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Register a new userConstructions of the soluble potentials for the non-relativistic quantum system by means of the Heun functions
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The Schr{o}dinger equation $psi(x)+kappa^2 psi(x)=0$ where $kappa^2=k^2-V(x)$ is rewritten as a more popular form of a second order differential equation through taking a similarity transformation $psi(z)=phi(z)u(z)$ with $z=z(x)$. The Schr{o}dinger invariant $I_{S}(x)$ can be calculated directly by the Schwarzian derivative ${z, x}$ and the invariant $I(z)$ of the differential equation $u_{zz}+f(z)u_{z}+g(z)u=0$. We find an important relation for moving particle as $ abla^2=-I_{S}(x)$ and thus explain the reason why the Schr{o}dinger invariant $I_{S}(x)$ keeps constant. As an illustration, we take the typical Heun differential equation as an object to construct a class of soluble potentials and generalize the previous results through choosing different $rho=z(x)$ as before. We get a more general solution $z(x)$ through integrating $(z)^2=alpha_{1}z^2+beta_{1}z+gamma_{1}$ directly and it includes all possibilities for those parameters. Some particular cases are discussed in detail.
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After a brief introduction to Heun type functions we note that the actual solutions of the eigenvalue equation emerging in the calculation of the one loop contribution to QCD from the Belavin-Polyakov-Schwarz-Tyupkin instanton and the similar calculation for a Dirac particle coupled to a scalar $CP^1$ model in two dimensions can be given in terms of confluent Heun equation in their original forms. These equations were previously modified to be solved by more elementary functions. We also show that polynomial solutions with discrete eigenvalues are impossible to find in the unmodified equations.
A simple 1-D relativistic model for a diatomic molecule with a double point interaction potential is solved exactly in a constant electric field. The Weyl-Titchmarsh-Kodaira method is used to evaluate the spectral density function, allowing the correct normalization of continuum states. The boundary conditions at the potential wells are evaluated using Colombeaus generalized function theory along with charge conjugation invariance and general properties of self-adjoint extensions for point-like interactions. The resulting spectral density function exhibits resonances for quasibound states which move in the complex energy plane as the model parameters are varied. It is observed that for a monotonically increasing interatomic distance, the ground state resonance can either go deeper into the negative continuum or can give rise to a sequence of avoided crossings, depending on the strength of the potential wells. For sufficiently low electric field strength or small interatomic distance, the behavior of resonances is qualitatively similar to non-relativistic results.
We discuss the possibility of generating spin squeezed states by means of driven superradiance, analytically affirming and broadening the finding in [Phys. Rev. Lett. 110, 080502 (2013)]. In an earlier paper [Phys. Rev. Lett. 112, 140402 (2014)] the authors determined that spontaneous purely-dissipative Dicke model superradiance failed to generate any entanglement over the course of the systems time evolution. In this article we show that by adding a driving field, however, the Dicke model system can be tuned to evolve toward an entangled steady state. We discuss how to optimize the driving frequency to maximize the entanglement. We show that the resulting entanglement is fairly strong, in that it leads to spin squeezing.
Unitary $t$-designs are `good finite subsets of the unitary group $U(d)$ that approximate the whole unitary group $U(d)$ well. Unitary $t$-designs have been applied in randomized benchmarking, tomography, quantum cryptography and many other areas of quantum information science. If a unitary $t$-design itself is a group then it is called a unitary $t$-group. Although it is known that unitary $t$-designs in $U(d)$ exist for any $t$ and $d$, the unitary $t$-groups do not exist for $tgeq 4$ if $dgeq 3$, as it is shown by Guralnick-Tiep (2005) and Bannai-Navarro-Rizo-Tiep (BNRT, 2018). Explicit constructions of exact unitary $t$-designs in $U(d)$ are not easy in general. In particular, explicit constructions of unitary $4$-designs in $U(4)$ have been an open problem in quantum information theory. We prove that some exact unitary $(t+1)$-designs in the unitary group $U(d)$ are constructed from unitary $t$-groups in $U(d)$ that satisfy certain specific conditions. Based on this result, we specifically construct exact unitary $3$-designs in $U(3)$ from the unitary $2$-group $SL(3,2)$ in $U(3),$ and also unitary $4$-designs in $U(4)$ from the unitary $3$-group $Sp(4,3)$ in $U(4)$ numerically. We also discuss some related problems.
Schrodingers equation (SE) and the information-optimizing principle based on Fishers information measure (FIM) are intimately linked, which entails the existence of a Legendre transform structure underlying the SE. In this comunication we show that the existence of such an structure allows, via the virial theorem, for the formulation of a parameter-free ground states SE-ansatz for a rather large family of potentials. The parameter-free nature of the ansatz derives from the structural information it incorporates through its Legendre properties.
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