On the asymptotics of spectrum of an even-order differential operator with a delta-function potential

S. I. Mitrokhin

Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, Tome 25 (2021) no. 4, pp. 634-662.

Résumé

We study a sequence of differential operators of high even order whose potentials converge to the Dirac delta-function. One of the types of separated boundary conditions is considered. At the points of potential discontinuity, it is necessary to study the conditions of gluing for the correct determination of the corresponding differential equations solutions. For large values of the spectral parameter, asymptotic solutions of differential equations are furnished by the Naimark method. The conditions of gluing are studied, the boundary conditions are investigated, the equation for the eigenvalues of the considered differential operator is derived. The method of successive approximations is used to find the asymptotics of spectrum of studied differential operators, the limit of which determines a spectrum of operator with a delta-function potential.

Détail

Keywords: differential operator, Dirac delta-function, asymptotics of solutions of differential equations, piecewise smooth potential, eigenvalues, asymptotics of the spectrum.

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