Related papers: Asymptotic Freedom, Dimensional Transmutation, and…
We study the relativistic version of Schr\"odinger equation for a point particle in 1-d with potential of the first derivative of the delta function. The momentum cutoff regularization is used to study the bound state and scattering states.…
A general method has been developed to solve the Schr\"odinger equation for an arbitrary derivative of the $\delta$-function potential in 1-d using cutoff regularization. The work treats both the relativistic and nonrelativistic cases. A…
A thorough analysis is presented of the class of central fields of force that exhibit: (i) dimensional transmutation and (ii) rotational invariance. Using dimensional regularization, the two-dimensional delta-function potential and the…
We study the solution of the relativistic Schr\"odinger equation for a point particle in 1-d under $\delta$-function potential by using cutoff regularization. We show that the problem is renormalizable, and the results are exactly the same…
The Schrodinger equation with a two-dimensional delta-function potential is a simple example of an asymptotically free theory that undergoes dimensional transmutation. Renormalization requires the introduction of a mass scale, which can be…
This is the first in a series of papers addressing the phenomenon of dimensional transmutation in nonrelativistic quantum mechanics within the framework of dimensional regularization. Scale-invariant potentials are identified and their…
Some aspects of asymptotic freedom are discussed in the context of a simple two-particle non-relativisitic confining potential model. In this model asymptotic freedom follows from the similarity of the free-particle and bound state radial…
We obtain for the attractive Dirac delta-function potential in two-dimensional quantum mechanics a renormalized formulation that avoids reference to a cutoff and running coupling constant. Dimensional transmutation is carried out before…
In this paper we consider the three-dimensional Schr\"{o}dinger operator with a $\delta$-interaction of strength $\alpha > 0$ supported on an unbounded surface parametrized by the mapping $\mathbb{R}^2\ni x\mapsto (x,\beta f(x))$, where…
We consider physical interpretations of non-trivial boundary conditions of self-adjoint extensions for one-dimensional Schr\"odinger operator of free spinless particle. Despite its model and rather abstract character this question is worth…
We investigate a two-dimensional Schr\"odinger operator, $-h^2 \Delta +iV(x)$, with a purely complex potential $iV(x)$. A rigorous definition of this non-selfadjoint operator is provided for bounded and unbounded domains with common…
This work presents a study on the nonrelativistic quantum motion of a charged particle in a rotating frame, considering the Aharonov-Bohm effect and a uniform magnetic field. We derive the equation of motion and the corresponding radial…
This paper investigates a two-dimensional Kemmer oscillator within relativistic quantum mechanics, incorporating minimal length and non-commutative phase space effects. We derive eigen solutions in configuration space $\{p\}$, examining the…
Exact solutions of two-particle relativistic equations of quantum field theory describing the scattering $s$-states and the bound $s$-states are found in the cases of delta-shell potential and superposition of delta-shell potentials. Some…
In this article we consider asymptotics for the spectral function of Schr\"odinger operators on the real line. Let $P:L^2(\mathbb{R})\to L^2(\mathbb{R})$ have the form $$ P:=-\tfrac{d^2}{dx^2}+W, $$ where $W$ is a self-adjoint first order…
We study the $d$-dimensional discrete nonlinear Schr\"odinger equation with general power nonlinearity and a delta potential. Our interest lies in the interplay between two localization mechanisms. On the one hand, the attractive…
We discuss the potential scattering on the noncompact star graph. The Schr\"{o}dinger operator with the short-range potential localizing in a neighborhood of the graph vertex is considered. We study the asymptotic behavior the corresponding…
The one-dimensional Schr\"odinger equation with the point potential in the form of the derivative of Dirac's delta function, $\lambda \delta'(x)$ with $\lambda$ being a coupling constant, is investigated. This equation is known to require…
In two recent papers, an isometric conformal transformation has been introduced that eliminates potential interaction terms from the Schr\"odinger equation for central potential problems. The method has been demonstrated for both the…
In the framework of the Moutard transformation formalism we find multi-point delta-type potentials of two-dimensional Schrodinger operators and their isospectral deformations on the zero energy level. In particular, these potentials are…