Model-Independent Normalization Condition for Gamow Vectors

Two comments on the theory of Gamow states are presented. First, it is shown that Gamow states may be viewed as two resonating states which correspond to two different eigenproblems for the same real value of energy. Second, a universal normalization condition for all Gamow states is derived.     

Gamow Vectors for Resonances: A Lax-Phillips Point of View

Results from the Lax-Phillips Scattering Theory are used to analyze quantum mechanical scattering systems, in particular to obtain spectral properties of their resonances which are defined to be the poles of the scattering matrix. For this approach the interplay between the positive energy projection and the Hardy-space projections is decisive. Among other things it turns out that the spectral properties of these poles can be described by the (discrete) eigenvalue spectrum of a so-called truncated evolution, whose eigenvectors can be considered as the Gamow vectors corresponding to these poles. Further an expansion theorem of the positive Hardy-space part of vectors Sg (S scattering operator) into a series of Gamow vectors is presented.     

Relativistic Gamow Vectors

Whereas in Dirac quantum mechanics and relativistic quantum field theory one uses Schwartz space distributions, the extensions of the Hilbert space that we propose uses Hardy spaces. The in- and out-Lippmann-Schwinger kets of scattering theory are functionals in two rigged Hilbert space extensions of the same Hilbert space. This hypothesis also allows to introduce generalized vectors corresponding to unstable states, the Gamow kets. Here the relativistic formulation of the theory of unstable states is presented. It is shown that the relativistic Gamow vectors of the unstable states, defined by a resonance pole of the S-matrix, are classified according to the irreducible representations of the semigroup of the Poincaré transformations (into the forward light cone). As an application the problem of the mass definition of the intermediate vector boson Z is discussed and it is argued that only one mass definition leads to the exponential decay law, and that is not the standard definition of the on-the-mass-shell renormalization scheme.     

Derivation of Gamow Vectors for Resonances in Cut-Off Potentials

Gamow vectors are rigourously derived for a realistic case including an infinite number of resonant poles. This is the case of resonances produced by cut-off potentials. These are three-dimensional spherically symmetric potentials which vanish outside a bounded region. We solely consider the case of particles without internal structure and work with l = 0.     

Mathematical Models for Unstable Quantum Systems and Gamow States

We review some results in the theory of non-relativistic quantum unstable systems. We account for the most important definitions of quantum resonances that we identify with unstable quantum systems. Then, we recall the properties and construction of Gamow states as vectors in some extensions of Hilbert spaces, called Rigged Hilbert Spaces. Gamow states account for the purely exponential decaying part of a resonance; the experimental exponential decay for long periods of time physically characterizes a resonance. We briefly discuss one of the most usual models for resonances: the Friedrichs model. Using an algebraic formalism for states and observables, we show that Gamow states cannot be pure states or mixtures from a standard view point. We discuss some additional properties of Gamow states, such as the possibility of obtaining mean values of certain observables on Gamow states. A modification of the time evolution law for the linear space spanned by Gamow shows that some non-commuting observables on this space become commuting for large values of time. We apply Gamow states for a possible explanation of the Loschmidt echo.     

The Physical Mechanism of Formation of Quantum Mechanical Gamow States

No Heading A new construction of quantum mechanical Gamow states is presented. The physical nature of these states is revealed without introducing the notion of complex energy. In the presented aproach the time reversal is a linear transformation instead of the customary antilinear one. Also the Fourier integral solution of the free nonrelativistic wave equation contains both positive and negative frequencies.     

Exactly Solvable Model of Quantum Diffusion

We study the transport property of diffusion in a finite translationally invariant quantum subsystem described by a tight-binding Hamiltonian with a single energy band. The subsystem interacts with its environment by a coupling expressed in terms of correlation functions which are delta-correlated in space and time. For weak coupling, the time evolution of the subsystem density matrix is ruled by a quantum master equation of Lindblad type. Thanks to the invariance under spatial translations, we can apply the Bloch theorem to the subsystem density matrix and exactly diagonalize the time evolution superoperator to obtain the complete spectrum of its eigenvalues, which fully describe the relaxation to equilibrium. Above a critical coupling which is inversely proportional to the size of the subsystem, the spectrum at given wave number contains an isolated eigenvalue describing diffusion. The other eigenvalues rule the decay of the populations and quantum coherences with decay rates which are proportional to the intensity of the environmental noise. An analytical expression is obtained for the dispersion relation of diffusion. The diffusion coefficient is proportional to the square of the width of the energy band and inversely proportional to the intensity of the environmental noise because diffusion results from the perturbation of quantum tunneling by the environmental fluctuations in this model. Diffusion disappears below the critical coupling     

Exactly Solvable Models and Spontaneous Symmetry Breaking

We study a few two-dimensional models with massless and massive fermions in the hamiltonian framework and in both conventional and light-front (LF) forms of field theory. The new ingredient is a modification of the canonical procedure by taking into account solutions of the operator field equations. After summarizing the main results for the derivative-coupling and the Thirring models, we briefly compare conventional and LF versions of the Federbush model including the massive current bosonization and a Bogoliubov transformation to diagonalize the Hamiltonian. Then we sketch an extension of our hamiltonian approach to the two-dimensional Nambu–Jona-Lasinio model and the Thirring-Wess model. Finally, we discuss the Schwinger model in a covariant gauge. In particular, we point out that the solution due to Lowenstein and Swieca implies the physical vacuum in terms of a coherent state of massive scalar field and suggest a new formulation of the model’s vacuum degeneracy.     

Saturation of Electrostatic Potential: Exactly Solvable 2D Coulomb Models

We test the concepts of renormalized charge and potential saturation, introduced within the framework of highly asymmetric Coulomb mixtures, on exactly solvable Coulomb models. The object of study is the average electrostatic potential induced by a unique “guest” charge immersed in a classical electrolyte, the whole system being in thermal equilibrium at some inverse temperature β. The guest charge is considered to be either an infinite hard wall carrying a uniform surface charge or a charged colloidal particle. The systems are treated as two-dimensional; the electrolyte is modelled by a symmetric two-component plasma (TCP) of point-like ±e charges with logarithmic Coulomb interactions. Two cases are solved exactly: the Debye–Hückel limit β e²→ 0 and the Thirring free-fermion point β e²=2. The results at the free-fermion point can be summarized as follows: (i) The induced electrostatic potential exhibits the asymptotic behavior, at large distances from the guest charge, whose form is different from that obtained in the Debye–Hückel (linear Poisson–Boltzmann) theory. This means that the concept of renormalized charge, developed within the nonlinear Poisson–Boltzmann (PB) theory to describe the screening effect of the electrolyte cloud, fails at the free-fermion point. (ii) In the limit of an infinite bare charge, the induced electrostatic potential saturates at a finite value in every point of the electrolyte region. This fact confirms the previously proposed hypothesis of potential saturation.     

Resonance expansions in quantum mechanics

The goal of this contribution is to discuss various resonance expansions that have been proposed in the literature. Presented at the 3rd International Workshop “Pseudo-Hermitian Hamiltonians in Quantum Physics”, Istanbul, Turkey, June 20–22, 2005.     

Entropy Production in Exactly Solvable Systems

The rate of entropy production by a stochastic process quantifies how far it is from thermodynamic equilibrium. Equivalently, entropy production captures the degree to which global detailed balance and time-reversal symmetry are broken. Despite abundant references to entropy production in the literature and its many applications in the study of non-equilibrium stochastic particle systems, a comprehensive list of typical examples illustrating the fundamentals of entropy production is lacking. Here, we present a brief, self-contained review of entropy production and calculate it from first principles in a catalogue of exactly solvable setups, encompassing both discrete- and continuous-state Markov processes, as well as single- and multiple-particle systems. The examples covered in this work provide a stepping stone for further studies on entropy production of more complex systems, such as many-particle active matter, as well as a benchmark for the development of alternative mathematical formalisms.     

Quantum Tunneling in Time-Dependent Potential Barrier: a General Formulation and Some Exactly Solvable Models

Quantum tunneling in one spatial dimension in the presence of time-dependent potentials is investigated theoretically. First, a general multichannel formulation of the problem is given for harmonic time-dependence. The case of an oscillatory delta-function potential of constant strength is discussed at length and specific numerical calculations are presented for the reflection and transmission coefficients. Then other exactly solvable time-dependent potentials are obtained by carrying out successive unitary transformations. As an example of this class, a delta-function potential with time-dependent strength and boundary conditions is studied. Finally tunneling time-delay for time-dependent potentials is formulated in the multichannel situation, and also the concept of first passage time for the decay of a wave packet confined to one well of a double well potential is generalized for the case of time-dependent barrier and boundary conditions.     

Gamow Vectors in a Periodically Perturbed Quantum System

Gamow vectors and resonances play an important role in scattering theory, especially in the physics of metastable states. We study Gamow vectors and resonances in a time-dependent setting using the Borel summation method. In particular, we analyze the behavior of the wave function ψ(x,t) for one dimensional time-dependent Hamiltonian \(H=-\partial_{x}^{2}\pm2\delta(x)(1+2r\cos\omega t)\) where ψ(x,0) is compactly supported.We show that ψ(x,t) has a Borel summable expansion containing finitely many terms of the form \(\sum_{n=-\infty}^{\infty}e^{i^{3/2}\sqrt{-\lambda_{k}+n\omega i}|x|}A_{k,n}e^{-\lambda_{k}t+n\omega it}\), where λ _k represents the associated resonance. This expression defines Gamow vectors and resonances in a rigorous and physically relevant way for all frequencies and amplitudes in a time-dependent model.For small amplitude (|r|?1) there is one resonance for generic initial conditions. We calculate the position of the resonance and discuss its physical meaning as related to multiphoton ionization. We give qualitative theoretical results as well as numerical calculations in the general case.     

Information-Theoretic Features of Many Fermion Systems: An Exploration Based on Exactly Solvable Models

Finite quantum many fermion systems are essential for our current understanding of Nature. They are at the core of molecular, atomic, and nuclear physics. In recent years, the application of information and complexity measures to the study of diverse types of many-fermion systems has opened a line of research that elucidates new aspects of the structure and behavior of this class of physical systems. In this work we explore the main features of information and information-based complexity indicators in exactly soluble many-fermion models of the Lipkin kind. Models of this kind have been extremely useful in shedding light on the intricacies of quantum many body physics. Models of the Lipkin kind play, for finite systems, a role similar to the one played by the celebrated Hubbard model of solid state physics. We consider two many fermion systems and show how their differences can be best appreciated by recourse to information theoretic tools. We appeal to information measures as tools to compare the structural details of different fermion systems. We will discover that few fermion systems are endowed by a much larger complexity-degree than many fermion ones. The same happens with the coupling-constants strengths. Complexity augments as they decrease, without reaching zero. Also, the behavior of the two lowest lying energy states are crucial in evaluating the system’s complexity.     

Remarks on Exactly Solvable Noncommutative Quantum Field

We study exactly the solvable noncommutative scalar quantum field models of （2n） or （2n ＋ 1） dimensions. By writing out an equivalent action of the noncommutative field, it is shown that the special condition B. 0 = 4-1 in field theoretic context means the full restoration of the maximal U（∞） gauge symmetries broken due to kinetic term. It is further shown that the model can be obtained by dimensional reduction of a 2n-dimensional exactly solvable noncommutative φ4 quantum field model closely related to the 1＋1-dimensional Moyal/matrix-valued nonlinear Schr6dinger （MNLS） equation. The corresponding quantum fundamental commutation relation of the MNLS model is also given explicitly.     

三维精确可解格点模型

对Baxter-Bazhanov三维精确可解格点模型玻尔兹曼权Sk间的变换关系作了仔细讨论,并给出了局域可积性条件──三维星-星关系的一个完整证明.     

空心圆柱微腔谐振模式研究

基于分离变量法研究了平面波斜入射时,空心无限长介质圆柱微腔的光学特性,通过计算散射场展开系数分析了入射角度、空心部分的尺寸等因素对谐振廓线的影响。研究结果表明,当空心部分的尺寸参数较大时将使微腔的形貌共振(Morphology Dependent Resonances,MDR)峰值产生偏移,当空心部分尺寸参数较小时,圆柱仍然可以产生MDR共振,且位置和均匀圆柱重合。当平面波斜入射时,MDR峰也会产生偏移,并且随着入射角度的减小,MDR峰值减小,直至完全消失。