Search Results (47)

We consider a multi-species mixture of interacting bosons, $N_1$ bosons of mass $m_1$, $N_2$ bosons of mass $m_2$, and $N_3$ bosons of mass $m_3$, in a harmonic trap with frequency $\omega$. The corresponding intra-species interaction strengths are ${\lambda }_{11}$, ${\lambda }_{22}$, and ${\lambda }_{33}$, and the inter-species interaction strengths are ${\lambda }_{12}$, ${\lambda }_{13}$, and ${\lambda }_{23}$. When the shape of all interactions is harmonic, the system corresponds to the generic multi-species harmonic-interaction model, which is exactly solvable. We start by solving the many-particle Hamiltonian and concisely discussing the ground-state wavefunction and energy in explicit forms as functions of all parameters, the masses, numbers of particles, and the intra-species and inter-species interaction strengths. We then explicitly compute the reduced one-particle density matrices for all the species and diagonalize them, thus generalizing the treatment by the authors earlier. The respective eigenvalues determine the degree of fragmentation of each species. As an application, we focus on phenomena that do not arise in the corresponding single-species or two-species systems. For instance, we consider a mixture of two kinds of bosons in a bath made by a third kind, controlling the fragmentation of the former by coupling to the latter. Another example exploits the possibility of different connectivities (i.e., which species interacts with which species) in the mixture, and demonstrates how the fragmentation of species 3 can be manipulated by the interaction between species 1 and species 2, when species 3 and 1 do not interact with each other. We highlight the properties of fragmentation that only appear in the multi-species mixture. Further applications are briefly discussed. Full article

In the context of the current lack of compatibility of the classical and quantum approaches to gravity, exactly solvable elementary pseudo-Hermitian quantum models are analyzed, supporting the acceptability of a point-like form of the Big Bang. The purpose is served by a hypothetical (non-covariant) identification of the “time of the Big Bang” with Kato’s exceptional-point parameter $t=0$. The consequences (including the ambiguity of the patterns of unfolding the singularity after the Big Bang) are studied in detail. In particular, singular values of the observables are shown to be useful in the analysis. Full article

This paper presents a two-qubit model derived from an SU(2)-symmetric 4×4 Hamiltonian. The resulting model is physically significant and, due to the SU(2) symmetry, is exactly solvable in both time-independent and time-dependent cases. Using the formal, general form of the related time evolution operator, the time dependence of the entanglement level for certain initial conditions is examined within the Rabi and Landau–Majorana–Stückelberg–Zener scenarios. The potential for applying this approach to higher-dimensional Hamiltonians to develop more complex exactly solvable models of interacting qubits is also highlighted. Full article

Let us consider a two-sided multi-species stochastic particle model with finitely many particles on $\mathbb{Z}$, defined as follows. Suppose that each particle is labelled by a positive integer l, and waits a random time exponentially distributed with rate 1. It then chooses the right direction to jump with probability p, or the left direction with probability $q=1-p$. If the particle chooses the right direction, it jumps to the nearest site occupied by a particle $l^{\prime }<l$ (with the convention that an empty site is considered as a particle with labelled 0). If the particle chooses the left direction, it jumps to the next site on the left only if that site is either empty or occupied by a particle $l^{\prime }<l$, and in the latter case, particles l and $l^{\prime }$ swap their positions. We show that this model is integrable, and provide the exact formula of the transition probability using the Bethe ansatz. Full article

For the first time, a self-consistent mathematical approach to describe economic processes with a general form of a memory function is proposed. In this approach, power-type memory is a special case of such general memory. The memory is described by pairs of memory functions that satisfy the Sonin and Luchko conditions. We propose using general fractional calculus (GFC) as a mathematical language that allows us to describe a general form of memory in economic processes. The existence of memory (non-locality in time) means that the process depends on the history of changes to this process in the past. Using GFC, exactly solvable economic models of natural growth with a general form of memory are proposed. Equations of natural growth with general memory are equations with general fractional derivatives and general fractional integrals for which the fundamental theorems of GFC are satisfied. Exact solutions for these equations of models of natural growth with general memory are derived. The properties of dynamic maps with a general form of memory are described in the general form and do not depend on the choice of specific types of memory functions. Examples of these solutions for various types of memory functions are suggested. Full article

A unitary-evolution process leading to an ultimate collapse and to a complete loss of observability alias quantum phase transition is studied. A specific solvable $N\mathrm{-}$state model is considered, characterized by a non-stationary non-Hermitian Hamiltonian. Our analysis uses quantum mechanics formulated in Schrödinger picture in which, in principle, only the knowledge of a complete set of observables (i.e., operators ${\mathrm{\Lambda }}_j$) enables one to guarantee the uniqueness of the related physical Hilbert space (i.e., of its inner-product metric $\mathrm{\Theta }$). Nevertheless, for the sake of simplicity, we only assume the knowledge of just a single input observable (viz., of the energy-representing Hamiltonian $H\equiv {\mathrm{\Lambda }}_1$). Then, out of all of the eligible and Hamiltonian-dependent “Hermitizing” inner-product metrics $\mathrm{\Theta }=\mathrm{\Theta }\left(H\right)$, we pick up just the simplest possible candidate. Naturally, this slightly restricts the scope of the theory, but in our present model, such a restriction is more than compensated for by the possibility of an alternative, phenomenologically better motivated constraint by which the time-dependence of the metric is required to be smooth. This opens a new model-building freedom which, in fact, enables us to force the system to reach the collapse, i.e., a genuine quantum catastrophe as a result of the mere conventional, strictly unitary evolution. Full article

Reinterpretation of mathematics behind the exactly solvable Calogero’s A-particle quantum model is used to propose its generalization. Firstly, it is argued that the strongly singular nature of Calogero’s particle–particle interactions makes the original permutation-invariant Hamiltonian tractable as a direct sum $H=⨁H_a$ of isospectral components, which are mutually independent. Secondly, after the elimination of the center-of-mass motion, the system is reconsidered as existing in the reduced Euclidean space ${\mathbb{R}}^{A-1}$ of relative coordinates and decaying into a union of subsets $W_a$ called Weyl chambers. The mutual independence of the related reduced forms of operators $H_a$ enables us to makes them nonisospectral. This breaks the symmetry and unfolds the spectral degeneracy of H. A new multiparametric generalization of the conventional A-body Calogero model is obtained. Its detailed description is provided up to $A=4$. Full article

We consider a well-known, exactly solvable model of an open quantum system with pure decoherence. The aim of this paper is twofold. Firstly, decoherence is a property of open quantum systems important for both quantum technologies and the fundamental question of the quantum–classical transition. It is worth studying how the long-term rate of decoherence depends on the spectral density characterising the system–bath interaction in this exactly solvable model. Secondly, we address a more general problem of the Markovian embedding of non-Markovian open system dynamics. It is often assumed that a non-Markovian open quantum system can be embedded into a larger Markovian system. However, we show that such embedding is possible only for Ohmic spectral densities (for the case of a positive bath temperature) and is impossible for both sub- and super-Ohmic spectral densities. On the other hand, for Ohmic spectral densities, an asymptotic large-time Markovianity (in terms of the quantum regression formula) takes place. Full article

This review delves into the utilization of a sextic oscillator within the $\beta$ degree of freedom of the Bohr Hamiltonian to elucidate critical-point solutions in nuclei, with a specific emphasis on the critical point associated with the $\beta$ shape variable, governing transitions from spherical to deformed nuclei. To commence, an overview is presented for critical-point solutions E(5), X(5), X(3), Z(5), and Z(4). These symmetries, encapsulated in simple models, all model the $\beta$ degree of freedom using an infinite square-well (ISW) potential. They are particularly useful for dissecting phase transitions from spherical to deformed nuclear shapes. The distinguishing factor among these models lies in their treatment of the $\gamma$ degree of freedom. These models are rooted in a geometrical context, employing the Bohr Hamiltonian. The review then continues with the analysis of the same critical solutions but with the adoption of a sextic potential in place of the ISW potential within the $\beta$ degree of freedom. The sextic oscillator, being quasi-exactly solvable (QES), allows for the derivation of exact solutions for the lower part of the energy spectrum. The outcomes of this analysis are examined in detail. Additionally, various versions of the sextic potential, while not exactly solvable, can still be tackled numerically, offering a means to establish benchmarks for criticality in the transitional path from spherical to deformed shapes. This review extends its scope to encompass related papers published in the field in the past 20 years, contributing to a comprehensive understanding of critical-point symmetries in nuclear physics. To facilitate this understanding, a map depicting the different regions of the nuclide chart where these models have been applied is provided, serving as a concise summary of their applications and implications in the realm of nuclear structure. Full article

We demonstrate that the Finite-Time-Path Field Theory is an adequate tool for calculating neutrino oscillations. We apply this theory using a mass-mixing Lagrangian which involves the correct Dirac spin and chirality structure and a Pontecorvo–Maki–Nakagawa–Sakata (PMNS)-like mixing matrix. The model is exactly solvable. The Dyson–Schwinger equations transform propagators of the input free (massless) flavor neutrinos into a linear combination of oscillating (massive) neutrinos. The results are consistent with the predictions of the PMNS matrix while allowing for extrapolation to early times. Full article

We extend the scope of the unified factorization method to the solution of conditionally and unconditionally exactly solvable models of quantum mechanics, proposed in a previous paper [R.R. Nigmatullin, A.A. Khamzin, D. Baleanu, Results in Physics 41 (2022) 105945]. The possibilities of applying the unified approach in the factorization method are demonstrated by calculating the energy spectrum of a potential constructed in the form of a second-order polynomial in many of the linearly independent functions. We analyze the solutions in detail when the potential is constructed from two linearly independent functions. We show that in the general case, such kinds of potentials are conditionally exactly solvable. To verify the novel approach, we consider several known potentials. We show that the shape of the energy spectrum is invariant to the number of functions from which the potential is formed and is determined by the type of differential equations that the potential-generating functions obey. Full article

This work scrutinizes, using statistical mechanics indicators, important traits displayed by quantum many-body systems. Our statistical mechanics quantifiers are employed, in the context of Gibbs’ canonical ensemble at temperature T. A new quantifier of this sort is also presented here. The present discussion focuses attention on the role played by the fermion number N in many-fermion dynamics, that is, N is our protagonist. We have discovered discovers particular values of N for which the thermal indicators exhibit unexpected abrupt variations. Such a fact reflects an unanticipated characteristic of fermionic dynamics. Full article

We analyze here through exact calculations the thermodynamical effects in depolarizing a quantum spin-bath initially at zero temperature through a quantum probe coupled to an infinite temperature bath by evaluating the heat and entropy changes. We show that the correlations induced in the bath during the depolarizing process does not allow for the entropy of the bath to increase towards its maximal limit. On the contrary, the energy deposited in the bath can be completely extracted in a finite time. We explore these findings through an exactly solvable central spin model, wherein a central spin-$1/2$ system is homogeneously coupled to a bath of identical spins. Further, we show that, upon destroying these unwanted correlations, we boost the rate of both energy extraction and entropy towards their limiting values. We envisage that these studies are relevant for quantum battery research wherein both charging and discharging processes are key to characterizing the battery performance. Full article

In this paper, we construct a solvable toy model of the quantum dynamics of the interior of a spherical black hole with falling spherical scalar field excitations. We first argue about how some aspects of the quantum gravity dynamics of realistic black holes emitting Hawking radiation can be modeled using Kantowski–Sachs solutions with a massless scalar field when one focuses on the deep interior region $r\ll M$ (including the singularity). Further, we show that in the $r\ll M$ regime, and in suitable variables, the KS model becomes exactly solvable at both the classical and quantum levels. The quantum dynamics inspired by loop quantum gravity is revisited. We propose a natural polymer quantization where the area a of the orbits of the rotation group is quantized. The polymer (or loop) dynamics is closely related to the Schroedinger dynamics away from the singularity with a form of continuum limit naturally emerging from the polymer treatment. The Dirac observable associated with the mass is quantized and shown to have an infinite degeneracy associated with the so-called $ϵ$-sectors. Suitable continuum superpositions of these are well-defined distributions in the fundamental Hilbert space and satisfy the continuum Schroedinger dynamics. Full article

We introduce and prove the “root theorem”, which establishes a condition for families of operators to annihilate all root states associated with zero modes of a given positive semi-definite k-body Hamiltonian chosen from a large class. This class is motivated by fractional quantum Hall and related problems, and features generally long-ranged, one-dimensional, dipole-conserving terms. Our theorem streamlines analysis of zero-modes in contexts where “generalized” or “entangled” Pauli principles apply. One major application of the theorem is to parent Hamiltonians for mixed Landau-level wave functions, such as unprojected composite fermion or parton-like states that were recently discussed in the literature, where it is difficult to rigorously establish a complete set of zero modes with traditional polynomial techniques. As a simple application, we show that a modified $V_1$ pseudo-potential, obtained via retention of only half the terms, stabilizes the $\nu =1/2$ Tao–Thouless state as the unique densest ground state. Full article