Logarithmic Potential Model of Quigg and Rosner as a Generalization of Naive Quark Model

Exploiting the explicit mass formulae for logarithmic potential model of Quigg and Rosner it is shown that at least on the level of mass-relations this model reproduces the naive quark model relations and generalizes the last one in case of highly non-trivial potential. Generalization includes the relations for higher values of orbital quantum numbers. In particular, preditions for recently discovered atom-like P-states are no worse than for any other potential models.The advantage consists in simplicity of approach.Comment: 5 page

On the Abnormal Type Anomalous Solutions of Quasipotential Equations

It is shown that there exist solutions of the quasipotential equations exhibiting the abnormal type behaviour of the Bethe-Salpeter equation.Comment: 8 pages, LaTex, no figure

On the Existence of Additional (Hydrino) states in the Dirac equation

In case of spinless particles there appear additional (singular) solutions in the framework of relativistic Klein-Gordon equation for Coulomb potential. These solutions obey to all requirements of quantum mechanical general principles. Observation of such states (hydrino, small hydrogen) should be important for manifestation of various physical phenomena. In this article the same problem is considered for spin-1/2 particle (electron) in the Dirac equation. It is shown that such kind of solutions really occurs, but the rate of singularity is more higher than in spinless case. By this reason we have no time- independence of total probability (norm). Moreover the orthogonality property is also failed, while the total probability is finite in the certain area of the model-parameters. Therefore, we are inclined to conclude that this additional solution in the Dirac equation must be ignored and restrict ourselves only by normal (standard) solutions.Comment: 6 page

Unusual Bound States in QFT Models

Homogeneous Bethe-Salpeter equation for simplest Wick-Cutkosky model is studied in the case when the mass of the two-body system is more then the sum of constituent particles masses. It is shown that there is always a small attraction between the like-sigh charged particles as a pure relativistic effect. If the coupling constant exceeds some critical values there arise discrete levels.The situation here is analogous to the so-called "abnormal" solutions.The signature of the norm of these discrete states coincides with the "time-parity".The states with the negative norms can be excluded from the physical sector-the one-time (quasipotential) wave-function corresponding to them vanishes identically.However the positive norm states survive and contribute to the total Green function (and the S-matrix) with the proper sign.Comment: 12 pages, 4 fugure

Some Problems of Self-Adjoint Extension in the Schrodinger equation

The Self-Adjoint Extension in the Schrodinger equation for potentials behaved as an attractive inverse square at the origin is critically reviewed. Original results are also presented. It is shown that the additional solutions must be retained for definite interval of parameters, which requires performing of Self-Adjoint Extension necessarily. The "Pragmatic approach" is used and some of its consequences are considered for wide class of transitive potentials. The problems of restriction of Self-Adjoint Extension parameter are also discussed. Various relevant applications are presented as well.Comment: 23 pages,submitted to J.Phys.

Pragmatic SAE procedure in the Schrodinger equation for the inverse-square-like potentials

The Self-Adjoint Extension in the Schrodinger equation for potentials behaved as an attractive inverse square at the origin is critically reviewed. Original results are also presented. It is shown that the additional non-regular solutions must be retained for definite interval of parameters, which requires a necessity of performing a Self-Adjoint Extension (SAE) procedure of radial Hamiltonian.The Pragmatic approach is used and some of its consequences are considered for wide class of transitive potentials. Our consideration is based on the established earlier by us a boundary condition for the radial wave function and the corresponding consequences are derived. Various relevant applications are presented as well. They are: inverse square potential in the Schrodinger equation is solved when the additional non-regular solution is retained. Valence electron model and the Klein-Gordon equation with the Coulomb potential is considered and the hydrino -like levels are discussed.Comment: 26 page

Supersymmetry in the Dirac equation for generalized Coulomb potential

We propose a symmetry of the Dirac equation under the interchange of signs of eigenvalues of the Dirac's $K$ operator. We show that the only potential which obeys this requirement is the Coulomb one for both vector and scalar cases. Spectrum of the Dirac Equation is obtained algebraically for arbitrary combination of Lorentz-scalar and Lorentz-vector Coulomb potentials using the Witten's Superalgebra approach. The results coincides with that, known from the explicit solution of the Dirac equation.Comment: 10 pages. Submitted to Phys. Rev.

Delta-like singularity in the Radial Laplace Operator and the Status of the Radial Schrodinger Equation

By careful exploration of separation of variables into the Laplacian in spherical coordinates, we obtain the extra delta-like singularity, elimination of which restricts the radial wave function at the origin. This constraint has the form of boundary condition for the radial Schrodinger equation.Comment: 8 page

Unexpected Delta-Function Term in the Radial Schrodinger Equation

Careful exploration of the idea that equation for radial wave function must be compatible with the full Schrodinger equation shows appearance of the delta-function while reduction of full Schrodinger equation in spherical coordinates. Elimination of this extra term produces a boundary condition for the radial wave function, which is the same both for regular and singular potentials.Comment: 5 page

Status of the Radial Schrodinger Equation

We show that equation for radial wave function in its traditional form is compatible with the full Schrodinger equation if and only if a definite additional constraint required. This constraint has a boundary condition form at the origin. Some of consequences are also discussed.Comment: 6 page