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We decorate the one-dimensional conic oscillator $frac{1}{2} left[-frac{d^{2} }{dx^{2} } + left|x right| right]$ with a point impurity of either $delta$-type, or local $delta$-type or even nonlocal $delta$-type. All the three cases are exactly solvable models, which are explicitly solved and analysed, as a first step towards higher dimensional models of physical relevance. We analyse the behaviour of the change in the energy levels when an interaction of the type $-lambda,delta(x)$ or $-lambda,delta(x-x_0)$ is switched on. In the first case, even energy levels (pertaining to antisymmetric bound states) remain invariant with $lambda$ although odd energy levels (pertaining to symmetric bound states) decrease as $lambda$ increases. In the second, all energy levels decrease when the form factor $lambda$ increases. A similar study has been performed for the so called nonlocal $delta$ interaction, requiring a coupling constant renormalization, which implies the replacement of the form factor $lambda$ by a renormalized form factor $beta$. In terms of $beta$, even energy levels are unchanged. However, we show the existence of level crossings: after a fixed value of $beta$ the energy of each odd level, with the natural exception of the first one, becomes lower than the constant energy of the previous even level. Finally, we consider an interaction of the type $-adelta(x)+bdelta(x)$, and analyse in detail the discrete spectrum of the resulting self-adjoint Hamiltonian.
We find supersymmetric partners of a family of self-adjoint operators which are self-adjoint extensions of the differential operator $-d^2/dx^2$ on $L^2[-a,a]$, $a>0$, that is, the one dimensional infinite square well. First of all, we classify these
We show the existence of Lorentz invariant Berry phases generated, in the Stueckleberg-Horwitz-Piron manifestly covariant quantum theory (SHP), by a perturbed four dimensional harmonic oscillator. These phases are associated with a fractional perturb
The Porter-Thomas (PT) distribution of resonance widths is one of the oldest and simplest applications of statistical ideas in nuclear physics. Previous experimental data confirmed it quite well but recent and more careful investigations show clear d
This paper is about the scattering theory for one-dimensional matrix Schrodinger operators with a matrix potential having a finite first moment. The transmission coefficients are analytically continued and extended to the band edges. An explicit expr
The Darwin-Howie-Whelan equations are commonly used to describe and simulate the scattering of fast electrons in transmission electron microscopy. They are a system of infinitely many envelope functions, derived from the Schrodinger equation. However