The Generalized Uncertainty Principle and the Semi-relativistic Harmonic Oscillator Abstract We study the Massless Semi-Relativistic Harmonic Oscillator within the framework of quantum mechanics with a Generalized Uncertainty Principle (GUP). The latter derives from the idea of minimal observable length, a quantity whose existence is expected to affect the energy eigenvalues and the eigenfunctions of the system. These effects are worked out, to the first order in the deformation parameter, using a perturbative approach based on Brau’s representation of position and momentum operators. Besides, we have discussed the impact of the GUP on the known duality between the considered model and the Schrödinger equation with a linear potential.

The Hankel Transform of the Hulthén Green’s Function Abstract It is shown that the Hankel transform of the s-wave Hulthén physical Green’s function satisfies a second-order differential equation. This equation is solved by applying the proper boundary conditions in association with the properties of the special functions of mathematics to get a closed form expression for the same. The Hankel transform of the physical Green’s function is exploited to extract off-shell solutions and Half- and off-shell T-matrices in the maximal reduced form. The check on our expressions with particular emphasis on their limiting behaviour is made and is found in order. The bound state spectrums of n–p and \(\hbox {n-C}^{12}\) systems are computed by exploiting the associated Jost function and found excellent agreement with experimental results.

Scalar Particle in New Type of the Extended Uncertainty Principle Abstract In the context of new type of the extended uncertainty principle using the displacement operator method, we present an exact solution of some problems such as: the Klein–Gordon particle confined in a one dimensional box, the scalar particle with linear vector and scalar potentials and the case of inversely linear vector and scalar potentials of Coulomb-type. The expressions of bound state energies and the associated wave functions are exactly determined for these three cases.

Trigonometric Rosen–Morse Potential as a Quark–Antiquark Interaction Potential for Meson Properties in the Non-relativistic Quark Model Using EAIM Abstract Trigonometric Rosen–Morse potential is suggested as a quark–antiquark interaction potential for studying thermodynamic properties and masses of heavy and heavy–light mesons. For this purpose, the N-radial Schrödinger equation is analytically solved using an exact-analytical iteration method. The energy eigenvalues and corresponding wave functions are obtained in the N-space. The present results are applied in calculating the mass of mesons such as charmonium c \({\bar{\hbox {c}}}\), bottomonium b \({\bar{\hbox {b}}}\), b \( {\bar{\hbox {c}}}, \) and c \({\bar{\hbox {s}}}\) mesons and thermodynamic properties such as the mean internal energy, the specific heat, the free energy, and the entropy. The effect of dimensional number is studied on the meson properties. The present results are improved in comparison with other recent works and are in good agreement in comparison with experimental data. Thus, the present potential provides satisfied results in comparison with other works and experimental data.

Relativistic Spin-0 Feshbach–Villars Equations for Polynomial Potentials Abstract We propose a solution method for studying relativistic spin-0 particles. We adopt the Feshbach–Villars formalism of the Klein–Gordon equation and express the formalism in an integral equation form. The integral equation is represented in the Coulomb–Sturmian basis. The corresponding Green’s operator with Coulomb and linear confinement potential can be calculated as a matrix continued fraction. We consider Coulomb plus short range vector potential for bound and resonant states and linear confining scalar potentials for bound states. The continued fraction is naturally divergent at resonant state energies, but we made it convergent by an appropriate analytic continuation.

Tests of the Envelope Theory in One Dimension Abstract The envelope theory is a simple technique to obtain approximate, but reliable, solutions of many-body systems with identical particles. The accuracy of this method is tested here for two systems in one dimension with pairwise forces. The first one is the fermionic ground state of the analytical Calogero model with linear forces supplemented by inverse-cube forces. The second one is the ground state of up to 100 bosons interacting via a Gaussian potential. Good bounds can be obtained depending on values of the model parameters.

Confinement Induced Resonance with Weak Bare Interaction in a Quasi 3+0 Dimensional Ultracold Gas Abstract Confinement induced resonance (CIR) is a useful tool for the control of the interaction between ultracold atoms. In most cases the CIR occurs when the characteristic length \(a_\mathrm{trap}\) of the confinement is similar as the scattering length \(a_{s}\) of the two atoms in the free three-dimensional (3D) space. If there is a CIR which can occur with weak bare interaction, i.e., under the condition \(a_\mathrm{trap}\gg a_s\), then it can be realized for much more systems, even without the help of a magnetic Feshbach resonance, and would be very useful. In a previous research by Massignan and Castin (Phys Rev A 74:013616, 2006), it was shown that it is possible to realize such a CIR in a quasi-(3+0)D system, where one ultracold atom is moving in the 3D space and another one is localized by a 3D harmonic trap. In this work we carefully investigate the properties of the CIRs in this system. We show that the CIR with \(a_\mathrm{trap}\gg a_s\) can really occur, and the number of the CIRs of this type increases with the mass ratio between the moving and localized atoms. However, when \(a_\mathrm{trap}\gg a_s\) the CIR becomes extremely narrow, and thus are difficult to be controlled in realistic experiments.

The Faddeev–Yakubovsky Symphony Abstract We briefly summarize the main steps leading to the Faddeev–Yakubovsky equations in configuration space for \(\hbox {N}=3, 4\) and 5 interacting particles.

Momentum-Space Probability Density of $${}^6$$6 He in Halo Effective Field Theory Abstract We compute the momentum-space probability density of \({}^6\)He at leading order in Halo EFT. In this framework, the \({}^6\)He nucleus is treated as a three-body problem with a \({}^4\)He core (\(c\)) and two valence neutrons (\(n\)). This requires the \(nn\) and \(n c\) t-matrices as well as a \(cnn\) force as input in the Faddeev equations. Since the \(n c\) t-matrix corresponds to an energy-dependent potential, we consider the consequent modifications to the standard normalization and orthogonality conditions. We find that these are small for momenta within the domain of validity of Halo EFT. In this regime, the \({}^6\)He probability density is regulator independent, provided the cutoff is significantly above the EFT breakdown scale.

Electromagnetic Transition Form Factor of the Nucleon $$\Delta $$Δ (1232) in The Nonrelativistic Constituent Quark Model Abstract The study of nucleon electromagnetic form factors has long been identified as a singular source of information for conception strong interactions in the extent of quark confinement. We have performed a calculation of the helicity amplitudes and the electromagnetic transition form factors of the electromagnetic excitation in \(\Delta \)(1232) resonances. In this paper, the electromagnetic interaction for N \((938)\rightarrow \Delta (1232)\) transitions at four-momenta transfer \(0 \le \hbox {Q2}(\hbox {GeV}^\mathrm{{2}}) \le 8\) in the nonrelativistic constituent quark model calculated. In comparison with present experimental, relativistic and non-relativistic data, our results are in good agreement with the experimental and the other theoretical results, in particular of the medium-high \(\hbox {Q}^{2}\) behavior.