On the Spectrum of the Differential Operators of Odd Order with $$\mathcal{PT}$$-Symmetric Coefficients

This paper is devoted to a study of the longtime behavior of the hyperbolic equations with an arbitrary internal damping, under sharp regularity assumptions that both the principal part coefficients and the boundary of the space domain (in which the system evolves) are continuously differentiable. For this purpose, we derive a new point-wise inequality for second differential operators with symmetric coefficients. Then, based on a global Carleman estimate, we establish an estimate on the underlying resolvent operator of the equation, via which, we show the logarithmic decay rate for solutions of the hyperbolic equations.

In this paper we establish well posedness of the Neumann problem with boundary data in $$L^2$$ or the Sobolev space $$\dot{W}^2_{-1}$$ , in the half space, for linear elliptic differential operators with coefficients that are constant in the vertical direction and in addition are self adjoint. This generalizes the well known well posedness result of the second order case and is based on a higher order and one sided version of the classic Rellich identity, and is the first known well posedness result for an elliptic divergence form higher order operator with rough variable coefficients and boundary data in a Lebesgue or Sobolev space.

The paper is devoted to the study of spectral properties of differential operators of arbitrary odd order with a summable potential and periodic boundary conditions. For large values of the spectral parameter the asymptotics of the solutions of the differential equation that defines the differential operator is obtained. The differential equation that defines the differential operator is reduced to the Volterra integral equation. The integral equation is solved by Picard's method of successive approximations. The method of studying of operators with a summable potential is an extension of the method of studying operators with piecewise smooth coefficients. The study of periodic boundary conditions leads to the study of the roots of the entire function represented in the form of an arbitrary odd-order determinant. To obtain the roots of this function, the indicator diagram has been examined. The roots of this equation are in the sectors of an infinitesimal angle, determined by the indicator diagram. In the paper the asymptotics of eigenvalues of the differential operator under consideration is found. The obtained formulas make it impossible to study the spectral properties of the eigenfunctions and to derive the formula for the first regularized trace of the differential operator under study. Keywords: The differential operator of odd order, spectral parameter, summable potential, periodic boundary conditions, indicator diagram, asymptotics of solutions, asymptotics of eigenvalues.

The present paper is the first one in a series of two papers devoted to a unified approach to the problem of completeness of the generalized eigenvectors (the root vectors) for a specific class of linear non‐selfadjoint unbounded differential operators. The list of the problems for which such operators are the dynamics generators includes the following: (a) initial boundary‐value problem (IBVP) for a non‐homogeneous string with both distributed and boundary damping; (b) IBVP for small vibrations of an ideal filament with dissipative boundary condition at one end and with a heavy load at the other end; (c) IBVP for a three‐dimensional damped wave equation with spherically symmetric coefficients and both distributed and boundary damping. In the second paper of the series the following problems will be considered: (a) IBVP for a system of two coupled hyperbolic equations constituting a Timoshenko beam model with variable coefficients and boundary damping; (b) IBVP for a coupled Euler–Bernoulli and Timoshenko beam model with boundary energy dissipation (the model known in engineering literature as bending‐torsion vibration model); (c) IBVP for a system of two Timoshenko beams coupled through linear Van der Waals forces. The model of (c) describes vibrational motion of a double‐walled carbon nanotube. In all of the above cases, the result has been obtained by using Krien’s Theorem on completeness of root vectors of a dissipative operator with a nuclear imaginary part. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim

Problems relating to the asymptotic behaviour in the neighbourhood of the point and in the neighbourhood of the origin of a solution of an equation of arbitrary (even or odd) order with complex-valued coefficients are studied. It is assumed here that the coefficients of the quasidifferential expression have the following property: if one reduces the equation to a system of first-order differential equations, then one can transform that system to a system of differential equations with regular singular point at or . The results obtained allow one to determine the deficiency indices of the corresponding minimal symmetric differential operators and the structure of the spectrum of self-adjoint extensions of these operators. In addition, on the basis of refined asymptotic formulae for solutions to the equation the deficiency numbers of a certain differential operator generated by a differential expression with leading coefficient vanishing in the interior of the interval in question are found.

An equivalent linearization technique to obtain the response of non‐linear multi‐degree‐of‐freedom dynamic systems to stationary gaussian excitations is developed. The non‐linearities are assumed to be single‐valued functions of accelerations, velocities and displacements. Using a property of gaussian vector processes, the closed forms of the coefficients of the equivalent linear system are obtained by the direct application of partial differentiation and expectation operators to the non‐linear terms. It is shown that when the non‐linearities possess potentials, the linear system has symmetric coefficient matrices. A geometrical interpretation of the linear coefficients, in connection with the original non‐linearities, is presented. The accuracy is investigated by means of examples.

We consider a boundary value problem with irregular boundary conditions for a differential operator of arbitrary odd order. The potential in this operator is assumed to be an integrable function. We suggest a method for studying the spectral properties of differential operators with integrable coefficients. We analyze the asymptotic behavior of solutions of the differential equation in question for large values of the spectral parameter. The eigenvalue asymptotics for the considered differential operator is obtained.

Let Ω ⊂ RN be a smooth bounded domain. Let be a second-order strongly elliptic differential operator with smooth symmetric coefficients. Let B denote the Dirichlet or the Neumann boundary operator. We prove the existence of a smooth potential a : Ω → R such that all sufficiently small vector fields on RN + 1 can be realized on the centre manifold of the semilinear parabolic equation by an appropriate nonlinearity f : ( x, s, w ) ∈ Ω x R x RN ↦ f ( x, s, w ) ∈ R.For N = 2, n, k ∈ N, we prove the existence of a smooth potential a : Ω → R such that all sufficiently small k-jets of vector fields on Rn can be realized on the centre manifold of the semilinear parabolic equation by an appropriate nonlinearity f : ( x, s ) ∈ Ω x R ↦ f (x, s ) ∈ R2 ( here, ‘·’ denotes the scalar product in R2).

We enlarge the class of regular odd order differential operators and find the self-adjoint boundary condition of every operator. In this paper, we give the characterization of self-adjoint domains for regular odd order C-symmetric differential operators with two-point boundary conditions. The previously known characterization of self-adjoint operators for regular odd order symmetric differential operators is the special case of this one.

One of the main problems of the spectral theory of linear ordinary differential operators is the study of their deficiency indices depending on the behavior of the coefficients of the corresponding differential expression ly . Such problems were investigated by Naimark [1], Fedoryuk [2], Sultanaev, and others (see the bibliography in [2]). As a rule, symmetric operators with real coefficients were studied. For the case of complex coefficients, Fedoryuk obtained asymptotic formulas for the fundamental system of solutions of the equation ly = 0 under the condition that the coefficients of the differential expression ly are polynomials subject to a rigid constraint on their degree. The paper [3] was devoted to the study of a symmetric differential operator of (2n+1)th order with complex-valued coefficients of the odd powers. Classes of differential operators with deficiency indices (m, m) , where 1 ≤ m ≤ 2n − 1 , were described. The papers [4, 5] were also devoted to the study of symmetric operators in the case of complex coefficients under regularity conditions of Titchmarsh– Levitan type; asymptotic formulas for the fundamental system of solutions of the equation ly = λy were obtained, and in certain special cases the deficiency indices of the corresponding minimum operator were found. It is well known that deficiency indices define the dimension of the solution spaces of the equations lu = iy and lu = −iy . The goal of the present paper is to study the deficiency indices of a class of differential operators generated on the semiaxis by a self-adjoint differential expression of odd order with complex-valued coefficients. We consider the equation

В последнее время интенсивно изучаются начально – граничные задачи в прямоугольной области для дифференциальных уравнений в частных производных как четного, так и нечетного порядка. При этом в качестве объекта исследования, в основном, берется не вырождающееся уравнение или уравнение, вырождающееся на одной стороне четырехугольника. Начально – граничные задачи (как локальные, так и нелокальные) для уравнений с двумя или тремя линиями вырождения остаются неизученными. В данной работе в прямоугольной области рассмотрено уравнение четвёртого порядка, вырождающееся на трех сторонах четырехугольника и содержащее оператор дробного дифференцирования Герасимова –Капуто. Для этого уравнения сформулирована и исследована одна начально – граничная задача с нелокальными условиями, связывающими значения искомой функции и её производных до третьего порядка (включительно), принимаемых на боковых сторонах прямоугольника. Сначала методом интегралов энергии доказана единственность решения поставленной задачи. Затем, исследована спектральная задача, возникающая при применении метода Фурье, основанном на разделении переменных, к поставленной начально – граничной задаче. Построена функция Грина спектральной задачи, с помощью чего она эквивалентно сведена к интегральному уравнению Фредгольма второго рода с симметричным ядром, откуда следует существование счетного числа собственных значений и собственных функций спектральной задачи. Доказана теорема разложения заданной функции в равномерно сходящийся ряд по системе собственных функций. С помощью найденного интегрального уравнения и теоремы Мерсера доказана равномерная сходимость некоторых билинейных рядов, зависящих от найденных собственных функций. Установлен порядок коэффициентов Фурье. Решение изучаемой задачи выписано в виде суммы ряда Фурье по системе собственных функций спектральной задачи. Исследована равномерная сходимость этого ряда и рядов, полученных из него почленным дифференцированием. Получена оценка для решения задачи, откуда следует его непрерывная зависимость от заданных функций. Recently, initial-boundary problems in a rectangular domain for differential equations in partial derivatives of both even and odd order have been intensively studied. In this case, non-degenerate equations or equations that degenerate on one side of the quadrilateral are taken as the object of study. But initialboundary problems (both local and non-local) for equations with two or three lines of degeneracy remain unexplored. In this paper, in a rectangular domain, a fourth-order equation degene-rating on three sides of the rectangular and contains the Gerasimov-Caputo fractional diffe-rentiation operator has been considered. For this equation, an initial-boundary problem is formulated and investigated, with non-local conditions connecting the values of the desired function and its derivatives up to the third order (inclusive), taken on the sides of the rectangle. From the beginning, the uniqueness of the solution of the formulated problem was proved by the method of energy integrals. Then, the spectral problem that arises when applying the Fourier method based on the separation of variables to the considered initial-boundary problem has been investigated. The Green’s function of the spectral problem was constructed, with the help of which it is equivalently reduced to an integral Fredholm equation of the second kind with a symmetric kernel, which implies the existence of a countable number of eigenvalues and eigenfunctions of the spectral problem. A theorem is proved for expanding a given function into a uniformly convergent series in terms of a system of eigenfunctions. Using the found integral equation and Mercer’s theorem, we prove the uniform convergence of some bilinear series depending on the found eigenfunctions. The order of the Fourier coeffi-cients have been established. The solution of the considered is written as the sum of a Fourier series with respect to the system of eigenfunctions of the spectral problem. The uniform convergence of this series and the series obtained from it by term-by-term differentiation is studied. An estimate for solution to problem is obtained, from which follows its continuous dependence on the given functions.

Initial-boundary-value problems in three different domains are considered for quasilinear evolution partial differential equations of an odd (not less than third) order with respect to spatial variables in the multidimensional case. The nonlinearity has the divergent form and at most a quadratic rate of growth. Assumptions on the differential operator of odd order provide global estimates on solutions in $L_2$ and a local smoothing effect. Results on existence and uniqueness of global weak solutions are established. The essential part of the study is the construction of special solutions to the corresponding linear equations of the "boundary potential" type, which ensures the results under natural smoothness assumptions on initial and boundary data provided we have certain relations between the dimension and the order of the equations.

It is shown that for the asymmetric diffusion operator the case when the characteristic determinant is identically equal to zero is impossible and the only possible degenerate boundary conditions are the Cauchy conditions. In the case of a symmetric diffusion operator, the characteristic determinant is identically equal to zero if and only if the boundary conditions are false–periodic boundary conditions and is identically equal to a constant other than zero if and only if its boundary conditions are generalized Cauchy conditions. All degenerate boundary conditions for a spectral problem with a third–order differential equation y'''(x) = λy(x) are described. The general form of degenerate boundary conditions for the fourth–order differentiation operator D4 is found. 12 classes of boundary value eigenvalue problems are described for the operator D4, the spectrum of which fills the entire complex plane. It is known that spectral problems whose spectrum fills the entire complex plane exist for differential equations of any even order. John Locker posed the following problem (eleventh problem): are there similar problems for odd–order differential equations? A positive answer is given to this question. It is proved that spectral problems, the spectrum of which fills the entire complex plane, exist for differential equations of any odd order. Thus, the problem of John Locker is resolved. John Locker posed a problem (tenth problem): can a spectral boundary–value problem have a finite spectrum? Boundary value problems with a polynomial occurrence of a spectral parameter in a differential equation are considered. It is shown that the corresponding boundary–value problem can have a predetermined finite spectrum in the case when the roots of the characteristic equation are multiple. If the roots of the characteristic equation are not multiple, then there can be no finite spectrum. Thus, John Locker’s tenth problem is resolved.

Some examples of orthogonal matrix polynomials satisfying odd order differential equations

In this paper we consider differential opeartor L=d^4_x + u(x). We find the commutativity condition for operator L with a differential operator M of order 4g+2, where L and M are operators of rank 2. Some examples are constructed. These examples don't commute with differential opeartors of odd order.