Spectral analysis of higher order differential operators with unbounded coefficients II

Behncke, Horst; Nyamwala, Fredrick Oluoch

doi:10.1002/mana.201000031

Cited by 4 publications

(7 citation statements)

References 18 publications

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“…Similarly, the author together with others have discussed the deficiency indices as well as the location of absolutely continous spectrum of both self‐adjoint extenstion operators of differential operators as well as for self adjoint extenstion subspaces for difference equations. For more details see . Therefore, the results of this paper can be considered as a continuation of the investigations carried out in the above mentioned papers and in particular, as an extension of some of the results in to the discretised version.…”

Section: Introductionmentioning

confidence: 53%

“…For more details see . Therefore, the results of this paper can be considered as a continuation of the investigations carried out in the above mentioned papers and in particular, as an extension of some of the results in to the discretised version. Here, we have applied asymptotic summation to compute the deficiency indices as well as the location of both essential and continuous spectrum.…”

Section: Introductionmentioning

confidence: 53%

“…This means that the x ‐roots are of the form

x_{n - k} \approx - 4 \frac{p_{k + 1} + q_{k + 1}}{p_{k} + q_{k}} + o false(1 false) .

An application of implicit function theorem shows that the correction term to the x ‐roots which is denoted by o (1) in (3.7) is given by

\frac{\partial normalΦ false(x_{k}, z false)}{\partial x}

with the leading term computed at the pivotal coefficient, that is,

p_{k} + q_{k}

since the application of (3.6) shows that the other terms of

\frac{\partial normalΦ false(x_{k}, z false)}{\partial x}

off the pivotal coefficient tend to zero as

t \to \infty

. Thus we have the correction term approximated by

4^{k} false(p_{k} + q_{k} false) {(() - 4 \frac{p_{k + 1} + q_{k + 1}}{p_{k} + q_{k}})}^{n - k - 1} .

In approximating the

x_{n - k}

‐roots of

Φ (x, z)

using the techniques of , one writes

Φ false(x, z false) = {normalΦ}_{(n - k) 0} false(x, z false) + {normalΦ}_{(n - k) 1} false(x, z false)

where

{normalΦ}_{(n - k}

…”

Section: Essential and Continuous Spectramentioning

confidence: 99%

“…In addition to the problem mentioned above, computation of the zeros of polynomials of degree five or more can be quite involving and there is no closed form formula for doing this. In order to avoid such a problem, we have employed the techniques in [, Section 3.3] which have also been used in to approximate the roots of the associated characteristic polynomial which are partinent ingredients in the analysis of the deficiency indices of the minimal subspace generated by (1.1) and also the essential and continuous spectrum of the selfadjoint extension subspace of this minimal subspace. We have managed to circumvent the problem of non‐densely defined minimal operators generated by (1.1) by using subspace theory to do our analysis.…”

Section: Introductionmentioning

confidence: 99%

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Essential and continuous spectrum of symmetric difference equations

Nyamwala

2017

Mathematische Nachrichten

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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

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Section: Introductionmentioning

confidence: 53%

Section: Introductionmentioning

confidence: 53%

“…This means that the x ‐roots are of the form

x_{n - k} \approx - 4 \frac{p_{k + 1} + q_{k + 1}}{p_{k} + q_{k}} + o false(1 false) .

An application of implicit function theorem shows that the correction term to the x ‐roots which is denoted by o (1) in (3.7) is given by

\frac{\partial normalΦ false(x_{k}, z false)}{\partial x}

with the leading term computed at the pivotal coefficient, that is,

p_{k} + q_{k}

since the application of (3.6) shows that the other terms of

\frac{\partial normalΦ false(x_{k}, z false)}{\partial x}

off the pivotal coefficient tend to zero as

t \to \infty

. Thus we have the correction term approximated by

4^{k} false(p_{k} + q_{k} false) {(() - 4 \frac{p_{k + 1} + q_{k + 1}}{p_{k} + q_{k}})}^{n - k - 1} .

In approximating the

x_{n - k}

‐roots of

Φ (x, z)

using the techniques of , one writes

Φ false(x, z false) = {normalΦ}_{(n - k) 0} false(x, z false) + {normalΦ}_{(n - k) 1} false(x, z false)

where

{normalΦ}_{(n - k}

…”

Section: Essential and Continuous Spectramentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Essential and continuous spectrum of symmetric difference equations

Nyamwala

2017

Mathematische Nachrichten

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“…In order to avoid the first difficulty mentioned above, we have employed the techniques in [15, Sect. 3.3] which have also been used in 10 to approximate the roots of the associated characteristic polynomial which are pertinent ingredients in the analysis of the deficiency indices of the minimal subspace generated by (1.1) and also the absolutely continuous spectrum of the selfadjoint extension subspace of this minimal subspace. We have managed to circumvent the second problem by doing our analysis through the subspaces generated by (1.1).…”

Section: Introductionmentioning

confidence: 99%

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

Agure

Ambogo

Nyamwala

2012

Mathematische Nachrichten

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Using subspace theory together with appropriate smoothness and decay conditions, we calculated the deficiency indices and absolutely continuous spectrum of fourth order difference equations with unbounded coefficients. In particular, we found the absolutely continuous spectrum to be \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}${\mathbb {R}}$\end{document} with a spectral multiplicity one.

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