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New trace formula for the matrix Sturm-Liouville equation with eigenparameter dependent boundary conditions

A regularized trace formula of first order for the matrix Sturm-Liouville equation with eigenparameter in the boundary conditions is obtained. | Turkish Journal of Mathematics http://journals.tubitak.gov.tr/math/ Research Article Turk J Math (2013) 37: 278 – 285 ¨ ITAK ˙ c TUB doi:10.3906/mat-1008-37 New trace formula for the matrix Sturm-Liouville equation with eigenparameter dependent boundary conditions Chuan-Fu YANG∗ Department of Applied Mathematics, Nanjing University of Science and Technology, Nanjing 210094, Jiangsu, People’s Republic of China Received: 12.08.2010 • Accepted: 10.01.2011 • Published Online: 19.03.2013 • Printed: 22.04.2013 Abstract: A regularized trace formula of first order for the matrix Sturm-Liouville equation with eigenparameter in the boundary conditions is obtained. Key words: Matrix Sturm-Liouville problem, eigenparameter dependent boundary conditions, trace formula 1. Introduction As is known, the trace of a finite-dimensional matrix is the sum of all the eigenvalues. But in an infinitedimensional space, in general, ordinary differential operators do not have a finite trace. Gelfand and Levitan [9] firstly obtained a trace formula for a self-adjoint Sturm-Liouville differential equation. After these studies several mathematicians were interested in developing trace formulae for different differential operators. For the scalar Sturm-Liouville problems, there is an enormous literature on estimates of large eigenvalues and regularized trace formulae which may often be computed explicitly in terms of the coefficients of operators and boundary conditions. A detailed list of publications related to the present aspect can be found in [13]. Note that the trace formulae are used in the numerical computation of the first eigenvalue of the SturmLiouville problem [6]. As a generalization of the scalar Sturm-Liouville equation, the matrix Sturm-Liouville equations were found to be important in the study of particle physics [16]. Starting with Faddeev’s study of the regularized trace formula [7], matrix Sturm-Liouville operators have raised some interesting new problems. Trace .