On the ``splitting'' effect for multipoint differential operators with summable potential

S. I. Mitrokhin

Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, Tome 21 (2017) no. 2, pp. 249-270.

Résumé

We study the differential operator of the fourth order with multipoint boundary conditions. The potential of the differential operator is summable function on a finite segment. For large values of spectral parameter the asymptotic behavior of solutions of differential equation which define the differential operator is found. The equation for eigenvalues of the studied operators is derived by studying the boundary conditions. The parameters of boundary conditions are selected in such a way that the main approach of the equation for eigenvalues has multiple roots. The author shows that for the studied operator the effect of “splitting” of multiple eigenvalues in the main approximation is observed. We derive all series of single eigenvalues of the investigated operator. The indicator diagram of the considered operator is studied. The asymptotic behavior of eigenvalues in all sectors of the indicator diagram is found. The obtained precision of the asymptotic formulas is enough for finding an asymptotics of eigenfunctions of the studied differential operator.

Détail

Keywords: differential operator, spectral parameter, summable potential, the equation for the eigenvalues, the indicator diagram, the asymptotic behavior of the eigenvalues.

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