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Estimates of large eigenvalues and trace formula for the vectorial Sturm-Liouville equations. (English) Zbl 1292.34082
This paper describes the \(N\)-dimensional vectorial Sturm-Liouville problem with coupled boundary conditions. The authors first derive the asymptotic expressions of large eigenvalues for the vectorial Sturm-Liouville operator with smooth coefficients. In addition, the regularized trace formula for the operator is calculated with residue techniques in complex analysis. These formulae are then used to obtain some results of inverse eigenvalue problems in the spirit of V. Ambarzumian [Z. Phys. 53, 690–695 (1929; JFM 55.0868.01)].
MSC:
34L20 Asymptotic distribution of eigenvalues, asymptotic theory of eigenfunctions for ordinary differential operators
34B24 Sturm-Liouville theory
34L05 General spectral theory of ordinary differential operators
47E05 General theory of ordinary differential operators (should also be assigned at least one other classification number in Section 47-XX)
34A55 Inverse problems involving ordinary differential equations
34L15 Eigenvalues, estimation of eigenvalues, upper and lower bounds of ordinary differential operators
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