The quantum lifetime of the Davydov soliton

Considering an infinite spine in the Alpha-helix, stationary states should be eigenstates of a translational operator. These states should be nonlocalized in contradiction to a localized soliton. The difference in energy between localized and nonlocalized (Bloch) states is due to zero point motion and gives information about the quantum stability of the Davydov soliton. We develop a theory of statinary states and show that only for the limiting case of a classical lattice the product ansatz by Davydov is exact. Finally, we calculate the width of the soliton band to get information on the lifetime of the localized soliton.     

Statistics of Yang-Mills solitons

We determine the statistics and the spin of isolated Yang-Mills monopoles in eigenstates of their electric and magnetic charge. Exchange of solitons is defined using the translation operator of a companion paper; and under exchange, state vectors representingN identical solitons change sign precisely when the angular momentum of each soliton is half-integral.     

Internal modes of discrete solitons near the anti-continuum limit of the dNLS equation

Discrete solitons of the discrete nonlinear Schrödinger (dNLS) equation are compactly supported in the anti-continuum limit of the zero coupling between lattice sites. Eigenvalues of the linearization of the dNLS equation at the discrete soliton determine its spectral stability. Small eigenvalues bifurcating from the zero eigenvalue near the anti-continuum limit were characterized earlier for this model. Here we analyze the resolvent operator and prove that it is bounded in the neighborhood of the continuous spectrum if the discrete soliton is simply connected in the anti-continuum limit. This result rules out the existence of internal modes (neutrally stable eigenvalues of the discrete spectrum) near the anti-continuum limit.     

CPT-Frames for Non-Hermitian Hamiltonians

Based on the PT-symmetric quantum theory, the concepts of PT-frame, PT-symmetric operator and CPT-frame on a Hilbert space K and for an operator on K are proposed. It is proved that the spectrum and point spectrum of a PT-symmetric linear operator are both symmetric with respect to the real axis and the eigenvalues of an unbroken PT-symmetric operator are real. For a linear operator H on C^d, it is proved that H has unbroken PT- symmetry if and only if it has d different eigenvalues and the corresponding eigenstates are eigenstates of PT. Given a CPT-frame on K, a new positive inner product on K is induced and called CPT-inner product. Te relationship between the CPT-adjoint and the Dirac adjoint of a densely defined linear operator is derived, and it is proved that an operator which has a bounded CPT-frame is CPT-Hermitian if and only if it is T-symmetric, in that case, it is similar to a Hermitian operator. The existence of an operator C consisting of a CPT-frame is discussed. These concepts and results will serve a mathematical discussion about PT-symmetric quantum mechanics.     

Chiral phase transition and Anderson localization in the instanton liquid model for QCD

We study the spectrum and eigenmodes of the QCD Dirac operator in a gauge background given by an instanton liquid model (ILM) at temperatures around the chiral phase transition. Generically we find the Dirac eigenvectors become more localized as the temperature is increased. At the chiral phase transition, both the low lying eigenmodes and the spectrum of the QCD Dirac operator undergo a transition to localization similar to the one observed in a disordered conductor. This suggests that Anderson localization is the fundamental mechanism driving the chiral phase transition. We also find an additional temperature dependent mobility edge (separating delocalized from localized eigenstates) in the bulk of the spectrum which moves toward lower eigenvalues as the temperature is increased. In both regions, the origin and the bulk, the transition to localization exhibits features of a 3D Anderson transition including multifractal eigenstates and spectral properties that are well described by critical statistics. Similar results are obtained in both the quenched and the unquenched case though the critical temperature in the unquenched case is lower. Finally we argue that our findings are not in principle restricted to the ILM approximation and may also be found in lattice simulations.     

Eigenvalue problem for arbitrary linear combinations of a boson annihilation and creation operator

The eigenvalue problem for arbitrary linear combinations kα + μα^? of a boson annihilation operator α and a boson creation operator α^? is solved. It is shown that these operators possess nondegenerate eigenstates to arbitrary complex eigenvalues. The expansion of these eigenstates into the basic set of number states | n >, (n = 0, 1, 2, …), is found. The eigenstates are normalizable and are therefore states of a Hilbert space for | ζ | < 1 with ζ ? μ/k and represent in this case squeezed coherent states of minimal uncertainty product. They can be considered as states of a rigged Hilbert space for | ζ | ? 1. A completeness relation for these states is derived that generalizes the completeness relation for the coherent states | α 〉. Furthermore, it is shown that there exists a dual orthogonality in the entire set of these states and a connected dual completeness of the eigenstates on widely arbitrary paths over the complex plane of eigenvalues. This duality goes over into a selfduality of the eigenstates of the hermitian operators kα + k* α^? to real eigenvalues. The usually as nonexistent considered eigenstates of the boson creation operator α^? are obtained by a limiting procedure. They belong to the most singular case among the considered general class of eigenstates with ζ ? μ/k as a parameter.     

Time-dependent mean field S-matrix theory

By using the path integral approach to many-body systems, we formulate a time-dependent mean field S-matrix theory of nuclear reactions. Many-body channel eigenstates are constructed by using projection techniques. In this way the S-matrix between the channel eigenstates is expressed as a superposition of S-matrix elements between wave-packet-like states localized in space and time. A field operator representation of the interaction picture S-matrix is derived which enables one to apply the path integral approach. Applying the stationary phase approximation to the path integral representation of the interaction picture S-matrix between the localized states an asymptotically constant time-dependent mean field approximation to this S-matrix is obtained. Finally, the S-matrix between the projected channel eigenstates is obtained by evaluating the integral, arising from the projections, over the space-time positions of the localized states in the stationary phase approximation. The stationary phase conditions select those localized states from the projected channel states for which the mean field values of energy and momentum coincide with their corresponding channel eigenvalues.     

Discreteness of area and volume in quantum gravity

We study the operator that corresponds to the measurement of volume, in non-perturbative quantum gravity, and we compute its spectrum. The operator is constructed in the loop representation, via a regularization procedure; it is finite, background independent, and diffeomorphism-invariant, and therefore well defined on the space of diffeomorphism invariant states (knot states). We find that the spectrum of the volume of any physical region is discrete. A family of eigenstates are in one to one correspondence with the spin networks, which were introduced by Penrose in a different context. We compute the corresponding component of the spectrum, and exhibit the eigenvalues explicitly. The other eigenstates are related to a generalization of the spin networks, and their eigenvalues can be computed by diagonalizing finite dimensional matrices. Furthermore, we show that the eigenstates of the volume diagonalize also the area operator. We argue that the spectra of volume and area determined here can be considered as predictions of the loop-representation formulation of quantum gravity on the outcomes of (hypothetical) Planck-scale sensitive measurements of the geometry of space.     

SUPERSYMMETRIC SU(1|2) GAUDIN MODEL

In this paper, we propose a supersymmetric SU(1|2) Gaudin model and have derived its eigenvalues. We also present the well-defined eigenstates through the quasi-classical limit of the eigenstates in the supersymmetric t-J model.     

Exact solutions for nonlinear Hamiltonians

We find the eigenvalues and eigenvectors of two nonlinear Hamiltonians describing the interaction between a two-level system and a quantized linear harmonic oscillator. In the first case we obtain exact isolated solutions for the Hamiltonian used as a model of an ion in a harmonic trap and interacting with a laser field, not restricted to the Lamb-Dicke limit. After projecting these eigenstates onto one of the levels of the two-level system the oscillator state is described by a finite superposition of Fock states. In the second case we consider a Hamiltonian, with a squeeze operator in the interaction part. We give perturbation results in the weak-coupling limit and results obtained by numerical diagonalization for the strong coupling limit. Non-classical results are pointed out also in this case.     

The length operator in Loop Quantum Gravity

The dual picture of quantum geometry provided by a spin network state is discussed. From this perspective, we introduce a new operator in Loop Quantum Gravity—the length operator. We describe its quantum geometrical meaning and derive some of its properties. In particular we show that the operator has a discrete spectrum and is diagonalized by appropriate superpositions of spin network states. A series of eigenstates and eigenvalues is presented and an explicit check of its semiclassical properties is discussed.     

Quantum Phase and Phase Difference for Fermion

In this paper, from the index theorem, we present the Hermitian phase operator for fermion, give its eigenvalues and orthogonal, normalized and complete eigenstates. Furthermore, we also get the phase difference operator for two fermions and discuss its quantum properties.     

Phonon spectra in one-dimensional quasicrystals

The propagation of phonons in one-dimensional quasicrystals is investigated. We use the projection method which has been recently proposed to generate almost periodic tilings of the line. We define a natural Laplace operator on these structures, which models phonon (and also tight-binding electron) propagation. The selfsimilarity properties of the spectrum are discussed, as well as some characteristic features of the eigenstates, which are neither extended nor localized. The long-wavelength limit is examined in more detail; it is argued that one is the lower critical dimension for this type of models.     

CPT-Frames for Non-Hermitian Hamiltonians

Based on the P T-symmetric quantum theory,the concepts of P T-frame,P T-symmetric operator and CPT-frame on a Hilbert space K and for an operator on K are proposed.It is proved that the spectrum and point spectrum of a P T-symmetric linear operator are both symmetric with respect to the real axis and the eigenvalues of an unbroken P T-symmetric operator are real.For a linear operator H on Cd,it is proved that H has unbroken P Tsymmetry if and only if it has d diferent eigenvalues and the corresponding eigenstates are eigenstates of P T.Given a C P T-frame on K,a new positive inner product on K is induced and called C P T-inner product.Te relationship between the CP T-adjoint and the Dirac adjoint of a densely defined linear operator is derived,and it is proved that an operator which has a bounded CP T-frame is CP T-Hermitian if and only if it is T-symmetric,in that case,it is similar to a Hermitian operator.The existence of an operator C consisting of a CP T-frame is discussed.These concepts and results will serve a mathematical discussion about P T-symmetric quantum mechanics.     

电磁场与等离子体互作用和广义压缩相干态

本文用量子力学方法研究在一腔中的等离子体与电磁场的相互作用,求得系统的哈密顿量的本征值和本征态,这本征态是一种广义压缩相干态。还求得在广义压缩相干态,场的压缩、反聚束、亚泊松统计等效应以及测量相算符的有关公式,这些效应均为时间守恒的。 关键词：     

Quantization of the Single-qubit Structure with SQUID

The quantization scheme of a single-qubit structure with Superconducting Quantum Interference Device (SQUID) is given. By introducing a unitary matrix and by means of spectral decomposition, the Hamiltonian operator of the system is exactly formulated in compact forms in spin-1/2 notation. The eigenvalues and eigenstates of the system are discussed.     

无限深阱势的非线性谱生成代数与新型相干态

利用对称一维无限深阱势的哈密顿算符和自然算符构造出该势场的非线性谱生成代数,并在此基础上得到了一种新的非线性相干态.该相干态具有时间稳定性,既可以看成本征值为算符函数的降算符本征态,也可以看成广义极小测不准状态的转动态.     

Nonlinear Spectrum and Generating Algebra for Infinitely Deep Square Well Potential and New Coherent State

Using the Hamiltonian of symmetrical one dimersional infinitely deep square well potential and natural operators,we obtain its nonlinear spectrum and generating algebra,and get a class of new nonlinear coherent states on the basis of the nonlinear algebra obfained. These coherent states are of temporal stability,and can be regarded as the eigenstates of the lower operator with the eigenvalues in an operator field and as the rotational states of the generalized minimal uncertainly states as well.     

The Characteristic Equation of q-Deformed Bosonic Oscillator

The main goal of this paper is to deduce the characteristic equation of [N]. From the fundamental relation of q-deformed bosonic oscillator, we deduce a strict algebraic equation. After introducing a moderate boundary condition, we obtain the characteristic equation of number operator [N].Using this equation, we extract all the eigenvalues of operator [N] and write out the corresponding eigenstates. Then we obtain the completeness of the Hilbert space.     

Time reversal for a single spherical scatterer.

We show that the time reversal operator for a planar time reversal mirror (TRM) can have up to four distinct eigenvalues with a small spherical acoustic scatterer. Each eigenstate represents a resonance between the TRM and an induced scattering moment of the sphere. Their amplitude distributions on the TRM are orthogonal superpositions of the radiation patterns from a monopole and up to three orthogonal dipoles. The induced monopole moment is associated with the compressibility contrast between the sphere and the medium, while the dipole moments are associated with density contrast. The number of eigenstates is related to the number of orthogonal orientations of each induced multipole. For hard spheres (glass, metals) the contribution of the monopole moment to the eigenvalues is much greater than that of the dipole moments, leading to a single dominant eigenvalue. The other eigenvalues are much smaller, making it unlikely multiple eigenvalues could have been observed in previous experiments using hard materials. However, for soft materials such as wood, plastic, or air bubbles the eigenvalues are comparable in magnitude and should be observable. The presence of multiple eigenstates breaks the one-to-one correspondence between eigenstates and distinguishable scatterers discussed previously by Prada and Fink [Wave Motion 20, 151-163 (1994)]. However, eigenfunctions from separate scatterers would have different phases for their eigenfunctions, potentially restoring the ability to distinguish separate scatterers. Since relative magnitudes of the eigenvalues for a single scatterer are governed by the ratio of the compressibility contrast to the density contrast, measurement of the eigenvalue spectrum would provide information on the composition of the scatterer.