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In this paper, we define Berezin-type and Odzijewicz-type quantizations on compact smooth manifolds. The method is as follows: we embed the smooth manifold of real dimension $n$ into $ℂP^n$ and induce the quantizations from there. The standard way by which reproducing kernel Hilbert spaces are defined on submanifolds gives a way to define the pullback coherent states. In Berezin-type quantization, the Hilbert space of quantization is the pullback (by the embedding) of the Hilbert space of geometric quantization of $ℂP^n$. In the Odzijewicz-type quantization, one has to consider a tensor product of the geometric quantization line bundle with holomorphic $n$-forms. In the Berezin case, the operators that are quantized are those induced from the ambient space $ℂP^n$. The Berezin-type quantization exhibited here is a generalization of an earlier work of the author and Ghosh. In both Berezin- and Odzijewicz-type quantizations, we first exhibit this quantization explicitly on $ℂP^n$ and we induce the quantization on the smooth compact embedded manifold from $ℂP^n$.

Quantum billiards are a key focus in quantum mechanics, offering a simple yet powerful model to study complex quantum features. While the development of algebras for quantum systems is traced from one-dimensional integrable models to quantum groups and the Generalized Heisenberg Algebra (GHA). The primary focus of this work is to extend the GHA to quantum billiards, showcasing its application to separable and non-separable billiards. We apply the formalism to a square billiard, first generating one-dimensional coherent states with specific quantum numbers and exploring their time evolution. Then, we extend this approach to develop two-dimensional coherent states for the square billiards. We also demonstrate its applicability in a non-separable equilateral triangle billiard, describing their algebra generators and associated one-dimensional coherent states.

Two non-commutative dynamical entropies are studied in connection with the classical limit. For systems with a strongly chaotic classical limit, the Kolmogorov–Sinai invariant is recovered on time scales that are logarithmic in the quantization parameter. The model of the quantized hyperbolic automorphisms of the 2-torus is examined in detail.

A representation of the Jacobi algebra 𝔥₁ ⋊ 𝔰𝔲(1, 1) by first-order differential operators with polynomial coefficients on the manifold is presented. The Hilbert space of holomorphic functions on which the holomorphic first-order differential operators with polynomials coefficients act is constructed.

For a subquadratic symbol H on ℝ^d × ℝ^d = T*(ℝ^d), the quantum propagator of the time dependent Schrödinger equation is a Semiclassical Fourier-Integral Operator when Ĥ = H(x, ℏD_x) (ℏ-Weyl quantization of H). Its Schwartz kernel is described by a quadratic phase and an amplitude. At every time t, when ℏ is small, it is "essentially supported" in a neighborhood of the graph of the classical flow generated by H, with a full uniform asymptotic expansion in ℏ for the amplitude.

In this paper, our goal is to revisit this well-known and fundamental result with emphasis on the flexibility for the choice of a quadratic complex phase function and on global L² estimates when ℏ is small and time t is large. One of the simplest choice of the phase is known in chemical physics as Herman–Kluk formula. Moreover, we prove that the semiclassical expansion for the propagator is valid for where δ > 0 is a stability parameter for the classical system.

We study the propagation of wave packets for nonlinear nonlocal Schrödinger equations in the semi-classical limit. When the kernel is smooth, we construct approximate solutions for the wave functions in subcritical, critical and supercritical cases (in terms of the size of the initial data). The validity of the approximation is proved up to Ehrenfest time. For homogeneous kernels, we establish similar results in subcritical and critical cases. Nonlinear superposition principle for two nonlinear wave packets is also considered.

We construct generalized coherent states for the rationally extended Scarf-I potential. Statistical and geometrical properties of these states are investigated. Special emphasis is given to the study of spatio-temporal properties of the coherent states via the quantum carpet structure and the auto-correlation function. Through this study, we aim to find the signature of the “rationalization” of the conventional potentials and the classical orthogonal polynomials.

We consider the quantum evolution $e^{-i\frac{t}{\hslash }{H}_{\beta }}{\psi }_{\xi }^{\hslash }$ of a Gaussian coherent state ${\psi }_{\xi }^{\hslash }\in L^2(ℝ)$ localized close to the classical state $\xi \equiv (q,p)\in {ℝ}^2$, where $H_{\beta }$ denotes a self-adjoint realization of the formal Hamiltonian $-\frac{{\hslash }^2}{2m}\frac{d^2}{d{x}^2}+\beta {\delta }_0^{\prime }$, with ${\delta }_0^{\prime }$ the derivative of Dirac’s 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 (with respect to the $L^2(ℝ)$-norm, uniformly for any $t\in ℝ$ away from the collision time) by $e^{\frac{i}{\hslash }{A}_t}{e}^{it{L}_B}{\varphi }_x^{\hslash }$, where $A_t=\frac{p^{2}t}{2m}$, ${\varphi }_x^{\hslash }(\xi ):={\psi }_{\xi }^{\hslash }(x)$ and $L_B$ is a suitable self-adjoint extension of the restriction to ${{𝒞}}_c^{\infty }({\mathcal{ℳ}}_0)$, ${\mathcal{ℳ}}_0:=\{(q,p)\in {ℝ}^2|q\ne 0\}$, 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 and A. Posilicano, The semi-classical limit with a delta potential, Ann. Mat. Pura Appl. 200 (2021) 453–489], in the present case the approximation gives a smaller error: it is of order ${\hslash }^{7/2-\lambda }$, $0<\lambda <1/2$, whereas it turns out to be of order ${\hslash }^{3/2-\lambda }$, $0<\lambda <3/2$, for the delta potential. We also provide similar approximation results for both the wave and scattering operators.

We construct two commuting sets of creation and annihilation operators for the PT-symmetric oscillator. We then build coherent states of the latter as eigenstates of such annihilation operators by employing a modified version of the normalization integral that is relevant to PT-symmetric systems. We show that the coherent states are normalizable only in the range (0,1) of the underlying coupling parameter α.

Firstly, the solvability of some quantum models like Eckart and Rosen–Morse II are explained on the basis of the shape invariance theory. Then, two generalized types of the Klauder–Perelomov and Gazeau–Klauder coherent states are calculated for the models. By means of calculating the Mandel parameter, it is shown that the weight distribution function of the first type coherent states obeys the Poissonian and super-Poissonian statistics, however, the weight distribution function of the second type coherent states obeys the Poissonian and sub-Poissonian statistics.

A new kind of q-deformed charged coherent states is constructed in Fock space of two-mode q-boson system with su_q(2) covariance and a resolution of unity for these states is derived. We also present a simple way to obtain these coherent states using state projection method.

Using the realization idea of simultaneous shape invariance with respect to two different parameters of the associated Legendre functions, the Hilbert space of spherical harmonics Y_{n m}(θ,φ) corresponding to the motion of a free particle on a sphere is split into a direct sum of infinite-dimensional Hilbert subspaces. It is shown that these Hilbert subspaces constitute irreducible representations for the Lie algebra u(1,1). Then by applying the lowering operator of the Lie algebra u(1,1), Barut–Girardello coherent states are constructed for the Hilbert subspaces consisting of Y_{m m}(θ,φ) and Y_{m+1 m}(θ,φ).

Using simultaneous shape invariance with respect to two different parameters, we introduce a pair of appropriate operators which realize shape invariance symmetry for the monomials on a half-axis. It leads to the derivation of rotational symmetry and dynamical symmetry group H₄ with infinite-fold degeneracy for the lowest Landau levels. This allows us to represent the Heisenberg–Lie algebra h₄ not only by the lowest Landau levels, but also by their corresponding standard coherent states.

We investigate a generalization of A_r statistics discussed recently in the literature. The explicit complete set of state vectors for the A_r statistics system is given. We consider a Bargmann or an analytic function description of the Fock space corresponding to A_r statistics of bosonic kind. This brings, in a natural way, the so-called Gazeau–Klauder coherent states defined as eigenstates of the Jacobson annihilation operators. The minimization of Robertson uncertainty relation is also considered.

The existence of degenerated quantum vacua (coherent states of zero modes), for N-level quantum systems, leads to a breakdown of the original unitary U(N) symmetry in the many-particle theory. The action of some spontaneously broken symmetry transformations destabilize these pseudo-vacua and make them radiate. We study the structure of this thermal radiation, which turns out to be of Fermi–Dirac type.

We construct a new model of the quantum oscillator, which is related to the discrete q-Hermite polynomials of the second type. The position and momentum operators in the model are appropriate operators of the Fock representation of a deformation of the Heisenberg algebra. These operators have a discrete non-degenerate spectra. These spectra are spread over the whole real line. Coordinate and momentum realizations of the model are constructed. Coherent states are explicitly given.

We show that the non-Hermitian Hamiltonians of the simple harmonic oscillator with and symmetries involve a pseudo generalization of the Heisenberg algebra via two pairs of creation and annihilation operators which are -pseudo-Hermiticity and -anti-pseudo-Hermiticity of each other. The non-unitary Heisenberg algebra is represented by each of the pair of the operators in two different ways. Consequently, the coherent and the squeezed coherent states are calculated in two different approaches. Moreover, it is shown that the approach of Schwinger to construct the su(2), su(1, 1) and sp(4, ℝ) unitary algebras is promoted so that unitary algebras with more linearly dependent number of generators are made.

In this paper, we address some of the issues raised in the literature about the conflict between a large vacuum energy density, a priori predicted by quantum field theory, and the observed dark energy which must be the energy of vacuum or include it. We present a number of arguments against this claim and in favor of a null vacuum energy. They are based on the following arguments: A new definition for the vacuum in quantum field theory as a frame-independent coherent state; results from a detailed study of condensation of scalar fields in Friedmann–Lemaître–Robertson–Walker (FLRW) background performed in a previous work; and our present knowledge about the Standard Model of particle physics. One of the predictions of these arguments is the confinement of nonzero expectation value of Higgs field to scales roughly comparable with the width of electroweak gauge bosons or shorter. If the observation of Higgs by the LHC is confirmed, accumulation of relevant events and their energy dependence in near future should allow us to measure the spatial extend of the Higgs condensate.

We have constructed coherent states for the higher derivative Pais–Uhlenbeck Oscillator (PUO). In the process, we have suggested a novel way to construct coherent states for the oscillator having only negative energy levels. These coherent states have negative energies in general but their coordinate and momentum expectation values and dispersions behave in an identical manner as that of normal (positive energy) oscillator. The coherent states for the PUO have constant dispersions and a modified Heisenberg Uncertainty Relation. Moreover, under reasonable assumptions on parameters these coherent states can have positive energies.

Since symmetry properties of coherent states (CS) on Möbius strip (MS) and fermions are closely related, CS on MS are naturally associated to the topological properties of fermionic fields. Here, we consider CS and superpositions of coherent states (SCS) on MS. We extend a recent propose of CS on MS (Cirilo-Lombardo, J. Phys. A: Math. Theor.45, 244026 (2012)), including the analysis of periodic behaviors of CS and SCS on MS and the uncertainty relations associated to angular momentum and the phase angle. The advantage of CS and SCS on MS with respect to the standard ones and potential applications in continuous variable quantum computation (CVQC) are also addressed.