The Ambarzumyan theorem connecting the Sturm–Liouville problem and the corresponding problem associated with the Fourier differential equation is extended to a class of second order matrix differential systems.

On an extension of the theorem of V. A. Ambarzumyan

Consider the system of second order differential equation y '' (x) + (� 2 R(x) + Q(x))y(x) = 0, 0 ≤ x ≤ �, where y(x) = (y1(x),y2(x)) T , Q(x)= � p(x) r(x)) s(x) = x m s1(x), t(x) = x m t1(x), s1(x) > 0, t1(x) > 0 and p(x), q(x), r(x), s1(x), t1(x) are real-valued functions having continuous second order derivatives at x, 0 ≤ x ≤ �, m being a real constant and �, a real parameter. In the present paper we consider m, a negative real number and determine the asymptotic solutions alongwith their derivatives for such a system for large values of the parameterand apply these to determine the asymptotic expres- sions for the distribution of the eigenvalues and the normalized eigenvectors under the Dirichlet boundary conditions.

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Asymptotic expressions for the eigenvalues and eigenvectors of a system of second order differential equations with a turning point (extension ii)

Square-integrable solutions of a matrix differential expression

In the present paper we generalize the Parseval theorem and the expansion theorem for a function f(x) = (f 1 (x), f 2 (x)) T satisfying f T (x)R(x)f(x) ∈ L (- ∞ , ∞ ) associated with the above system of second order differential equations under the general boundary conditions, where the elements of R(x) i.e. s(x), t(x) are assumed to be greater than zero for x ∈ (a, b), a, b being finite or infinite.

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On the expansion problem of a function associated with a system of second order differential equations

In this paper we determine the resolution of the identity of the operator LA generated by the matrix differential operator L under the general boundary conditions where s(x), t(x) are assumed to be greater than zero for x \in (a,b), a, b being finite or infinite.

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Resolution of the identity of the operator associated with a system of second order differential equations

The paper deals with the nature of the spectrum associated with the type of second-ordcr matrix differential operator with catain boundary conditions. It is found that under certain conditions. satisfied by the co-efficients of the differential system, the spectrum is discrete. Some results are then obtained giving distributions of the eigenvalues on the real axis. The method employed depends, among others upon some of the ideas and techniques of E. C. Titchmarsh.

On the distribution of the eigenvalues of a matrix differential operator

In the present paper we investigate the asymptotic formulae involving ~(x,y,l+n) and ~(x,y,n) as a tends to  infinity, where~(x,y,1) stands for DP(H(x,y, 2) - fl(x,y,A)), H(.),X(.) being the resolution matrices associated with  two different second-order dinerential systems with the same boundary conditions at two arbitrary points o and  b. Replacing fl(x,y, A) by the resolution matrix HF(x, y.2.) of the Fourier system and then making r -y we derive  some special asymptotic formulae. A modified form of a Tauberian theorem due to Wiener plays a key role in the investigation that follows.

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On the spectral resolution of a differential operator II

In the present paper we obtain certain asymptotic formula for the spectral matrix of a self-adjoint second-order differential system We base our derivation of the formula on a Tauberian theorem due to N. Wiener.

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N K Chakravarty

Papers

On the distribution of the eigenvalues of a matrix differential operator

On the spectral resolution of a differential operator II

On the asymptotic formulae for the spectral matrix of a differential operator