Gamow Vectors and Time Asymmetry

We prove that Gamow vectors are important toolsin the quantum theory of irreversibility. We use themathematical formalism of rigged Hilbert spaces. Wediscuss some spectral formulas that include Gamow vectors as well as some results concerningGamow vectors. The role of the time-reversal operator isstudied. The formalism can be applied to formulate asense of irreversibility in cosmology.     

Gamow Temperature in Tsallis and Kaniadakis Statistics

Relying on the quantum tunnelling concept and Maxwell–Boltzmann–Gibbs statistics, Gamow shows that the star-burning process happens at temperatures comparable to a critical value, called the Gamow temperature (T) and less than the prediction of the classical framework. In order to highlight the role of the equipartition theorem in the Gamow argument, a thermal length scale is defined, and then the effects of non-extensivity on the Gamow temperature have been investigated by focusing on the Tsallis and Kaniadakis statistics. The results attest that while the Gamow temperature decreases in the framework of Kaniadakis statistics, it can be bigger or smaller than T when Tsallis statistics are employed.     

Resonances and Gamow States in Non-Local Potentials

For a large class of non-local, non separable potentials with non-compact support, the solution of the radial integrodifferential equation may be reduced to the solution of a homogeneous linear integral equation of Fredholm type with a quadratically integrable kernel. In this way we derive expansions of the wave functions and the Green's function of the Schrödinger equation with a non-local potential in terms of bound states, resonant states and a continuum of scattering functions with complex wave number. The rules of normalization, orthogonality and completeness satisfied by the eigenstates of the Schrödinger equation belonging to complex eigenvalues with Im E_n < 0, (Gamow or resonant states) are also derived. Finally, by means of a realistic example, it is shown how to use these expansions to exhibit the resonant behaviour of the differential cross section. Explicit expressions for the transition amplitudes and the partial widths in terms of expectation values of operators computed with Gamow functions are given.     

The Gamow vectors and the Schwinger effect

We introduce a ‘proper time’ formalism to study the instability of the vacuum in a uniform external electric field due to particle production. This formalism allows us to reduce a quantum field-theoretic problem to a quantum mechanical one in a higher dimension. The instability results from the inverted oscillator structure which appears in the Hamiltonian. We show that the ‘proper time’ unitary evolution splits into two semigroups. The semigroup associated with decaying Gamov vectors is related to the Feynman boundary conditions for the Green functions and the semigroup associated with growing Gamov vectors is related to the Dyson boundary conditions.     

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.     

Jordan blocks and exponentially decaying higher-order Gamow states

In the framework of the rigged Hilbert space, unstable quantum systems associated with first-order poles of the analytically continued S-matrix can be described by Gamow vectors which are generalized vectors with exponential decay and a Breit-Wigner energy distribution. This mathematical formalism can be generalized to quasistationary systems associated with higher-order poles of the S-matrix, which leads to a set of Gamow vectors of higher order with a non-exponential time evolution. One can define a state operator from the set of higher-order Gamow vectors which obeys the exponential decay law. We shall discuss to what extent the requirement of an exponential time evolution determines the form of the state operator for a quasistationary microphysical system associated with a higher-order pole of the S-matrix. Dedicated to Professor L. P. Horwitz on the occasion of his 65th birthday, October 14, 1995.     

Effective Field Theory and the Gamow Shell Model

We combine halo/cluster effective field theory (H/CEFT) and the Gamow shell model (GSM) to describe the 0⁺ ground state of ⁶He as a three-body halo system. We use two-body interactions for the neutron-alpha particle and two-neutron pairs obtained from H/CEFT at leading order, with parameters determined from scattering in the p_3/2 and s₀ channels, respectively. The three-body dynamics of the system is solved using the GSM formalism, where the continuum states are incorporated in the shell model valence space. We find that in the absence of three-body forces the system collapses, since the binding energy of the ground state diverges as cutoffs are increased. We show that addition at leading order of a three-body force with a single parameter is sufficient for proper renormalization and to fix the binding energy to its experimental value.     

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.     

Gamow Vectors and Borel Summability in a Class of Quantum Systems

We analyze the detailed time dependence of the wave function ψ(x,t) for one dimensional Hamiltonians \(H=-\partial_{x}^{2}+V(x)\) where V (for example modeling barriers or wells) and ψ(x,0) are compactly supported.We show that the dispersive part of ψ(x,t) is the Borel sum of its asymptotic series in powers of t ^?1/2, t→∞. The remainder, the difference between ψ and the Borel sum, i.e., the exponential part of the transseries of ψ, is a convergent expansion of the form \(\sum_{k=0}^{\infty}g_{k}\Gamma_{k}(x)e^{-\gamma_{k} t}\), where Γ _k are the Gamow vectors of H, and iγ _k are the associated resonances; generically, all g _k are nonzero. For large k, γ _k ～const?klog?k+k ² π ² i/4. The effect of the Gamow vectors is visible when time is not very large, and the decomposition defines rigorously resonances and Gamow vectors in a nonperturbative regime, in a physically relevant way.The decomposition allows for calculating ψ for moderate and large t, to any prescribed exponential accuracy, using optimal truncation of power series plus finitely many Gamow vectors contributions.The analytic structure of ψ is perhaps surprising: in general (even in simple examples such as square wells), ψ(x,t) turns out to be C ^∞ in t but nowhere analytic on ?⁺. In fact, ψ is t-analytic in a sector in the lower half plane and has the whole of ?⁺ a natural boundary. In the dual space, we analyze the resurgent structure of ψ.     

Gamow states in a realistic two-center potential and nucleon emission in heavy-ion collisions

For a realistic finite-depth two-center potential with fixed spherical Woods-Saxon wells adiabatic bound and Gamow states have been calculated by representing each potential in a harmonic oscillator basis in momentum representation. Numerical results for the real and complex energy eigenvalues of the single-particle states as a function of the distance between the centers of the wells are presented for a neutron in a ¹⁶O + ¹⁶O potential. Due to the neglect of volume conservation and of any shape degrees of freedom the considerations are restricted to separation regions with a small overlap of the density distributions only. The matrix elements for the coupling of adiabatic states due to non-adiabatic effects in the collective relative motion of the potentials during a heavy-ion collision have been calculated for bound-bound as well as bound-continuum transitions. Neglecting the rotational coupling, from the population of Gamow states a differential neutron emission spectrum has been computed, which for a peripheral ¹⁷O + ¹⁶O reaction shows distinct peaks at the position of low-lying asymptotic adiabatic states and a decreasing high-energy tail connected with emission from quasistationary states, pushed up in energy when the collision partners are coming in contact. An appreciable fraction of the particle emission appears as a sequential decay of the excited fragments after separation.     

The Gamow shell model with realistic interactions: a theoretical framework for ab initio nuclear structure at drip-lines

Ab initio approaches are among the most advanced models to solve the nuclear many-body problem. In particular, the no-core–shell model and many-body perturbation theory have been recently extended to the Gamow shell model framework, where the harmonic oscillator basis is replaced by a basis bearing bound, resonance and scattering states, i.e. the Berggren basis. As continuum coupling is included at basis level and as configuration mixing takes care of inter-nucleon correlations, halo and resonance nuclei can be properly described with the Gamow shell model. The development of the no-core Gamow shell model and the introduction of the $\hat{\bar{Q}}$-box method in the Gamow shell model, as well as their first ab initio applications, will be reviewed in this paper. Peculiarities compared to models using harmonic oscillator bases will be shortly described. The current power and limitations of ab initio Gamow shell model will also be discussed, as well as its potential for future applications.     

Scientific Review: Spectroscopy at IPNS: Recent Instrumental Upgrade and Scientific Highlights

IPNS has four neutron spectrometers: two chopper spectrometers, HRMECS and LRMECS, and two crystal-analyzer spectrometers, QENS and CHEX. At the outset, HRMECS, LRMECS, and QENS (preceded by CAS in the 1980s) were fully users-dedicated instruments. CHEX was an add-on and has been a prototype for pilot experiments and for further development. Over the years, steady incremental improvements have been made on these spectrometers. Recently, QENS and HRMECS have undergone substantial upgrade both in hardware and software. Here, we describe the design and operation of QENS and HRMECS and illustrate their scientific capabilities by selected examples. Other details concerning IPNS' spectrometers are given elsewhere [1].     

乔治·格林及格林函数法

乔治.格林是一位重要的数学物理学家.简要介绍了乔治.格林的生平和科学研究经历,揭示了帮助他取得重要科学研究成果的主要因素,介绍了格林函数法的建立过程及其重要性.