Deficiency indices and spectrum of fourth order difference equations with unbounded coefficients

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We have studied the absolutely continuous spectrum of a selfadjoint subspace extension generated by a three‐term fourth order difference equation with bounded coefficients using subspace theory. In particular, we have shown that the absolutely continuous spectrum exists outside a certain bounded interval. In addition, we have computed the spectral multiplicity as well as the location of absolutely continuous spectrum of selfadjoint subspace extension under certain asymptotic conditions.

Section: Introductionmentioning

confidence: 58%

“…Our starting point is (1.1) and we study the solutions of (1.1) with

p (t)

and

q (t)

real valued. We analyse the case where

p (t)  0 as t  \infty, q (t) = O (w (t)), w (t) = 1 .

Similarly, we will assume that

\frac{p}{{(- (q - z))}^{\frac{1}{2}}}  0, \frac{p^{2}}{(q - z)} \in ℓ^{1} .

In order to simplify notations, let

ν = {(- (q - z))}^{\frac{1}{2}}

, then we make similar assumptions like those in , (2.1)], which are necessary for asymptotic summation

Δ ν, \frac{ν Δ p}{q}, \frac{Δ p}{q} \in ℓ^{2}

Δ^{2} ν, {((), Δ, ν)}^{2}, {((), \frac{ν Δ p}{q})}^{2}, \frac{ν Δ^{2} p}{q}, {((), \frac{Δ p}{q})}^{2}, \frac{Δ^{2} p}{q} \in ℓ^{1} .

Section: First Order System and Subspacesmentioning

confidence: 99%

“…If

α > 0

, then

b t^{α} \to \infty

t \to \infty

. If

b t^{α}

is unbounded, then different techniques are needed to approximate the eigenvalues and for more information on how to do that, see .…”

Section: Absolutely Continuous Spectrummentioning

confidence: 99%

“…In order to simplify notations, let ν = (−(q − z)) 1 2 , then we make similar assumptions like those in [1], (2.1)], which are necessary for asymptotic summation…”

Section: First Order System and Subspacesmentioning

confidence: 99%

Section: First Order System and Subspacesmentioning

confidence: 99%

See 3 more Smart Citations

Absolutely continuous spectrum of fourth order difference equations

Nyamwala

2014

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Essential spectrum of singular discrete linear Hamiltonian systems

Sun

Kong

Shi

2015

The paper is concerned with the essential spectral points of singular discrete linear Hamiltonian systems. Several sufficient conditions for a real point to be in the essential spectrum are obtained in terms of the number of linearly independent square-summable solutions of the corresponding homogeneous linear system, and a sufficient and necessary condition for a real point to be in the essential spectrum is given in terms of the number of linearly independent square-summable solutions of the corresponding nonhomogeneous linear system. As a direct consequence, the corresponding results for singular higher-order symmetric vector difference expressions are given.

Essential and continuous spectrum of symmetric difference equations

Nyamwala

2017

Self Cite

Essential and continuous spectrum of symmetric difference equations have been investigated. It has been shown that the deficiency indices and the existence of these components of the spectrum are determined by the growth conditions of the coefficients. In particular, the deficiency indices are superimposition of those clusters determined by the coefficient growth. Finally, we have proved the neccessary and sufficient conditions for the existence of essential spectrum of selfadjoint subspace extensions using subspace theory and asymtotic summation. K E Y W O R D Ssubspaces, deficiency indices, essential spectrum, continuous spectrum M S C ( 2 0 1 0 ) Primary: 39A20, Secondary: 47A55 www.mn-journal.org