Deficiency indices and spectrum of self-adjoint extensions of some classes of differential operators

Долгих, Ирина Николаевна; Mirzoev, K. A.

doi:10.1070/sm2006v197n04abeh003769

Cited by 3 publications

(4 citation statements)

References 3 publications

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“…The validity of this lemma follows directly from the definition of quasi-derivatives and in case p ≤ m is included in [10], p > m is in [11].…”

Section: Auxiliary Resultsmentioning

confidence: 94%

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Spectral analysis of one-term symmetric differential operators of even order with interior singularity

Braeutigam

2015

Preprint

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In this paper we discuss the spectral properties of one-term symmetric differential operators of even order with interior singularity, namely, we determine the deficiency numbers, describe its self-adjoint extensions and their spectrum. It is assumed that the operators are generated by the differential expressionthe coefficient c(x) has one zero on the set I and the orders of this zero on the right side and the left side are not necessarily equal.

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“…The validity of this lemma follows directly from the definition of quasi-derivatives and in case p ≤ m is included in [10], p > m is in [11].…”

Section: Auxiliary Resultsmentioning

confidence: 94%

“…In this paper, this fact will also form the basis of the assumption that a(z) and b(z) are analytic functions, as in that case, it is already possible to use methods of the analytical theory of differential equations. Such approach was also used in [11] and [12].…”

Section: Introductionmentioning

confidence: 99%

Spectral analysis of one-term symmetric differential operators of even order with interior singularity

Braeutigam

2015

Preprint

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“…such that the eigenvalues of the matrix coincide, as > 0, with the roots of this polynomial, for more details on the polynomials F 2 and F 2 +1 see [6], [7].…”

Section: 2mentioning

confidence: 99%

“…Moreover, the Shin-Zettl matrices 2 and 2 +1 constructed in [1] and generating expression (7)) are such that if we replace the smoothness of the entries in these matrices by their local integrability and if the derivatives in formulae (1) and (2) are treated in the distribution sense, then we can open the brackets and a regular generalized function in (2) for ∈ ( ) is represented as (7) in terms of the theory of generalized functions. At that we stress that the coefficients and in expression (7) should be only locally integrable (see [6], [7]). …”

Section: Introductionmentioning

confidence: 99%

Asymptotics of solutions to a class of linear differential equations

Konechnaja¹,

Mirzoev²

2017

Ufimskii Matematicheskii Zhurnal

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Abstract. In the paper we find the leading term of the asymptotics at infinity for some fundamental system of solutions to a class of linear differential equations of arbitrary order = , where is a fixed complex number. At that we consider a special class of ShinZettl type and is a quasi-differential expression generated by the matrix in this class. The conditions we assume for the primitives of the coefficients of the quasi-differential expression , that is, for the entries of the corresponding matrix, are not related with their smoothness but just ensures a certain power growth of these primitives at infinity. Thus, the coefficients of the expression can also oscillate. In particular, this includes a wide class of differential equations of arbitrary even or odd order with distribution coefficients of finite order. Employing the known definition of two quasi-differential expressions with nonsmooth coefficients, in the work we propose a method for obtaining asymptotic formulae for the fundamental system of solutions to the considered equation in the case when the left hand side of this equations is represented as a product of two quasi-differential expressions.The obtained results are applied for the spectral analysis of the corresponding singular differential operators. In particular, assuming that the quasi-differential expression is symmetric, by the known scheme we define the minimal closed symmetric operator generated by this expression in the space of Lebesgue square-integrable on [1, +∞) functions (in the Hilbert space ℒ 2 [1, +∞)) and we calculate the deficiency indices for this operator.Keywords: Quasi-derivative, quasi-differential expression, the main term of asymptotic of the fundamental system of solutions, minimal closed symmetric differential operator, deficiency numbers.

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The resolvent equation of the one-dimensional Schrödinger operator on the whole axis

Алиев

Eyvazov

2012

Sib Math J

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Deficiency indices and spectrum of self-adjoint extensions of some classes of differential operators

Cited by 3 publications

References 3 publications

Spectral analysis of one-term symmetric differential operators of even order with interior singularity

Spectral analysis of one-term symmetric differential operators of even order with interior singularity

Asymptotics of solutions to a class of linear differential equations

The resolvent equation of the one-dimensional Schrödinger operator on the whole axis

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