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Register a new userTwo-point one-dimensional $delta$-$delta^prime$ interactions: non-abelian addition law and decoupling limit
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Added by
Jose M Munoz-Castaneda
Publication date
2015
fields
Physics
and research's language is
English
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In this contribution to the study of one dimensional point potentials, we prove that if we take the limit $qto 0$ on a potential of the type $v_0delta({y})+{2}v_1delta({y})+w_0delta({y}-q)+ {2} w_1delta({y}-q)$, we obtain a new point potential of the type ${u_0} delta({y})+{2 u_1} delta({y})$, when $ u_0$ and $ u_1$ are related to $v_0$, $v_1$, $w_0$ and $w_1$ by a law having the structure of a group. This is the Borel subgroup of $SL_2({mathbb R})$. We also obtain the non-abelian addition law from the scattering data. The spectra of the Hamiltonian in the exceptional cases emerging in the study are also described in full detail. It is shown that for the $v_1=pm 1$, $w_1=pm 1$ values of the $delta^prime$ couplings the singular Kurasov matrices become equivalent to Dirichlet at one side of the point interaction and Robin boundary conditions at the other side.
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In this paper we construct $mathcal{N}=2$ supersymmetric (SUSY) quantum mechanics over several configurations of Dirac-$delta$ potentials from one single delta to a Dirac comb rqrq. We show in detail how the building of supersymmetry on potentials with delta interactions placed in two or more points on the real line requires the inclusion of quasi-square wells. Therefore, the basic ingredient of a supersymmetric Hamiltonian containing two or more Dirac-$delta$s is the singular potential formed by a Dirac-$delta$ plus a step ($theta$) at the same point. In this $delta/theta$ SUSY Hamiltonian there is only one singlet ground state of zero energy annihilated by the two supercharges or a doublet of ground states paired by supersymmetry of positive energy depending on the relation between the Dirac well strength and the height of the step potential. We find a scenario of either unbroken supersymmetry with Witten index one or supersymmetry breaking when there is one bosonicrqrq and one fermionicrqrq ground state such that the Witten index is zero. We explain next the different structure of the scattering waves produced by three $delta/theta$ potentials with respect to the eigenfunctions arising in the non-SUSY case. In particular, many more bound states paired by supersymmetry exist within the supersymmetric framework compared with the non-SUSY problem. An infinite array of equally spaced $delta$-interactions of the same strength but alternatively attractive and repulsive are susceptible of being promoted to a ${cal N}=2$ supersymmetric system...
We consider the quantum evolution $e^{-ifrac{t}{hbar}H_{beta}} psi_{xi}^{hbar}$ of a Gaussian coherent state $psi_{xi}^{hbar}in L^{2}(mathbb{R})$ localized close to the classical state $xi equiv (q,p) in mathbb{R}^{2}$, where $H_{beta}$ denotes a self-adjoint realization of the formal Hamiltonian $-frac{hbar^{2}}{2m},frac{d^{2},}{dx^{2}} + beta,delta_{0}$, with $delta_{0}$ the derivative of Diracs delta distribution at $x = 0$ and $beta$ a real parameter. We show that in the semi-classical limit such a quantum evolution can be approximated (w.r.t. the $L^{2}(mathbb{R})$-norm, uniformly for any $t in mathbb{R}$ away from the collision time) by $e^{frac{i}{hbar} A_{t}} e^{it L_{B}} phi^{hbar}_{x}$, where $A_{t} = frac{p^{2}t}{2m}$, $phi_{x}^{hbar}(xi) := psi^{hbar}_{xi}(x)$ and $L_{B}$ is a suitable self-adjoint extension of the restriction to $mathcal{C}^{infty}_{c}({mathscr M}_{0})$, ${mathscr M}_{0} := {(q,p) in mathbb{R}^{2},|,q eq 0}$, of ($-i$ times) the generator of the free classical dynamics. While the operator $L_{B}$ here utilized is similar to the one appearing in our previous work [C. Cacciapuoti, D. Fermi, A. Posilicano, The semi-classical limit with a delta potential, Annali di Matematica Pura e Applicata (2020)] regarding the semi-classical limit with a delta potential, in the present case the approximation gives a smaller error: it is of order $hbar^{7/2-lambda}$, $0 < lambda < 1/2$, whereas it turns out to be of order $hbar^{3/2-lambda}$, $0 < lambda < 3/2$, for the delta potential. We also provide similar approximation results for both the wave and scattering operators.
The spherically symmetric potential $a ,delta (r-r_0)+b,delta (r-r_0)$ is generalised for the $d$-dimensional space as a characterisation of a unique selfadjoint extension of the free Hamiltonian. For this extension of the Dirac delta, the spectrum of negative, zero and positive energy states is studied in $dgeq 2$, providing numerical results for the expectation value of the radius as a function of the free parameters of the potential. Remarkably, only if $d=2$ the $delta$-$delta$ potential for arbitrary $a>0$ admits a bound state with zero angular momentum.
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J. Mateos Guilarte
, Jose M. Munoz-Castaneda
, Irina Pirozhenko andn Lucia Santamaria-Sanz
2019
We study the spectrum of the 1D Dirac Hamiltonian encompassing the bound and scattering states of a fermion distorted by a static background built from $delta$-function potentials. We distinguish between mass-spike and electrostatic $delta$-potentials. Differences in the spectra arising depending on the type of $delta$-potential studied are thoroughly explored.
An almost non-abelian extension of the Rieffel deformation is presented in this work. The non-abelicity comes into play by the introduction of unitary groups which are dependent of the infinitesimal generators of $SU(n)$. This extension is applied to quantum mechanics and quantum field theory.
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