Abstract:
This paper is devoted to the study of the chaotic properties of some specific backward shift unbounded operators ; realized as differential operators in Bargmann space, where and are the standard Bose annihilation and creation operators such that . 1. Introduction It is well known that linear operators in finite-dimensional linear spaces cannot be chaotic but the nonlinear operator may be. Only in infinite-dimensional linear spaces can linear operators have chaotic properties. These last properties are based on the phenomenon of hypercyclicity or the phenomen of nonwandercity. The study of the phenomenon of hypercyclicity originates in the papers by Birkoff [1] and Maclane [2] who show, respectively, that the operators of translation and differentiation, acting on the space of entire functions, are hypercyclic. The theories of hypercyclic operators and chaotic operators have been intensively developed for bounded linear operators; we refer to [1, 3–5] and references therein. For a bounded operator, Ansari asserts in [6] that powers of a hypercyclic bounded operator are also hypercyclic. For an unbounded operator, Salas exhibits in [7] an unbounded hypercyclic operator whose square is not hypercyclic. The result of Salas shows that one must be careful in the formal manipulation of operators with restricted domains. For such operators, it is often more convenient to work with vectors rather than with operators themselves. Now, let be an unbounded operator on a separable infinite dimensional Banach space . A point is called wandering if there exists an open set containing such that for some and for all one has . (In other words, the neighbourhood eventually never returns). A point is called nonwandering if it is not wandering. A closed subspace has hyperbolic structure if: , , and , where (the unstable subspace) and (the stable subspace) are closed. In addition, there exist constants and , such that: (i) for any , (ii) for any . is said to be a nonwandering operator relative to which has hyperbolic structure if the set of periodic points of is dense in . For the nonwandering operators, they are new linear chaotic operators. They are relative to hypercyclic operators, but different from them in the sense that some hypercyclic operators are not non-wandering operators and there also exists a non-wandering operator, which does not belong to hypercyclic operators (see [8], Remark？？ 3.5). In fact, suppose is a bounded linear operator and is invertible; if is a hypercyclic operator, then (see [9], Remark？？ 4.3) but if is a non-wandering operator, then where is

Abstract:
In Communications in Mathematical Physics, no. 199, (1998), we have considered the Heun operator $\displaystyle{ H = a^* (a + a^*)a}$ acting on Bargmann space where $a$ and $a^{*}$ are the standard Bose annihilation and creation operators satisfying the commutation relation $[a, a^{*}] = I$. We have used the boundary conditions at infinity to give a description of all maximal dissipative extensions in Bargmann space of the minimal Heun's operator $H$. The characteristic functions of the dissipative extensions have be computed and some completeness theorems have be obtained for the system of generalized eigenvectors of this operator. In this paper we study the deficiency numbers of the generalized Heun's operator $\displaystyle{ H^{p,m} = a^{*^{p}} (a^{m} + a^{*^{m}})a^{p}; (p, m=1, 2, .....)}$ acting on Bargmann space. In particular, here we find some conditions on the parameters $p$ and $m$ for that $\displaystyle{ H^{p,m}}$ to be completely indeterminate. It follows from these conditions that $\displaystyle{ H^{p,m}}$ is entire of the type minimal. And we show that $\displaystyle{H^{p,m}}$ and $\displaystyle{ H^{p,m}+ H^{*^{p,m}}}$ (where $H^{*^{p,m}}$ is the adjoint of the $H^{p,m}$) are connected at the chaotic operators. We will give a description of all maximal dissipative extensions and all selfadjoint extensions of the minimal generalized Heun's operator $H^{p,m}$ acting on Bargmann space in separate paper.

Abstract:
In this article, we obtain a regularized trace formula for magic Gribov operator\\ $ H = \lambda{''}G + H_{\mu,\lambda}$ acting on Bargmann space where $$G = a^{*3}a^{3} \quad \quad and \quad \quad H_{\mu,\lambda} = \mu a^{*}a + i\lambda a^{*}(a + a^{*})a$$ Here $a$ and $a^{*}$ are the standard Bose annihilation and creation operators and in Reggeon field theory, the real parameters $\lambda{''}$ is the magic coupling of Pomeron, $\mu$ is Pomeron intercept, $\lambda$ is the triple coupling of Pomeron and $i^{2} = -1$. An exact relation is established between the degree of subordination of the perturbation operator $H_{\mu,\lambda}$ to the unperturbed operator $G$ and the number of corrections necessary for the existence of finite formula of the trace.

Abstract:
{\it In Bargmann representation, the reggeon's field theory{\color{blue} [5]} is caracterized by the non symmetrical Gribov operator $\displaystyle{H_{\lambda',\mu,\lambda} = \lambda' A^{*^{2}}A^{2} + \mu A^{*}A + i\lambda A^{*}(A + A^{*})A}$ where $A^{*}$ and $A$ are the creation and annihilation operators; $[A, A^{*}] = I $.\\ $(\lambda',\mu, \lambda) \in \mathbb{R}^{3}$ are respectively the four coupling, the intercept and the triple coupling of Pomeron and $i^{2} = -1$. For $\lambda' > 0 ,\mu > 0$, let $\sigma (\lambda',\mu) \neq 0$ be the smallest eigenvalue of $H_{\lambda',\mu,\lambda}$, we show in this paper that $\sigma (\lambda',\mu)$ is positive, increasing and analytic function on the whole real line with respect to $\mu$ and that the spectral radius of $H_{\lambda',\mu,\lambda}^{-1}$ converges to that of $H_{0,\mu,\lambda}^{-1}$ as $\lambda'$ goes to zero.\\ The above results can be derived from the method used in ({\color{blue} [2]} Commun. Math. Phys. 93, (1984), p:123-139) by Ando-Zerner to study the smallest eigenvalue $\sigma (0,\mu)$ of $H_{0,\mu,\lambda}$, however as $H_{\lambda',\mu,\lambda}$ is regular perturbation of $H_{0,\mu,\lambda}$ then its study is much more easily. We can exploit the structure of $H_{\lambda',\mu,\lambda}^{-1}$ to deduce the results of Ando-Zerner established on the function $\sigma (0,\mu)$ as $\lambda'$ goes to zero.\\}

Abstract:
It is shown that the generalized creation and annihilation operators on Bargmann space of infinite order in a direction $a=(a_1,a_2,\ldots) \in l^2$ are inductive limits of the creation and annihilation operator acting on Bargmann space of $n$-th order.

Abstract:
Consider the set $\chi^0_{\mathrm{nw}}$ of non-wandering continuous flows on a closed surface. Then such a flow can be approximated by regular non-wandering flows without heteroclinic connections nor locally dense orbits in $\chi^0_{\mathrm{nw}}$. Using this approximation, we show that a non-wandering continuous flow on a closed connected surface is topologically stable if and only if the orbit space of it is homeomorphic to a closed interval. Moreover we state the non-existence of topologically stable non-wandering flows on closed surfaces which are not neither $\mathbb{S}^2$, $\mathbb{P}^2$, nor $\mathbb{K}^2$.

Abstract:
Some positive results about the composition of Toeplitz operators on the Segal-Bargmann space are presented. A Wick symbol where it is not possible to construct its associated Toeplitz operator is given.

Abstract:
The projection-operator formalism of Feshbach is applied to resonance scattering in a single-channel case. The method is based on the division of the full function space into two segments, internal (localized) and external (infinitely extended). The spectroscopic information on the resonances is obtained from the non-Hermitian effective Hamilton operator $H_{\rm eff}$ appearing in the internal part due to the coupling to the external part. As well known, additional so-called cut-off poles of the $S$-matrix appear, generally, due to the truncation of the potential. We study the question of spurious $S$ matrix poles in the framework of the Feshbach formalism. The numerical analysis is performed for exactly solvable potentials with a finite number of resonance states. These potentials represent a generalization of Bargmann-type potentials to accept resonance states. Our calculations demonstrate that the poles of the $S$ matrix obtained by using the Feshbach projection-operator formalism coincide with both the complex energies of the physical resonances and the cut-off poles of the $S$-matrix.

Abstract:
Momentum space of a gapped quantum system is a metric space: it admits a notion of distance reflecting properties of its quantum ground state. By using this quantum metric, we investigate geometric properties of momentum space. In particular, we introduce a non-local operator which represents distance square in real space and show that this corresponds to the Laplacian in curved momentum space, and also derive its path integral representation in momentum space. The quantum metric itself measures the second cumulant of the position operator in real space, much like the Berry gauge potential measures the first cumulant or the electric polarization in real space. By using the non-local operator and the metric, we study some aspects of topological phases such as topological invariants, the cumulants and topological phase transitions. The effect of interactions to the momentum space geometry is also discussed.

Abstract:
In recent years, a new approach to the theory of nuclear reactions leading to a break-down of the interacting subsytems into various channels has been developed. This approach was named the Antisymmetrized Molecular Dynamic (AMD), and its main idea lies in the description of the nucleons by wave packets in which the antisymmetrization effects (but not other quantum effects) are accounted for. In the present review, the basic principles of AMD are illustrated on the examples of simplest nuclear systems, and the results are compared with those provided by an exact quantum-mechanical description in the Fock-Bargmann space. The applicability region of AMD is discussed, in particular, in the cases of systems with discrete spectrum, and a relation between the classical AMD trajectories and the quantum distributions is established. At the same time, a new interpretation of Brink orbitals and Slater determinants built on them as eigenfunctions of the coordinate operator defined in the Fock-Bargmann space is proposed. It is shown that these functions form the cluster geometry of nuclear systems.