Related papers: On the Abnormal Type Anomalous Solutions of Quasip…
We have generated, using an sl(2,R) formalism, several new classes of quasi-solvable elliptic potentials, which in the appropriate limit go over to the exactly solvable forms. We have obtained exact solutions of the corresponding spectral…
We study the limit behaviour of solutions of a class of solutions of nonlinear parabolic equations with a degenerate strong absorption. We prove that two types of phenomena can occur: the pointwise singularity or the formation of razor…
In this paper, we study the asymptotic behavior of radial solutions for several weighted elliptic equations with power type or exponential type nonlinearities on an annulus.
For two bound-state equations derived as simplified forms of the Bethe-Salpeter equation with confining interaction, stability of all solutions is rigorously shown.
We establish the existence and uniqueness of solutions for quasilinear singular Lane-Emden type systems subjected to Neumann boundary conditions. The approach is chiefly based on sub-supersolutions method.
Asymptotic behavior of solutions to heat equations with spatially singular inverse-square potentials is studied. By combining a parabolic Almgren type monotonicity formula with blow-up methods, we evaluate the exact behavior near the…
In this paper, we study the Schr\"odinger equation with a new quasi-exactly solvable double-well potential. Exact expressions for the energies, the corresponding wave functions and the allowed values of the potential parameters are obtained…
By using the theory of maximal $L^{q}$-regularity and methods of singular analysis, we show a Taylor's type expansion--with respect to the geodesic distance around an arbitrary point--for solutions of quasilinear parabolic equations on…
We show that the complex $\cal PT$-symmetric periodic potential $V(x) = - ({\rm i} \xi \sin 2x + N)^2$, where $\xi$ is real and $N$ is a positive integer, is quasi-exactly solvable. For odd values of $N \ge 3$, it may lead to exceptional…
In supersymmetric quantum mechanics, shape invariance is a sufficient condition for solvability. We show that all conventional additive shape invariant superpotentials that are independent of $\hbar$ obey two partial differential equations.…
The work is devoted to the construction of the asymptotic behavior of the solution of a singularly perturbed system of equations of parabolic type, in the case when the limit equation has a regular singularity as the small parameter tends…
A Hamiltonian is said to be quasi-exactly solvable (QES) if some of the energy levels and the corresponding eigenfunctions can be calculated exactly and in closed form. An entirely new class of QES Hamiltonians having sextic polynomial…
A set of quasi-exactly solvable quantum mechanical potentials associated with the Poeschl-Teller potential, the generalized Poeschl-Teller potential, the Scarf potential, and the harmonic oscillator potential have been studied. Solutions of…
We consider a certain ultrahyperbolic equation in a Euclidean space being a generalization of Klein-Gordon-Fock equation. The behavior of solutions at points tending to infinity along timelike directions is studied. We examine the issue of…
Explicit solutions of the quantum Yang-Baxter equation are given corresponding to the non-unitary solutions of the classical Yang-Baxter equation for sl(5).
Motivated by the prevalence of non-smooth, possibly non-periodic signals in real-world applications, the output regulation of linear systems subject to non-smooth non-periodic exogenous signals has emerged as a challenging problem. A…
The behaviour of solutions to fourth order problems is studied through the decomposition into a system of second order ones, which leads to relaxed formulations with the introduction of measure terms. This allows to solve a shape…
In this article we have discovered a close relationship between the (algebraic) Bethe Ansatz equations of the spin $s$ XXZ model of a finite size and the $q$-Sturm-Liouville problem. We have demonstrated that solutions of the Bethe Ansatz…
Quasipatterns (two-dimensional patterns that are quasiperiodic in any spatial direction) remain one of the outstanding problems of pattern formation. As with problems involving quasiperiodicity, there is a small divisor problem. In this…
Bethe-Salpeter equation, for massless exchange and large fine structure constant $\alpha>\pi/4$, in addition to the Balmer series, provides another (abnormal) series of energy levels which are not given by the Schr\"odinger equation. So…