Boundary Value Problems and Symplectic Algebra for Ordinary Differential and Quasi-differential Operators

Everitt, W. N.; Markus, Lawrence

doi:10.1090/surv/061

Cited by 89 publications

(114 citation statements)

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“…Таким образом, для выраже-ния реализуется случай предельного круга тогда и только тогда, когда все решения уравнения (1.3) принадлежат пространству ℒ 2 (R + ). Приведенные выше сведения хорошо известны и извлечены нами из работ [5], [6]. …”

Section: теперь докажем что справедливаunclassified

Об Асимптотическом Интегрировании Симметрических Квазидифференциальных Уравнений Второго Порядка

Конечная¹,

Konechnaja²

2011

Mat. Zametki

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Section: теперь докажем что справедливаunclassified

Об Асимптотическом Интегрировании Симметрических Квазидифференциальных Уравнений Второго Порядка

Конечная¹,

Konechnaja²

2011

Mat. Zametki

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“…, z n -характеристические числа матрицы A, т.е. корни уравнения F n (z, ν) = 0, если ν < 0, и F n (z, 0)−λ = 0, если ν = 0, расположенные в порядке Re z 1 Re z 2 · · · Re z n ; а q ij , i, j = 1, 2, . .…”

Section: пусть выполненоunclassified

Индексы Дефекта И Спектр Самосопряженных Расширений Некоторых Классов Дифференциальных Операторов

Долгих¹,

Mirzoev²

2006

Матем. сб.

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“…Following work [2] (see also [4]), in Section 4 we define the product of two quasi-differential expression and we propose a method allowing us to obtain asymptotic formulae for the solutions to equation (3) in the case, when the left hand side of this equations is represented as the product of two expressions in the class defined in Section 2. Posing additional assumptions for the matrix ensuring the symmetricity (formal self-adjointness) of the expression , see [1,Sect. I], in Section 5 we define the minimal closed symmetric operator generated by this expression in the space of Lebesgue square integrable functions in [1, +∞) (in the Hilbert space ℒ 2 [1, +∞)).…”

Section: Introductionmentioning

confidence: 99%

“…A]. 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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Boundary Value Problems and Symplectic Algebra for Ordinary Differential and Quasi-differential Operators

Cited by 89 publications

References 0 publications

Об Асимптотическом Интегрировании Симметрических Квазидифференциальных Уравнений Второго Порядка

Об Асимптотическом Интегрировании Симметрических Квазидифференциальных Уравнений Второго Порядка

Индексы Дефекта И Спектр Самосопряженных Расширений Некоторых Классов Дифференциальных Операторов

Asymptotics of solutions to a class of linear differential equations

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