On Basis Properties of a Part of Eigenfunctions of the Problem of Vibrations of a Smooth Inhomogeneous String Damped at the Midpoint |
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Authors: | Alexander Gomilko Vyacheslav Pivovarchik |
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Abstract: | The boundary problem is considered which occurs in the theory of small transversal vibrations of a smooth inhomogeneous string. The ends of the string assumed to be fixed and the midpoint of the string is damped by a pointwise force. The problem is reduced to a spectral problem for a nonmonic quadratic operator pencil. The spectrum of the pencil (i. e. the set of normal eigenvalues) can be presented as a union of two subsequences. One of the subsequences approaches the real axis. Under an additional condition the second branch approaches a horizontal line located in the upper half–plane. The basis properties of the sets of projections (onto the corresponding subintervals) of eigenfunctions corresponding to each of the subsequences are investigated. |
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Keywords: | Sturm– Liouville equation parameter– dependent boundary conditions Riesz basis |
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