Mark Kac [“Can one hear the shape of drums?” Am. Math. Monthly 73, l-23 (1966)] asked if the shape of a region fiCR” could be determined from its sound (spectrum of the Laplacian A,). He proved the conjecture for special classes of domains, including polygons and balls in W”. Similar problems could be raised in other geometric contexts, including “shape of metric” for Laplacians on manifolds or...

We develop novel methods to compute auto-correlation functions, or power spectral densities, for chaotic dynamical systems generated by an inverse method whose starting point is an invariant distribution and a two-form. In general, the inverse method makes some aspects of chaotic dynamics calculable by methods familiar in quantum field theory. This approach has the numerical advantage of being ...

Dynamic graphs are useful objects to describe a network which evolves over time. We propose a signal theory approach to analyze them which consists of transforming the graph at each time step into a collection of signals and analyze these signals using spectral decomposition. An inverse transformation is also proposed and makes it possible to reduce the dimension of the graph and select the mos...

The positive-definiteness of the Hamiltonian operator for Yang–Mills theory in four dimensions is studied by using the Coulomb gauge. It was indeed well known that the Coulomb gauge Hamiltonian involves integration of matrix elements of an operator P built from infinitely many inverse powers of the Laplacian. If space-time is replaced by a compact Riemannian four-manifold without boundary, on w...

‎in this study‎, ‎properties of spectral characteristic are investigated for‎ ‎singular sturm-liouville operators in the case where an eigen‎ ‎parameter not only appears in the differential equation but is‎ ‎also linearly contained in the jump conditions‎. ‎also weyl function‎ ‎for considering operator has been defined and the theorems which‎ ‎related to uniqueness of solution of inverse proble...

The inverse spectral problem of recovering pencils of second-order differential operators on the half-line from the Weyl function is studied. We establish properties of the spectral characteristics, give a formulation of the inverse problem, prove an uniqueness theorem and provide a constructive procedure for the solution of the inverse problem by the method of spectral mappings AMS Subject Cla...

We provide a complete spectral characterization of a new method of constructing isospectral (in fact, unitary) deformations of general Schrrdinger operators H =-d2/d.x 2 + V in L2(/l~). Our technique is connected to Dirichlet data, that is, the spectrum of the operator H D on L2((-~,x0)) @ L2((x0, oo)) with a Difichlet boundary condition at x 0. The transformation moves a single eigenvalue of H...

We use an inverse scattering approach to study multi-peakon solutions of the Degasperis–Procesi (DP) equation, an integrable PDE similar to the Camassa–Holm shallow water equation. The spectral problem associated to the DP equation is equivalent under a change of variables to what we call the cubic string problem, which is a third order non-selfadjoint generalization of the well-known equation ...