In this paper we study Schrodinger operators with absolutely integrable potentials on metric graphs. Uniform bounds-i.e. depending only on the graph and the potential-on the difference between the eigenvalues of the Laplace and Schrodinger operators are obtained. This in turn allows us to prove an extension of the classical Ambartsumian Theorem which was originally proven for Schrodinger operators with Neumann conditions on an interval. We also extend a previous result relating the spectrum of a Schrodinger operator to the Euler characteristic of the underlying metric graph.

Let CUnknown control sequence '\tt' denote a closed convex cone in \mathbb R^{d}Rd with apex at 0. We denote by E¢(C)Unknown control sequence '\tt' the set of distributions on \mathbb R^{d}Rd having compact support contained in CUnknown control sequence '\tt'. Then E¢(C)Unknown control sequence '\tt' is a ring with the usual addition and with convolution. We give a necessary and sufficient analytic condition on [^(f)]_{1},..., [^(f)]_{n}f1fn for f_{1},... ,f_{n} Î E¢(C)Unknown control sequence '\tt' to generate the ring E¢(C)Unknown control sequence '\tt'. (Here [^( · )] denotes Fourier-Laplace transformation.) This result is an application of a general result on rings of analytic functions of several variables by Lars Hörmander. En route we answer an open question posed by Yutaka Yamamoto.

We introduce a class of doubly infinite complex Jacobi matrices determined by a simple convergence condition imposed on the diagonal and off-diagonal sequences. For each Jacobi matrix belonging to this class, an analytic function, called a characteristic function, is associated with it. It is shown that the point spectrum of the corresponding Jacobi operator restricted to a suitable domain coincides with the zero set of the characteristic function. Also, coincidence regarding the order of a zero of the characteristic function and the algebraic multiplicity of the corresponding eigenvalue is proved. Further, formulas for the entries of eigenvectors, generalized eigenvectors, a summation identity for eigenvectors, and matrix elements of the resolvent operator are provided. The presented method is illustrated by several concrete examples.