Algebraic Lq-norms and complexity-like properties of Jacobi polynomials-Degree and parameter asymptotics

The Jacobi polynomials $\hat{P}_n^{(\alpha,\beta)}(x)$ conform the canonical family of hypergeometric orthogonal polynomials (HOPs) with the two-parameter weight function $(1-x)^\alpha (1+x)^\beta, \alpha,\beta>-1,$ on the interval $[-1,+1]$. The spreading of its associated probability density (i.e., the Rakhmanov density) over the orthogonality support has been quantified, beyond the dispersion measures (moments around the origin, variance), by the algebraic $\mathfrak{L}_{q}$-norms (Shannon and R\’enyi entropies) and the monotonic complexity-like measures of Cram\’er-Rao, Fisher-Shannon and LMC (L\’opez-Ruiz, Mancini and Calbet) types. These quantities, however, have been often determined in an analytically highbrow, non-handy way; specially when the degree or the parameters $(\alpha,\beta)$ are large. In this work, we determine in a simple, compact form the leading term of the entropic and complexity-like properties of the Jacobi polynomials in the two extreme situations: ($n\rightarrow \infty$; fixed $\alpha,\beta$) and ($\alpha\rightarrow \infty$; fixed $n,\beta$). These two asymptotics are relevant \textit{per se} and because they control the physical entropy and complexity measures of the high energy (Rydberg) and high dimensional (pseudoclassical) states of many exactly, conditional exactly and quasi-exactly solvable quantum-mechanical potentials which model numerous atomic and molecular systems.

Comments on two conditionally solvable quantum-mechanical models

We analyze the results published in this journal about two conditionally solvable quantum-mechanical models. We show that the authors failed to derive the spectrum correctly because they did not take into consideration all the roots of the truncation condition used for that aim and did not interpret them correctly.

The anti-Jaynes-Cummings model is solvable : quantum Rabi model in rotating and counter-rotating frames ; following the experiments

This article is a response to the continued assumption, cited even in reports and reviews of recent experimental breakthroughs and advances in theoretical methods, that the antiJaynes-Cummings (AJC)interaction is an intractable energy non-conserving component of the quantum Rabi model (QRM). We present three key features of QRM dynamics : (a) the AJC interaction component has a conserved excitation number operator and is exactly solvable (b) QRM dynamical space consists of a rotating frame (RF) dominated by an exactly solved Jaynes-Cummings (JC) interaction specied by a conserved JC excitation number operator which generates the U(1) symmetry of RF and a correlated counter-rotating frame (CRF) dominated by an exactly solved antiJaynes-Cummings (AJC) interaction specied by a conserved AJC excitation number operator which generates the U(1) symmetry of CRF (c) for QRM dynamical evolution in RF, the initial atom-eld state je0i is an eigenstate of the effective AJC Hamiltonian HAJC, while the effective JC Hamiltonian HJC drives this initial state je0i into a time evolving entangled state, and, in a corresponding process for QRM dynamical evolution in CRF, the initial atom-eld state jg0i is an eigenstate of the effective JC Hamiltonian, while the effective AJC Hamiltonian drives this initial state jg0i into a time evolving entangled state, thus addressing one of the long-standing challenges of theoretical and experimental QRM dynamics; consistent generalizations of the initial states je0i , jg0i to corresponding n 0 entangled eigenstates j+en i , j 􀀀g ni of the AJC in RF and JC in CRF, respectively, provides general dynamical evolution of QRM characterized by collapses and revivals in the time evolution of the atomic, eld mode, JC and AJC excitation numbers for large initial photon numbers ; the JC and AJC excitation numbers are conserved in the respective frames RF, CRF, but each evolves with time in the alternate frame.

Numerical study of the SWKB condition of novel classes of exactly solvable systems

The supersymmetric WKB (SWKB) condition is supposed to be exact for all known exactly solvable quantum mechanical systems with the shape invariance. Recently, it was claimed that the SWKB condition was not exact for the extended radial oscillator, whose eigenfunctions consisted of the exceptional orthogonal polynomial, even the system possesses the shape invariance. In this paper, we examine the SWKB condition for the two novel classes of exactly solvable systems: one has the multi-indexed Laguerre and Jacobi polynomials as the main parts of the eigenfunctions, and the other has the Krein–Adler Hermite, Laguerre and Jacobi polynomials. For all of them, one can always remove the [Formula: see text]-dependency from the condition, and it is satisfied with a certain degree of accuracy.

Emergence of generalized hydrodynamics in the non-local Luttinger model

We propose the Luttinger model with finite-range interactions as a simple tractable example in 1+1 dimensions to analytically study the emergence of Euler-scale hydrodynamics in a quantum many-body system. This non-local Luttinger model is an exactly solvable quantum field theory somewhere between conformal and Bethe-ansatz integrable models. Applying the recent proposal of generalized hydrodynamics, we show that the model allows for fully explicit yet non-trivial solutions of the resulting Euler-scale hydrodynamic equations. Comparing with exact analytical non-equilibrium results valid at all time and length scales, we show perfect agreement at the Euler scale when the interactions are short range. A formal proof of the emergence of generalized hydrodynamics in the non-local Luttinger model is also given, and effects of long-range interactions are briefly discussed.

Operator Entanglement in Local Quantum Circuits I: Chaotic Dual-Unitary Circuits

The entanglement in operator space is a well established measure for the complexity of quantum many-body dynamics. In particular, that of local operators has recently been proposed as dynamical chaos indicator, i.e. as a quantity able to discriminate between quantum systems with integrable and chaotic dynamics. For chaotic systems the local-operator entanglement is expected to grow linearly in time, while it is expected to grow at most logarithmically in the integrable case. Here we study the dynamics of local-operator entanglement in dual-unitary quantum circuits, a class of "statistically solvable" quantum circuits that we recently introduced. We identify a class of ``completely chaotic" dual-unitary circuits where the local-operator entanglement grows linearly and we provide a conjecture for its asymptotic behaviour which is in excellent agreement with the numerical results. Interestingly, our conjecture also predicts a ``phase transition" in the slope of the local-operator entanglement when varying the parameters of the circuits.