We study the quantum mechanical Liouville model with attractive potential which is obtained by Hamiltonian symmetry reduction from the system of a free particle on SL(2,RI ). The classical reduced system consists of a pair of Liouville subsystems which are ‘glued together’ in such a way that the singularity of the Hamiltonian flow is regularized. It is shown that the quantum theory of this redu...

We write down an asymptotic expression for action coordinates in an integrable Hamiltonian system with a focus-focus equilibrium. From the singularity in the actions we deduce that the Arnol’d determinant grows infinitely large near the pinched torus. Moreover, we prove that it is possible to globally parametrise the Liouville tori by their frequencies. If one perturbs this integrable system, t...

Abstract The goal of this study is to analyse the eigenvalues and weak eigenfunctions a new type multi-interval Sturm-Liouville problem (MISLP) which differs from standard problems (SLPs) in that Strum-Liouville equation defined on finite number non-intersecting subintervals boundary conditions are set not only at endpoints but also internal points interaction. For self-adjoint treatment consid...

In this work, we study a system of parabolic inequalities. Using the method of test functions, we establish an integral estimate and get a Liouville-type theorem for the solutions of this system. As an application, we observe the well-known Fujita blow-up phenomena for the corresponding parabolic system. Here, the importance of this observation is that we do not impose any regularity assumption...

We derive a Liouville type result for special Lagrangian equations with certain “convexity” and restricted linear growth assumptions on the solutions.

We study the existence and multiplicity of positive solutions for a system Riemann–Liouville fractional differential equations with sequential derivatives, parameters sign-changing singular nonlinearities, subject to nonlocal coupled boundary conditions which contain Riemann–Stieltjes integrals various derivatives. In proof our main results we use nonlinear alternative Leray–Schauder type Guo–K...

In this paper, we apply spectral method based on the Bernstein polynomials for solving a class of optimal control problems with Jumarie’s modified Riemann-Liouville fractional derivative. In the first step, we introduce the dual basis and operational matrix of product based on the Bernstein basis. Then, we get the Bernstein operational matrix for the Jumarie’s modified Riemann-Liouville fractio...

‎In this paper‎, ‎we find matrix representation of a class of sixth order Sturm-Liouville problem (SLP) with separated‎, ‎self-adjoint boundary conditions and we show that such SLP have finite spectrum‎. ‎Also for a given matrix eigenvalue problem $HX=lambda VX$‎, ‎where $H$ is a block tridiagonal matrix and $V$ is a block diagonal matrix‎, ‎we find a sixth order boundary value problem of Atkin...

The Liouville equation, first Bogoliubov hierarchy and Vlasov equations with derivatives of non-integer order are derived. Liouville equation with fractional derivatives is obtained from the conservation of probability in a fractional volume element. This equation is used to obtain Bogoliubov hierarchy and fractional kinetic equations with fractional derivatives. Statistical mechanics of fracti...

Solving an inverse Sturm-Liouville problem requires a mathematical process to determine unknown function in the Sturm-Liouville operator from given data in addition to the boundary values. In this paper, we identify a Sturm-Liouville potential function by using the data of one eigenfunction and its corresponding eigenvalue, and identify a spatial-dependent unknown function of a SturmLiouville d...