The zeros of the eigenfunctions of self-adjoint Sturm–Liouville eigenvalue problems interlace. For these problems interlacing is crucial for completeness. For the complex Sturm–Liouville problem associated with the Schrödinger equation for a non-Hermitian PT-symmetric Hamiltonian, completeness and interlacing of zeros have never been examined. This paper reports a numerical study of the Sturm– ...

Liouville-type theorems are powerful tools in partial differential equations. Boundedness assumption of solutions are often imposed in deriving such Liouville-type theorems. In this paper, we establish some Liouville-type theorems without the boundedness assumption of nonnegative solutions to certain classes of elliptic equations and systems. Using a rescaling technique and doubling lemma devel...

We show how a number of NP-complete as well as NP-hard problems can be reduced to the Sturm-Liouville eigenvalue problem in the quantum setting with queries. We consider power queries which are derived from the propagator of a system evolving with a Hamiltonian obtained from the discretization of the Sturm-Liouville operator. We use results of our earlier paper concering the complexity of the S...

(One may modify the question by putting various other restrictions on (L); for example, one can consider only steady-state solutions, or solutions with finite rate of dissipation or belonging to various other function spaces, etc.) We have proved a positive result for dimension n = 2 which we will discuss below, but let us begin by mentioning why the basic problem is interesting. Generally spea...

In [15], a Feller process called Liouville Brownian motion onR has been introduced. It can be seen as a Brownian motion evolving in a random geometry given formally by the exponential of a (massive) Gaussian Free Field e X and is the right diffusion process to consider regarding 2d-Liouville quantum gravity. In this note, we discuss the construction of the associated Dirichlet form, following e...

in this article, an analytical approximate solution of nonlinear fractional convection-diffusion with modifiedriemann-liouville derivative was obtained with the help of fractional variational iteration method (fvim). a newapplication of fractional variational iteration method (fvim) was extended to derive analytical solutions in theform of a series for this equation. it is indicated that the so...

In [14], a Feller process called Liouville Brownian motion on R2 has been introduced. It can be seen as a Brownian motion evolving in a random geometry given formally by the exponential of a (massive) Gaussian Free Field e X and is the right diffusion process to consider regarding 2d-Liouville quantum gravity. In this note, we discuss the construction of the associated Dirichlet form, following...

where the potentials are given functions. Under various boundary conditions, Sturm and Liouville established that solutions of problem (1) can exist only for particular values of the real parameter λ, which is called an eigenvalue. Relevant examples of linear Sturm-Liouville problems are the Bessel equation and the Legendre equation. The classical Sturm-Liouville theory does not depend upon the...

We extend the idea of Jodeit and Levitan [3] for constructing isospectral problems of the classical scalar Sturm-Liouville differential equations to the vectorial Sturm-Liouville differential equations. Some interesting relations are found.

We define the Liouville distribution on the tangent bundle of a pseudo-Finsler manifold and prove that it is integrable. Also, we find geometric properties of both leaves of Liouville distribution and the vertical distribution.