Self-adjoint extensions for discrete linear Hamiltonian systems

Ren, Guojing; Shi, Yuming

doi:10.1016/j.laa.2014.04.016

Cited by 26 publications

(55 citation statements)

References 15 publications

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“…Note that (2.2) is a special discrete Hamiltonian system. By Theorems 5.5 and 7.1 in [12], T is a self-adjoint operator. Now, we show that T has a pure discrete spectrum and bounded from below.…”

Section: Preliminariesmentioning

confidence: 94%

“…For more discussions on discrete Hamiltonian systems, we refer the reader to [11]. Recently, some important results have been obtained about the spectral theory of linear discrete Hamiltonian systems [12][13][14]. There are many good results on the existence of periodic, homoclinic, and heteroclinic orbits of discrete Hamiltonian systems [15][16][17][18][19][20][21].…”

Section: Introductionmentioning

confidence: 99%

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Homoclinic orbits of a class of second-order difference equations

Zhang

Shi

2012

Journal of Mathematical Analysis and Applications

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a b s t r a c tIn this paper, we apply the variational method and the spectral theory of difference operators to investigate the existence of homoclinic orbits of the second-order differencesuperquadratic and subquadratic. Under the assumptions that L(t) is positive definite for sufficiently large |t| ∈ Z, we show that there exists at least one non-trivial homoclinic orbit of the difference equation. Further, if V (t, x) is superquadratic and even with respect to x, then it has infinitely many different non-trivial homoclinic orbits. At the end, two illustrative examples are provided.

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Section: Preliminariesmentioning

confidence: 94%

Section: Introductionmentioning

confidence: 99%

Homoclinic orbits of a class of second-order difference equations

Zhang

Shi

2012

Journal of Mathematical Analysis and Applications

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“…The following theorem which is from 27, Thm. 4.1] gives a relationship between the spectral properties of a selfadjoint extension subspace to those of the corresponding selfadjoint extension operator if the minimal operator generated by (1.1) is densely defined and the maximal operator is single‐valued.…”

Section: Hamiltonian Systems and Subspacesmentioning

confidence: 99%

“…Another difficulty that is always encountered in the difference systems is that the minimal operator generated by (1.1) may be neither densely defined nor single valued, its maximal operator may not be well defined and thus the selfadjoint extension operator for minimal operator cannot be discussed by application of von Neumann theory for densely defined Hermitian operators. This, therefore, requires the theory of Hermitian subspaces where the von Neumann theory has been extended in order to discuss the selfadjoint extension of the minimal Hermitian subspaces and for more details see 25–27, 29 and the references cited therein.…”

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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“…Recently, the first author of the present paper applied the above Coddington results in [14] to extend the GlazmanKrein-Naimark (GKN) theory for symmetric operators to Hermitian subspaces [17]. Applying this GKN theory, she with Sun, and with Ren gave complete characterizations of self-adjoint extensions for second-order symmetric linear difference equations and general linear discrete Hamiltonian systems in both regular and singular cases, separately [6,18]. For more results about non-densely defined Hermitian operators or Hermitian subspaces, we refer to [11][12][13][19][20][21][22][23][24][25] and some references cited therein.…”

Section: Introductionmentioning

confidence: 99%

Spectral properties of self-adjoint subspaces

Shi

Shao

Ren

2013

Linear Algebra and its Applications

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AMS classification: 47A06 47A10 47B15 47B25 39A70 Keywords: Spectrum Hermitian subspace Self-adjoint subspace Singular symmetric linear difference equationThis paper focuses on spectral properties of self-adjoint subspaces in the product space of a Hilbert space. Several classifications of the spectrum of a self-adjoint subspace are introduced, including discrete spectrum, essential spectrum, continuous spectrum, singular continuous spectrum, absolutely continuous spectrum, and singular spectrum. Their relationships between the subspace and its operator part are established, respectively. It is shown that all self-adjoint subspace extensions of a closed Hermitian subspace have the same essential spectrum. As a simple application, it is shown that every self-adjoint subspace extension of the corresponding minimal subspace to a singular second-order symmetric linear difference equation has a bounded spectrum under some conditions. This shows that there is an essential difference between spectral properties of singular symmetric difference operators and classical regular or singular symmetric differential operators.

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Self-adjoint extensions for discrete linear Hamiltonian systems

Cited by 26 publications

References 15 publications

Homoclinic orbits of a class of second-order difference equations

Homoclinic orbits of a class of second-order difference equations

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

Spectral properties of self-adjoint subspaces

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