liouville ◽  
Image Position ◽  
Liouville Operators

We derive Povzner–Wienholtz-type self-adjointness results for m × m matrix-valued Sturm–Liouville operators in L2((a, b);R dx)m, m ∈ N, for (a, b) a half-line or R.

Dissipative Sturm-Liouville operators

1981 ◽  
Vol 88 (3-4) ◽  
pp. 329-343 ◽  
Author(s):  
Ian Knowles
Keyword(s):  
Hilbert Space ◽  
Differential Expression ◽  
Minimal Operator ◽  
Sturm Liouville ◽  
Image Position ◽  
Complex Valued ◽  
Liouville Operators

SynopsisConsider the differential expressionwherepandw> 0 are real-valued andqis complex-valued onI. A number of criteria are established for certain extensions of the minimal operator generated by τ in the weighted Hilbert spaceto be maximal dissipative.

scholarly journals Description of the scattering data for Sturm-Liouville operators on the half-line

2019 ◽  
Vol 39 (4) ◽  
pp. 557-576
Author(s):  
Yaroslav Mykytyuk ◽  
Nataliia Sushchyk
Keyword(s):  
Nondecreasing Function ◽  
Scattering Data ◽  
Half Line ◽  
Sturm Liouville ◽  
Liouville Operators

We describe the set of the scattering data for self-adjoint Sturm-Liouville operators on the half-line with potentials belonging to \(L_1(\mathbb{R}_+,\rho(x)\,\text{d} x)\), where \(\rho:\mathbb{R}_+\to\mathbb{R}_+\) is a monotonically nondecreasing function from some family \(\mathscr{R}\). In particular, \(\mathscr{R}\) includes the functions \(\rho(x)=(1+x)^{\alpha}\) with \(\alpha\geq 1\).

An inverse problem for Sturm–Liouville operators on the half-line with complex weights

2019 ◽  
Vol 27 (3) ◽  
pp. 439-443
Author(s):  
Vjacheslav Yurko
Keyword(s):  
Inverse Problem ◽  
Inverse Problems ◽  
Differential Operators ◽  
Spectral Characteristics ◽  
Half Line ◽  
Sturm Liouville ◽  
The Given ◽  
Complex Valued ◽  
The Uniqueness Theorem ◽  
Liouville Operators

Abstract Second order differential operators on the half-line with complex-valued weights are considered. Properties of spectral characteristics are established, and the inverse problem of recovering operator’s coefficients from the given Weyl-type function is studied. The uniqueness theorem is proved for this class of nonlinear inverse problems, and a number of examples are provided.

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.

Sturm-Liouville problems for the p-Laplacian on a half-line

2010 ◽  
Vol 53 (2) ◽  
pp. 271-291 ◽  
Author(s):  
Paul Binding ◽  
Patrick J. Browne ◽  
Illya M. Karabash
Keyword(s):  
Eigenvalue Problem ◽  
Nonlinear Eigenvalue Problem ◽  
Half Line ◽  
Sturm Liouville ◽  
Image Position ◽  
Nonlinear Eigenvalue

AbstractThe nonlinear eigenvalue problemfor 0 ≤ x < ∞, fixed p ∈ (1, ∞), and with y′(0)/y(0) specified is studied under various conditions on the coefficients s and q, leading to either oscillatory or non-oscillatory situations.

Resolvent Means and Inverting Generalized Fourier Transforms

1986 ◽  
Vol 38 (4) ◽  
pp. 861-877 ◽  
Author(s):  
Louise A. Raphael
Keyword(s):  
Boundary Conditions ◽  
Weight Function ◽  
Eigenfunction Expansion ◽  
Resolvent Operator ◽  
Fourier Transforms ◽  
Half Line ◽  
Positive Spectrum ◽  
Sturm Liouville ◽  
Image Position ◽  
The Resolvent Operator

Let S-L denote a singular Sturm-Liouville system on the half line with homogeneous boundary conditions, possessing a discrete negative and continuous positive spectrum. Let A be the S-L operator and Sα(f; x) the S-L eigenfunction expansion associated with the resolvent operator (z – A)–1, z complex. That is, Sα(f; x) denotes the resolvent summability means with weight function z(z – λ)–1 (or (1 + tλ)–1 where t = – 1/z).We first study the problem of determining when(1)where is the Green's function associated with a certain perturbation of our system.

scholarly journals An inverse problem for Sturm-Liouville operators on the half-line having Bessel-type singularity in an interior point

2013 ◽  
Vol 11 (12) ◽  
Author(s):  
Alexey Fedoseev
Keyword(s):  
Inverse Problem ◽  
Interior Point ◽  
Sufficient Conditions ◽  
Weyl Function ◽  
Half Line ◽  
Type Singularity ◽  
Sturm Liouville ◽  
Necessary And Sufficient ◽  
The Given ◽  
Liouville Operators

AbstractWe study the inverse problem of recovering Sturm-Liouville operators on the half-line with a Bessel-type singularity inside the interval from the given Weyl function. The corresponding uniqueness theorem is proved, a constructive procedure for the solution of the inverse problem is provided, also necessary and sufficient conditions for the solvability of the inverse problem are obtained.

Lower Bounds for the Essential Spectrum of Fourth-Order Differential Operators

1969 ◽  
Vol 21 ◽  
pp. 460-465
Author(s):  
Kurt Kreith
Keyword(s):  
Lower Bound ◽  
Essential Spectrum ◽  
Lower Bounds ◽  
Fourth Order ◽  
Differential Operators ◽  
Intimate Connection ◽  
Sturm Liouville ◽  
Image Position ◽  
Oscillation Properties ◽  
Liouville Operators

In this paper, we seek to determine the greatest lower bound of the essential spectrum of self-adjoint singular differential operators of the form1where 0 ≦ x < ∞. In the event that this bound is + ∞, our results will yield criteria for the discreteness of the spectrum of (1).Such bounds have been established by Friedrichs (3) for Sturm-Liouville operators of the formand our techniques will be closely related to those of (3). However, instead of studying the solutions of2directly, we shall exploit the intimate connection between the infimum of the essential spectrum of (1) and the oscillation properties of (2).

scholarly journals Necessary and sufficient conditions for the solvability of the inverse problem for non-self-adjoint pencils of Sturm-Liouville operators on the half-line

2011 ◽  
Vol 42 (3) ◽  
pp. 247-258 ◽  
Author(s):  
Vjacheslav Yurko
Keyword(s):  
Boundary Condition ◽  
Inverse Problem ◽  
Differential Operators ◽  
Sufficient Conditions ◽  
Weyl Function ◽  
Half Line ◽  
Method Of Spectral Mappings ◽  
Sturm Liouville ◽  
Necessary And Sufficient ◽  
Liouville Operators

Non-self-adjoint Sturm-Liouville differential operators on the half-line with a boundary condition depending polynomially on the spectral parameter are studied. We investigate the inverse problem of recovering the operator from the Weyl function. For this inverse problem we provide necessary and suffcient conditions for its solvability along with a procedure for constructing its solution by the method of spectral mappings.

Uniqueness of positive solutions of some semilinear Sturm–Liouville problems on the half line

1984 ◽  
Vol 97 ◽  
pp. 259-263 ◽  
Author(s):  
J. F. Toland
Keyword(s):  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Positive Solutions ◽  
Simple Proof ◽  
Boundary Value ◽  
Half Line ◽  
Linear Boundary ◽  
Autonomous Case ◽  
Sturm Liouville ◽  
Image Position

SynopsisThis note gives a simple proof of uniqueness for positive solutions of certain non-linear boundary value problems on ℝ+ which are typified by the equationwith boundary conditions u′(0) = u(+∞) = 0. In the autonomous case (r ≡ 1), this is easy to see, by quadrature. The proof here supposes r to be non-increasing on ℝ+.

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