Search Results (8)

The Hamiltonian of Reggeon field theory is defined by ${\displaystyle H_{\mu ,\lambda }=\mu A^{*}A + }$${\displaystyle i\lambda A^{*}(A+A^{*})A}$, where A and $A^{*}$ are the annihilation and creation operators satisfying $[A,A^{*}]=I$ and $\mu$, $\lambda$ are real parameters, and $i^2=-1$. This operator acts on Bargmann space $\mathbb{B}$ where $\mathbb{B}$ is a Hilbert space of holomorphic square integrable functions with respect to the Gaussian-weighted Lebesgue measure. In this work, we consider the operator ${\displaystyle H_{\lambda }=i\lambda A^{*}(A+A^{*})A}$ with maximum domain ${\displaystyle D\left(H_{\lambda }\right)=\{\phi \in \mathbb{B};H_{\lambda }\phi \in \mathbb{B}\}}$. If we limit the domain to polynomials and take the closure of the obtained operator, we denote it by ${\displaystyle H_{\lambda }^{min},}$ of which $H_{\lambda }$ is obviously an extension. Contrary to what happens for $\mu \ne 0$, it is well known that these two operators are different. The main purpose of the present work is to show that ${\displaystyle H_{\lambda }}$ admits a right-inverse ${\displaystyle K_{\lambda }}$, i.e., ${\displaystyle H_{\lambda }{K}_{\lambda }=I}$ on negative imaginary axis and that ${\displaystyle K_{\lambda }}$ is compact. Full article

We construct the Green functions (or Feynman’s propagators) for the Schrödinger equations of the form $i{\psi }_t+{\displaystyle \frac{1}{4}}{\psi }_{xx}\pm t{x}^2\psi =0$ (for the wave function $\psi$ and its time (t) and x-space derivatives) in terms of Airy functions and solve the Cauchy initial value problem in the coordinate and momentum representations. Particular solutions of the corresponding nonlinear Schrödinger equations with variable coefficients are also found. A special case of the quantum parametric oscillator is studied in detail first. The Green function is explicitly given in terms of Airy functions and the corresponding transition amplitudes are found in terms of a hypergeometric function. The general case of the quantum parametric oscillator is considered then in a similar fashion. A group theoretical meaning of the transition amplitudes and their relation with Bargmann’s functions is established. The relevant bibliography, to the best of our knowledge, is addressed. Full article

In 1998, we gave a complete scattering analysis of the cubic Heun operator ${\displaystyle H=a^{\ast }(a+a^{\ast })a}$ acting on Bargmann space, where a and $a^{\ast }$ are the standard Bose annihilation and creation operators satisfying the commutation relation $[a,a^{\ast }]=I$. We 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 were computed, and some completeness theorems were obtained for the system of generalized eigenvectors of this operator. In this paper, we study the deficiency numbers of the generalized Heun’s operator ${\displaystyle H^{p,m}=a^{{\ast }^p}(a^m+a^{{\ast }^m})a^p;(p,m=1,2,\dots )}$ acting on Bargmann space. In particular, here we find some conditions on the parameters p and m such that ${\displaystyle H^{p,m}}$ is completely indeterminate. It follows from these conditions that ${\displaystyle H^{p,m}}$ is entirely of minimal type. Then, we show that ${\displaystyle H^{p,m}}$ and ${\displaystyle H^{p,m}+H^{{\ast }^{p,m}}}$ (where $H^{{\ast }^{p,m}}$ is the adjoint of $H^{p,m}$) are connected to the chaotic operators. Full article

This paper is concerned with a study of a special integral equation. This integral equation arises in many applied problems, including transmutation theory, inverse scattering problems, the solution of singular Sturm–Liouville and Shrödinger equations, and the representation of solutions of singular Sturm–Liouville and Shrödinger equations. A special integral equation is derived and formulated using the Riemann function of a singular hyperbolic equation. In the paper, the existence of a unique solution to this equation is proven by the method of successive approximations. The results can be applied, for example, to representations of solutions to Sturm–Liouville equations with singular potentials, such as Bargmann and Miura potentials, and similiar. The treatment of problems with such potentials are very important in mathematical physics, and inverse, scattering and related problems. The estimates received do not contain any undefined constants, and for transmutation kernels all estimates are explicitly written. Full article

We review several results in the theory of weighted Bergman kernels. Weighted Bergman kernels generalize ordinary Bergman kernels of domains $\mathsf{\Omega }\subset {\mathbb{C}}^n$ but also appear locally in the attempt to quantize classical states of mechanical systems whose classical phase space is a complex manifold, and turn out to be an efficient computational tool that is useful for the calculation of transition probability amplitudes from a classical state (identified to a coherent state) to another. We review the weighted version (for weights of the form $\gamma =|\phi {|}^m$ on strictly pseudoconvex domains $\mathsf{\Omega }=\{\phi <0\}\subset {\mathbb{C}}^n$ ) of Fefferman’s asymptotic expansion of the Bergman kernel and discuss its possible extensions (to more general classes of weights) and implications, e.g., such as related to the construction and use of Fefferman’s metric (a Lorentzian metric on $\partial \mathsf{\Omega }\times S^1$ ). Several open problems are indicated throughout the survey. Full article

A relation is established in the present paper between Dicke states in a d-dimensional space and vectors in the representation space of a generalized Weyl–Heisenberg algebra of finite dimension d. This provides a natural way to deal with the separable and entangled states of a system of $N=d-1$ symmetric qubit states. Using the decomposition property of Dicke states, it is shown that the separable states coincide with the Perelomov coherent states associated with the generalized Weyl–Heisenberg algebra considered in this paper. In the so-called Majorana scheme, the qudit (d-level) states are represented by N points on the Bloch sphere; roughly speaking, it can be said that a qudit (in a d-dimensional space) is describable by a N-qubit vector (in a N-dimensional space). In such a scheme, the permanent of the matrix describing the overlap between the N qubits makes it possible to measure the entanglement between the N qubits forming the qudit. This is confirmed by a Fubini–Study metric analysis. A new parameter, proportional to the permanent and called perma-concurrence, is introduced for characterizing the entanglement of a symmetric qudit arising from N qubits. For $d=3$ ( $\iff N=2$ ), this parameter constitutes an alternative to the concurrence for two qubits. Other examples are given for $d=4$ and 5. A connection between Majorana stars and zeros of a Bargmmann function for qudits closes this article. Full article

Using torus gauge fixing, Hahn in 2008 wrote down an expression for a Chern-Simons path integral to compute the Wilson Loop observable, using the Chern-Simons action \(S_{CS}^\kappa\), \(\kappa\) is some parameter. Instead of making sense of the path integral over the space of \(\mathfrak{g}\)-valued smooth 1-forms on \(S^2 \times S^1\), we use the Segal Bargmann transform to define the path integral over \(B_i\), the space of \(\mathfrak{g}\)-valued holomorphic functions over \(\mathbb{C}^2 \times \mathbb{C}^{i-1}\). This approach was first used by us in 2011. The main tool used is Abstract Wiener measure and applying analytic continuation to the Wiener integral. Using the above approach, we will show that the Chern-Simons path integral can be written as a linear functional defined on \(C(B_1^{\times^4} \times B_2^{\times^2}, \mathbb{C})\) and this linear functional is similar to the Chern-Simons linear functional defined by us in 2011, for the Chern-Simons path integral in the case of \(\mathbb{R}^3\). We will define the Wilson Loop observable using this linear functional and explicitly compute it, and the expression is dependent on the parameter \(\kappa\). The second half of the article concentrates on taking \(\kappa\) goes to infinity for the Wilson Loop observable, to obtain link invariants. As an application, we will compute the Wilson Loop observable in the case of \(SU(N)\) and \(SO(N)\). In these cases, the Wilson Loop observable reduces to a state model. We will show that the state models satisfy a Jones type skein relation in the case of \(SU(N)\) and a Conway type skein relation in the case of \(SO(N)\). By imposing quantization condition on the charge of the link \(L\), we will show that the state models are invariant under the Reidemeister Moves and hence the Wilson Loop observables indeed define a framed link invariant. This approach follows that used in an article written by us in 2012, for the case of \(\mathbb{R}^3\). Full article