Higher order differential operators on graphs
Abstract: This thesis consists of two papers, enumerated by Roman numerals. The main focus is on the spectral theory of -Laplacians. Here, an -Laplacian, for integer , refers to a metric graph equipped with a differential operator whose differential expression is the -th derivative.In Paper I, a classification of all vertex conditions corresponding to self-adjoint -Laplacians is given, and for these operators, a secular equation is derived. Their spectral asymptotics are analysed using the fact that the secular function is close to a trigonometric polynomial, a type of almost periodic function. The notion of the quasispectrum for -Laplacians is introduced, identified with the positive roots of the associated trigonometric polynomial, and is proved to be unique. New results about almost periodic functions are proved, and using these it is shown that the quasispectrum asymptotically approximates the spectrum, counting multiplicities, and results about asymptotic isospectrality are deduced. The results obtained on almost periodic functions have wider applications outside the theory of differential operators.Paper II deals more specifically with bi-Laplacians (), and a notion of standard conditions is introduced. Upper and lower estimates for the spectral gap --- the difference between the two lowest eigenvalues - for these standard conditions are derived. This is achieved by adapting the methods of graph surgery used for quantum graphs to fourth order differential operators. It is observed that these methods offer stronger estimates for certain classes of metric graphs. A geometric version of the Ambartsumian theorem for these operators is proved.
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