Related papers: The Infinite Square Well with a Point Interaction:…
Motivated by the analysis of the tunneling effect for the magnetic Laplacian, we introduce an abstract framework for the spectral reduction of a self-adjoint operator to a hermitian matrix. We illustrate this framework by three…
We consider the one-dimensional Dirac equation for the harmonic oscillator and the associated second order separated operators giving the resonances of the problem by complex dilation. The same operators have unique extensions as closed…
Eigenvalue interlacing is a useful tool in linear algebra and spectral analysis. In its simplest form, the interlacing inequality states that a rank-one positive perturbation shifts each eigenvalue up, but not further than the next…
We give precise meaning to piecewise constant potentials in non-commutative quantum mechanics. In particular we discuss the infinite and finite non-commutative spherical well in two dimensions. Using this, bound-states and scattering can be…
Coupled pair of PT-symmetric square wells is studied as a prototype of a quantum system characterized by two manifestly non-Hermitian commuting observables. We demonstrate that there exists a domain of couplings where both the respective…
Two alternative scenarios are shown possible in Quantum Mechanics working with non-Hermitian $PT-$symmetric form of observables. While, usually, people assume that $P$ is a self-adjoint indefinite metric in Hilbert space (and that their…
The theory of self-adjoint extensions of first and second order elliptic differential operators on manifolds with boundary is studied via its most representative instances: Dirac and Laplace operators. The theory is developed by exploiting…
The method of self-adjoint extensions is employed to determine the vacuum quantum numbers induced by a singular static magnetic vortex in $2+1$-dimensional spinor electrodynamics. The results obtained are gauge-invariant and, for certain…
We determine explicitly a boundary triple for the Dirac operator $H:=-i\alpha\cdot \nabla + m\beta + \mathbb V(x)$ in $\mathbb R^3$, for $m\in\mathbb R$ and $\mathbb V(x)= |x|^{-1} ( \nu \mathbb{I}_4 +\mu \beta -i \lambda…
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…
We show that the delta function potential can be exploited along with perturbation theory to yield the result of certain infinite series. The idea is that any exactly soluble potential if coupled with a delta function potential remains…
We compare two alternative expansions for finite attractive wells. One of them is known from long ago and is given in terms of powers of the strength parameter. The other one is based on the solution of the equations of the…
Using a recent reformulation of quantum mechanics where the potential function is not required, we are able to obtain the energy spectrum and wave function associated with the infinite square well analytically. Therefore, this work…
In this paper, we study self-adjointness and spectrum of operators of the form $$H=\displaystyle -\frac{d^2}{dx^2}+Fx, F>0 \quad\text{on} \quad \mathcal{H}=L^{2}(-L,L).$$ $H$ is called Stark operator and describes a quantum particle in a…
We apply the $\delta$-expansion perturbation scheme to the $\lambda \phi^{4}$ self-interacting scalar field theory in 3+1 D at finite temperature. In the $\delta$-expansion the interaction term is written as $\lambda (\phi^{2})^{ 1 +…
The exact solutions of the (2+1) dimensional Dirac equation on the torus and the new extension and generalization of the trigonometric Poschl-Teller potential families in terms of the torus parameters are obtained. Supersymmetric quantum…
We obtain an exact solution of the 1D Dirac equation for a square well potential of depth greater then twice the particle's mass. The energy spectrum formula in the Klein zone is surprisingly very simple and independent of the depth of the…
We discuss spectral properties of the Laplacian with multiple ($N$) point interactions in two-dimensional bounded regions. A mathematically sound formulation for the problem is given within the framework of the self-adjoint extension of a…
An approximation method, namely, the Extended Wronskian Determinant Approach, is suggested to study the one-dimensional Dirac equation. An integral equation which can be solved by iterative procedure to find the wave functions is…
We investigate the simple resonances of a 2 by 2 matrix of n-dimensional semiclassical Shr\"odinger operators that interact through a first order differential operator. We assume that one of the two (analytic) potentials admits a well with…