A regularized trace formula of first order for the matrix Sturm-Liouville equation with eigenparameter in the boundary conditions is obtained.

The paper describes relations between Liouville type theorems for solutions of a periodic elliptic equation (or a system) on an abelian cover of a compact Riemannian manifold and the structure of the dispersion relation for this equation at the edges of the spectrum. Here one says that the Liouville theorem holds if the space of solutions of any given polynomial growth is finite dimensional. Th...

An exact differential equation is derived for the evolution of the Liouville effective action with the mass parameter. This derivation is based on properties of the exponential potential and some consequences of the equation are discussed. Liouville theory has been studied for a long time [1] and is currently relevant to two-dimensional quantum gravity or non critical string theory. For a recen...

We construct a class of numerical schemes for the Liouville equation of geometric optics coupled with the Geometric Theory of Diffractions to simulate the high frequency linear waves with a discontinuous index of refraction. In the work [26], a Hamiltonian-preserving scheme for the Liouville equation was constructed to capture partial transmissions and reflections at the interfaces. These schem...

We derive the high frequency limit of the Helmholtz equation with source term when the source is the sum of two point sources. We study it in terms of Wigner measures (quadratic observables). We prove that the Wigner measure associated with the solution satisfies a Liouville equation with, as source term, the sum of the source terms that would be created by each of the two point sources taken s...

Mode-coupling theory (MCT) of sheared dense granular liquids is formulated. Starting from the Liouville equation of granular particles, the generalized Langevin equation is derived with the aid of the projection operator technique. The MCT equation for the density correlation function obtained from the generalized Langevin equation is almost equivalent to MCT equation for elastic particles unde...

In this paper, a new fractional sub-equation method is proposed for finding exact solutions of fractional partial differential equations (FPDEs) in the sense of modified Riemann-Liouville derivative. With the aid of symbolic computation, we choose the space-time fractional Zakharov-Kuznetsov-Benjamin-Bona-Mahony (ZKBBM) equation in mathematical physics with a source to illustrate the validity a...

We derive a Feynman-Kac formula for functionals of a stochastic hybrid system evolving according to a piecewise deterministic Markov process. We first derive a stochastic Liouville equation for the moment generator of the stochastic functional, given a particular realization of the underlying discrete Markov process; the latter generates transitions between different dynamical equations for the...

in this paper, a new fractional sub-equation method is proposed for finding exact solutions of fractional partial differential equations (fpdes) in the sense of modified riemann-liouville derivative. with the aid of symbolic computation, we choose the space-time fractional zakharov-kuznetsov-benjamin-bona-mahony (zkbbm) equation in mathematical physics with a source to illustrate the validity a...

We consider the quantum Liouville equation and give a characterization of the solutions which satisfy the Heisenberg uncertainty relation. We analyze three cases. Initially we consider a particular solution of the quantum Liouville equation: the Wigner transform f x,v,t of a generic solution ψ x;t of the Schrödinger equation. We give a representation of ψ x, t by the Hermite functions. We show ...