Spectral analysis of wave propagation on branching strings

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In this paper the classical Hill problem with complex potentials are extended to the star graph. The definition of the Hill operator on such graph is discussed. The operator is defined with complex, periodic potentials and using special boundary conditions connecting values of the functions at the vertices. An explicit description of the resolvent is given and the spectrum is described exactly, the inverse problem with respect to the reflection coefficients is solved.

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