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The Schrodinger equation for stationary states in a central potential is studied in an arbitrary number of spatial dimensions, say q. After transformation into an equivalent equation, where the coefficient of the first derivative vanishes,…
The quantum mechanics description of a physical object stretched in space and stable in time from the relativistic space-time properties point of view, introduced in special theory of relativity, is considered and analysed. The mathematical…
The internal energy of hard spheres (HS) is the same as that of an ideal gas, so that the energy route to thermodynamics becomes useless. This problem can be avoided by taking an interaction potential that reduces to the HS one in certain…
The $\Lambda$-term in Einstein's equations is a fundamental building block of the `concordance' $\Lambda$CDM model of cosmology. Even though the model is not free of fundamental problems, they have not been circumvented by any alternative…
Let $H=-\Delta+V$ be a Schr\"odinger operator on $L^2(\mathbb R^n)$ with real-valued potential $V$ for $n > 4$ and let $H_0=-\Delta$. If $V$ decays sufficiently, the wave operators $W_{\pm}=s-\lim_{t\to \pm\infty} e^{itH}e^{-itH_0}$ are…
Wave functions and energy eigenvalues of the path integral Hamiltonian are studied in Lorentz frame moving with velocity $v$. The instantaneous interaction produced by the Wilson loop is shown to be reduced by an overall factor…
An analysis of the analytical solution of the Schr\"{o}dinger equation (which is a second order differential equation) for $H_2^+$ shows that the second linear independent solution of this equation is a square integrable function and…
An application of a quantum wave impedance method for a study of quantum-mechanical systems which con\-tain singular zero-range potentials is considered. It was shown how to reformulate the problem of an investigation of mentioned systems…
To quantify the effect of decoherence in quantum measurements, it is desirable to measure not merely the square modulus of the spatial wavefunction, but the entire density matrix, whose phases carry information about momentum and how pure…
We study integrability by quadrature of a spatially flat Friedmann model containing both a minimally coupled scalar field $\phi$ with an exponential potential $V(\phi)\sim\exp[-\sqrt{6}\sigma\kappa\phi]$, $\kappa=\sqrt{8\pi G_N}$, of…
An exact quantization rule for the Schr\"{o}dinger equation is presented. In the exact quantization rule, in addition to $N\pi$, there is an integral term, called the quantum correction. For the exactly solvable systems we find that the…
This is the first of a sequence of four papers \cite{param1}, \cite{param2}, \cite{param3}, \cite{param4} dedicated to the construction and the control of a parametrix to the homogeneous wave equation $\square_{\bf g} \phi=0$, where ${\bf…
We consider quantum general relativity in three dimensions with a positive cosmological constant. The Hartle-Hawking wave function is computed as a function of metric data at asymptotic future infinity. The analytic continuation from…
In this paper, we establish some important results for the impulsive wave equation. We begin by proving the existence of a solution. Then, we study the impulse approximate controllability where the control function acts on a subdomain…
We provide a comprehensive picture for the formulation of the perfect fluid in the modern effective field theory formalism at both the classical and quantum level. Due to the necessity of decomposing the hydrodynamical variables $(\rho, p,…
In this paper we consider weighted $L^2$ integrability for solutions of the wave equation. For this, we obtain some weighed $L^2$ estimates for the solutions with weights in Morrey-Campanato classes. Our method is based on a combination of…
It is well known that any given density rho(x)can be realized by a determinantal wave function for N particles. The question addressed here is whether any given density rho(x) and current density j(x) can be simultaneously realized by a…
Electromagnetic wave scattering by many parallel infinite cylinders is studied asymptotically as $a\to 0$. Here $a$ is the radius of the cylinders. It is assumed that the points $\hat{x}_m$ are distributed so that…
An invariant differential-geometric approach to the integrability of (2+1)-dimensional systems of hydrodynamic type u_t+A(u)u_x+B(u)u_y=0 is developed. It is proved that the existence of special solutions known as `double waves' is…
An inertia-gravity wave (IGW) propagating in a vertically sheared, rotating stratified fluid interacts with the pair of inertial levels that surround the critical level. An exact expression for the form of the IGW is derived here in the…