Topological Invariants of Eigenvalue Intersections and Decrease of Wannier Functions in Graphene

We investigate the asymptotic decrease of the Wannier functions for the valence and conduction band of graphene, both in the monolayer and the multilayer case. Since the decrease of the Wannier functions is characterised by the structure of the Bloch eigenspaces around the Dirac points, we introduce a geometric invariant of the family of eigenspaces, baptised eigenspace vorticity. We compare it with the pseudospin winding number. For every value $n \in \mathbb {Z}$ of the eigenspace vorticity, we exhibit a canonical model for the local topology of the eigenspaces. With the help of these canonical models, we show that the single band Wannier function $w$ satisfies $|w(x)| \le {\mathrm {const}} \cdot |x|^{-2}$ as $|x| \rightarrow \infty $ , both in monolayer and bilayer graphene.     

Pentaquark decay is suppressed by chirality conservation

It is shown that, if the pentaquark \(\Theta ^ + = uudd\bar s\) baryon can be represented by the local quark current η_Θ, its decay Θ⁺→nK⁺(pK⁰) is forbidden in the limit of chirality conservation. The Θ⁺ decay width Γ is proportional to \(\alpha _s^2 \left\langle {0\left| {\bar qq} \right|0} \right\rangle ^2\), where \(\left\langle {0\left| {\bar qq} \right|0} \right\rangle ,q = u,d,s\), is quark condensate, and, therefore, is strongly suppressed. The polarization operator of the pentaquark current is calculated using the operator product expansion. The Θ⁺ mass found by the QCD sum rules method is in reasonable agreement with experiment.     

Domain and Range of the Modified Wave Operator for Schrödinger Equations with a Critical Nonlinearity

We study the final problem for the nonlinear Schrödinger equation

$i{\partial }_{t}u+\frac{1}{2}\Delta u=\lambda|u|^{\frac{2}{n}}u,\quad (t,x)\in {\mathbf{R}}\times \mathbf{R}^{n},$

where\(\lambda \in{\bf R},n=1,2,3\). If the final data\(u_{+}\in {\bf H}^{0,\alpha }=\left\{ \phi \in {\bf L}^{2}:\left( 1+\left\vert x\right\vert \right) ^{\alpha }\phi \in {\bf L}^{2}\right\} \) with\(\frac{ n}{2} < \alpha < \min \left( n,2,1+\frac{2}{n}\right) \) and the norm\(\Vert \widehat{u_{+}}\Vert _{{\bf L}^{\infty }}\) is sufficiently small, then we prove the existence of the wave operator in L ². We also construct the modified scattering operator from H ^0,α to H ^0,δ with\(\frac{n}{2} < \delta < \alpha\).

     

Analyticity properties of the weakly coupled double-well model

In this note we study lattice Φ⁴-models with Hamiltonian $$H = \tfrac{1}{2}(\varphi , - \Delta \varphi ) + \lambda \Sigma \left( {\varphi _i^2 - \frac{{m^2 }}{{8\lambda }}} \right)^2$$ and Gaussian boundary conditions. Using the polymer expansion we obtain analyticity of the pressure and the correlation functions in the infinite volume limit in a region $$\left\{ {\left. \lambda \right| \left| \lambda \right|< \varepsilon ,\left| {arg } \right.\left. \lambda \right|< \frac{\pi }{2} - \delta } \right\}$$ for every δ>0.     

Optimal Eigenvalue Estimate for the Dirac–Witten Operator on Bounded Domains with Smooth Boundary

Eigenvalue estimate for the Dirac–Witten operator is given on bounded domains (with smooth boundary) of spacelike spin hypersurfaces satisfying the dominant energy condition, under four natural boundary conditions (MIT, APS, modified APS and chiral conditions). Roughly speaking, any eigenvalue of the Dirac–Witten operator satisfies

Entangled Projector and a New Approach for Obtaining the Entangled Wigner Operator

Like the coordinate projector |q〉〈q|=δ(q?Q), where Q is coordinate operator, we find that $\pi\delta( \eta_{1}-\frac{Q_{1}-Q_{2}}{\sqrt{2}}) \delta( \eta_{2}-\frac{P_{1}+P_{2}}{\sqrt{2}}) $ is an entangled projector |η〉〈η|, where |η〉 is the bipartite entangled state and η=η ₁+iη ₂. We then derive the entangled Wigner operator in terms of the properties of the entangled projector. This seems a new approach for obtaining the entangled Wigner operator.     

Dirac multimode ket-bra operators’ \mathfrak{Q}-ordered and \mathfrak{P}-ordered integration theory and general squeezing operator

We develop quantum mechanical Dirac ket-bra operator’s integration theory in $\mathfrak{Q}$ -ordering or $\mathfrak{P}$ -ordering to multimode case, where $\mathfrak{Q}$ -ordering means all Qs are to the left of all Ps and $\mathfrak{P}$ -ordering means all Ps are to the left of all Qs. As their applications, we derive $\mathfrak{Q}$ -ordered and $\mathfrak{P}$ -ordered expansion formulas of multimode exponential operator $e^{ - iP_l \Lambda _{lk} Q_k } $ . Application of the new formula in finding new general squeezing operators is demonstrated. The general exponential operator for coordinate representation transformation $\left| {\left. {\left( {_{q_2 }^{q_1 } } \right)} \right\rangle \to } \right|\left. {\left( {_{CD}^{AB} } \right)\left( {_{q_2 }^{q_1 } } \right)} \right\rangle $ is also derived. In this way, much more correpondence relations between classical coordinate transformations and their quantum mechanical images can be revealed.     

Fractal wavelet dimensions and localization

In this paper we want to give a new definition of fractal dimensions as small scale behavior of theq-energy of wavelet transforms. This is a generalization of previous multi-fractal approaches. With this particular definition we will show that the 2-dimension (=correlation dimension) of the spectral measure determines the long time behavior of the time evolution generated by a bounded self-adjoint operator acting in some Hilbert space ?. It will be proved that for φ, ψ∈? we have $$\mathop {\lim \inf }\limits_{T \to \infty } \frac{{\log \int_0^T {d\omega \left| {\left\langle {\psi \left| {e^{ - iA\omega } } \right.\phi } \right\rangle } \right|^2 } }}{{\log T}} = - \kappa ^ + (2)$$ and that $$\mathop {\lim \sup }\limits_{T \to \infty } \frac{{\log \int_0^T {d\omega \left| {\left\langle {\psi \left| {e^{ - iA\omega } } \right.\phi } \right\rangle } \right|^2 } }}{{\log T}} = - \kappa ^ - (2),$$ wherek ^±(2) are the upper and lower correlation dimensions of the spectral measure associated with ψ and ?. A quantitative version of the RAGE theorem shall also be given.     

Evolution Law of the Optical Field of Degenerate Parametric Amplifier in Dissipative Channel

We explore the time-evolution law of the optical field of degenerate parametric amplifier (DPA) in dissipative channel. It turns out that its density operator at initial time ρ ₀ = A exp(E ^? a ^?2) exp(a ^? alnλ) exp(E a ²) evolves into \(\rho (t)= \frac {A}{\lambda ^{\prime }}\) \(\exp \left (\frac {E^{\ast }e^{-2\kappa t}a^{\dag 2}}{ \lambda ^{\prime 2}}\right )\exp \left \{a^{\dag }a\ln \frac {[\lambda -(\lambda ^{2}-4|E|^{2})T]e^{-2\kappa t}}{\lambda ^{\prime 2}}\right \} \exp \left (\frac { Ee^{-2\kappa t}a^{2}}{\lambda ^{\prime 2}}\right ),\) where κ is the damping constant of the channel, T = 1 ? e ^?2κt , and \(\lambda ^{\prime }\equiv \sqrt {(1-\lambda T)^{2}-4|E|^{2}T^{2}}.\) We employ the method of integration (or summation) within an ordered (normally ordered or antinormally ordered) of operators to overcome the obstacles in the process of calculation.     

Time Delay for the Dirac Equation

We consider time delay for the Dirac equation. A new method to calculate the asymptotics of the expectation values of the operator \({\int\limits_{0} ^{\infty}{\rm e}^{iH_{0}t}\zeta(\frac{\vert x\vert }{R}) {\rm e}^{-iH_{0}t}{\rm d}t}\), as \({R \rightarrow \infty}\), is presented. Here, H₀ is the free Dirac operator and \({\zeta\left(t\right)}\) is such that \({\zeta\left(t\right) = 1}\) for \({0 \leq t \leq 1}\) and \({\zeta\left(t\right) = 0}\) for \({t > 1}\). This approach allows us to obtain the time delay operator \({\delta \mathcal{T}\left(f\right)}\) for initial states f in \({\mathcal{H} _{2}^{3/2+\varepsilon}(\mathbb{R}^{3};\mathbb{C}^{4})}\), \({\varepsilon > 0}\), the Sobolev space of order \({3/2+\varepsilon}\) and weight 2. The relation between the time delay operator \({\delta\mathcal{T}\left(f\right)}\) and the Eisenbud–Wigner time delay operator is given. In addition, the relation between the averaged time delay and the spectral shift function is presented.     

Krein formula with compensated singularities for the ND-mapping and the generalized Kirchhoff condition at the Neumann Schrödinger junction

The Neumann Schrödinger operator \(\mathcal{L}\) is considered on a thin 2D star-shaped junction, composed of a vertex domain Ω_int and a few semi-infinite straight leads ω _m , m = 1, 2, ..., M, of width δ, δ ? diam Ω_int, attached to Ω_int at Γ ? ?Ω_int. The potential of the Schrödinger operator l ^ω on the leads vanishes, hence there are only a finite number of eigenvalues of the Neumann Schrödinger operator L _int on Ω_int embedded into the open spectral branches of l ^ω with oscillating solutions χ _±(x, p) = \(e^{ \pm iK_ + x} e_m \) of l ^ω χ _± = p ² χ _±. The exponent of the open channels in the wires is

$K_ + (\lambda ) = p\sum\limits_{m = 1}^M {e^m } \rangle \langle e^m = \sqrt \lambda P_ + $

, with constant e _m , on a relatively small essential spectral interval Δ ? [0, π ² δ ^?2). The scattering matrix of the junction is represented on Δ in terms of the ND mapping

$\mathcal{N} = \frac{{\partial P_ + \Psi }}{{\partial x}}(0,\lambda )\left| {_\Gamma \to P_ + \Psi _ + (0,\lambda )} \right|_\Gamma $

as

$S(\lambda ) = (ip\mathcal{N} + I_ + )^{ - 1} (ip\mathcal{N} - I_ + ), I_ + = \sum\limits_{m = 1}^M {e^m } \rangle \langle e^m = P_ + $

. We derive an approximate formula for \(\mathcal{N}\) in terms of the Neumann-to-Dirichlet mapping \(\mathcal{N}_{\operatorname{int} } \) of L _int and the exponent K _? of the closed channels of l ^ω . If there is only one simple eigenvalue λ ₀ ∈ Δ, L _intφ0 = λ _0φ0 then, for a thin junction, \(\mathcal{N} \approx |\vec \phi _0 |^2 P_0 (\lambda _0 - \lambda )^{ - 1} \) with

$\vec \phi _0 = P_ + \phi _0 = (\delta ^{ - 1} \int_{\Gamma _1 } {\phi _0 (\gamma )} d\gamma ,\delta ^{ - 1} \int_{\Gamma _2 } {\phi _0 (\gamma )} d\gamma , \ldots \delta ^{ - 1} \int_{\Gamma _M } {\phi _0 (\gamma )} d\gamma )$

and \(P_0 = \vec \phi _0 \rangle |\vec \phi _0 |^{ - 2} \langle \vec \phi _0 \),

$S(\lambda ) \approx \frac{{ip|\vec \phi _0 |^2 P_0 (\lambda _0 - \lambda )^{ - 1} - I_ + }}{{ip|\vec \phi _0 |^2 P_0 (\lambda _0 - \lambda )^{ - 1} + I_ + }} = :S_{appr} (\lambda )$

. The related boundary condition for the components P ₊Ψ(0) and P ₊Ψ′(0) of the scattering Ansatz in the open channel \(P_ + \Psi (0) = (\bar \Psi _1 ,\bar \Psi _2 , \ldots ,\bar \Psi _M ), P_ + \Psi '(0) = (\bar \Psi '_1 , \bar \Psi '_2 , \ldots , \bar \Psi '_M )\) includes the weighted continuity (1) of the scattering Ansatz Ψ at the vertex and the weighted balance of the currents (2), where

$\frac{{\bar \Psi _m }}{{\bar \phi _0^m }} = \frac{{\delta \sum\nolimits_{t = 1}^M { \bar \Psi _t \bar \phi _0^t } }}{{|\vec \phi _0 |^2 }} = \frac{{\bar \Psi _r }}{{\bar \phi _0^r }} = :\bar \Psi (0)/\bar \phi (0), 1 \leqslant m,r \leqslant M$

(1)

,

$\sum\limits_{m = 1}^M {\bar \Psi '_m } \bar \phi _0^m + \delta ^{ - 1} (\lambda - \lambda _0 )\bar \Psi /\bar \phi (0) = 0$

(1)

. Conditions (1) and (2) constitute the generalized Kirchhoff boundary condition at the vertex for the Schrödinger operator on a thin junction and remain valid for the corresponding 1D model. We compare this with the previous result by Kuchment and Zeng obtained by the variational technique for the Neumann Laplacian on a shrinking quantum network.

     

Solution of the Dirac equation with magnetic monopole and pseudoscalar potentials

The Dirac equation in the presence of the Dirac magnetic monopole potential, the Aharonov-Bohm potential, a Coulomb potential and a pseudo-scalar potential, is solved by separation of variables using the spinweighted spherical harmonics. The energy spectrum and the form of the spinor functions are obtained. It is shown that the number j in spin-weighted spherical harmonics must be greater than $\left| q \right| - \tfrac{1} {2}$ .     

Absence of Negative Energy Spectrum for N-Particle Hamiltonians

We consider the spectrum of the quantum Hamiltonian H for a system of N one-dimensional particles. H is given by $H = \sum\nolimits_{i = 1}^n { - \frac{1}{{2m_i }}\frac{{\partial ^2 }}{{\partial x_i^2 }}} + \sum {_{1 \leqslant i < j \leqslant N} } V_{ij} \left( {x_i - x_j } \right)$ acting in L ²(R ^N ). We assume that each pair potential is a sum of a hard core for |x|≤a, a>0, and a function V _ij (x), |x|>a, with $\smallint _a^\infty \left| {x - a} \right|\left| {V_{ij} \left( x \right)} \right|dx < \infty $ . We give conditions on V ^? _ij (x), the negative part of V _ij (x), which imply that H has no negative energy spectrum for all N. For example, this is the case if V ^? _ij (x) has finite range 2a and $$2m_i \smallint _a^{2a} \left| {x - a} \right|\left| {V_{ij}^ - \left( x \right)} \right|dx < 1.$$ If V ^? _ij is not necessarily small we also obtain a thermodynamic stability bound inf?σ(H)≥?cN, where 0<c<∞, is an N-independent constant.     

The Essence of Operators’ Weyl Ordering Scheme and New Operator Identities for Converting Q−P Ordering to Weyl Ordering

We find new operator formulas for converting Q?P and P?Q ordering to Weyl ordering, where Q and P are the coordinate and momentum operator. In this way we reveal the essence of operators’ Weyl ordering scheme, e.g., Weyl ordered operator polynomial ${_{:}^{:}}\;Q^{m}P^{n}\;{_{:}^{:}}$ , $$\begin{aligned} {_{:}^{:}}\;Q^{m}P^{n}\;{_{:}^{:}} =&\sum_{l=0}^{\min (m,n)} \biggl( \frac{-i\hbar }{2} \biggr) ^{l}l!\binom{m}{l}\binom{n}{l}Q^{m-l}P^{n-l} \\ =& \biggl( \frac{\hbar }{2} \biggr) ^{ ( m+n ) /2}i^{n}H_{m,n} \biggl( \frac{\sqrt{2}Q}{\sqrt{\hbar }},\frac{-i\sqrt{2}P}{\sqrt{\hbar }} \biggr) \bigg|_{Q_{\mathrm{before}}P} \end{aligned}$$ where ${}_{:}^{:}$ ${}_{:}^{:}$ denotes the Weyl ordering symbol, and H _m,n is the two-variable Hermite polynomial. This helps us to know the Weyl ordering more intuitively.     

New General Approach for Normally Ordering Coordinate-Momentum Operator Functions

By virtue of integration technique within ordered product of operators and Dirac’s representation theory we find a new general formula for normally ordering coordinate-momentum operator functions, that is \(f(g\hat {{Q}}+h\hat {P})= :\exp [\textstyle {g^{2}+h^{2} \over 4}\textstyle {{\partial ^{2}} \over {\partial (g\hat {{Q}}+h\hat {P})^{2}}}]f(g\hat {{Q}}+h\hat {P})\):, where \(\hat {Q}\) and \(\hat {P}\) are the coordinate operator and momentum operator respectively, the symbol :: denotes normal ordering. Using this formula we can derive a series of new relations about Hermite polynomial and Laguerre polynomial, as well as some new differential relations.     

Uncertainty Relation on Generalized Skew Information with aMonotone Pair

In this paper, we first define a generalized (f,g)-skew information \(\left |I_{ \rho }^{(f, g)}\right |(A)\) and two related quantity \(\left |J_{ \rho }^{(f, g)}\right |(A)\) and \(\left |U_{ \rho }^{(f, g)}\right |(A)\) for any non-Hermitian Hilbert-Schmidt operator A and a density operator ρ on a Hilbert space H and discuss some properties of them. And then, we obtain the following uncertainty relation in terms of \(\left |U_{ \rho }^{(f, g)}\right |(A)\):

$$\begin{array}{@{}rcl@{}} \left|U_{ \rho}^{(f, g)}\right|(A)\left|U_{ \rho}^{(f, g)}\right|(B)\geq \beta_{(f, g)}\left|Tr\left( f(\rho)g(\rho)[A, B]^{0}\right)\right|^{2}, \end{array} $$

which is a generalization of a known uncertainty relation in Ko and Yoo (J. Math. Anal. Appl. 383, 208–214, 11).

     

One-dimensional random Ising systems with interaction decayr −(1+ɛ): A convergent cluster expansion

WE consider a one-dimensional random Ising model with Hamiltonian $$H = \sum\limits_{i\ddag j} {\frac{{J_{ij} }}{{\left| {i - j} \right|^{1 + \varepsilon } }}S_i S_j } + h\sum\limits_i {S_i } $$ , where ε>0 andJ _ij are independent, identically distributed random variables with distributiondF(x) such that i) $$\int {xdF\left( x \right) = 0} $$ , ii) $$\int {e^{tx} dF\left( x \right)< \infty \forall t \in \mathbb{R}} $$ . We construct a cluster expansion for the free energy and the Gibbs expectations of local observables. This expansion is convergent almost surely at every temperature. In this way we obtain that the free energy and the Gibbs expectations of local observables areC ^∞ functions of the temperature and of the magnetic fieldh. Moreover we can estimate the decay of truncated correlation functions. In particular for every ε′>0 there exists a random variablec(ω)m, finite almost everywhere, such that $$\left| {\left\langle {s_0 s_j } \right\rangle _H - \left\langle {s_0 } \right\rangle _H \left\langle {s_j } \right\rangle _H } \right| \leqq \frac{{c\left( \omega \right)}}{{\left| j \right|^{1 + \varepsilon - \varepsilon '} }}$$ , where 〈 〉 _H denotes the Gibbs average with respect to the HamiltonianH.     

Absence of symmetry breaking for systems of rotors with random interactions

We prove that Gibbs states for the Hamiltonian , with thes _x varying on theN-dimensional unit sphere, obtained with nonrandom boundary conditions (in a suitable sense), are almost surely rotationally invariant if withJ _xy i.i.d. bounded random variables with zero average, 1 in one dimension, and 2 in two dimensions.     

An expansion of Hartree-Fock and correlation energies,relativistic corrections and the dipole matrix element in the powers of of 1/Z

Feynman diagrammatic technique was used for the calculation of Hartree-Fock and correlation energies, relativistic corrections, dipole matrix element. The whole energy of atomic system was defined as a polen-electron Green function. Breit operator was used for the calculation of relativistic corrections. The Feynman diagrammatic technique was developed for 〈H_B>. Analytical expressions for the contributions from diagrams were received. The calculations were carried out for the terms of such configurations as 1s² 2sⁿ¹ 2pⁿ² (2 ≧n₁≧ 0, 6≧ n₂ ≧ 0). Numerical results are presented for the energies of the terms in the form $$E = E_0 Z^2 + \Delta {\rm E}_2 + \frac{1}{Z}\Delta {\rm E}_3 + \frac{{\alpha ^2 }}{4}(E_0^r + \Delta {\rm E}_1^r Z^3 )$$ and for fine structure of the terms in the form $$\begin{gathered} \left\langle {1s^2 2s^{n_1 } 2p^{n_2 } LSJ|H_B |1s^2 2s^{n_1 \prime } 2p^{n_2 \prime } L\prime S\prime J} \right\rangle = \hfill \\ = ( - 1)^{\alpha + S\prime + J} \left\{ {\begin{array}{*{20}c} {L S J} \\ {S\prime L\prime 1} \\ \end{array} } \right\}\frac{{\alpha ^2 }}{4}(Z - A)^3 [E^{(0)} (Z - B) + \varepsilon _{co} ] + \hfill \\ + ( - 1)^{L + S\prime + J} \left\{ {\begin{array}{*{20}c} {L S J} \\ {S\prime L\prime 2} \\ \end{array} } \right\}\frac{{\alpha ^2 }}{4}(Z - A)^3 \varepsilon _{cc} . \hfill \\ \end{gathered} $$ Dipole matrix elements are necessary for calculations of oscillator strengths and transition probabilities. For dipole matrix elements two members of expansion by 1/Z have been obtained. Numerical results were presented in the form P(a,a′) = a/Z(1+τ/Z).     

Spherical Model on a Cayley Tree: Large Deviations

We study the spherical model of a ferromagnet on a Cayley tree and show that in the case of empty boundary conditions a ferromagnetic phase transition takes place at the critical temperature \(T_\mathrm{c} =\frac{6\sqrt{2}}{5}J\), where J is the interaction strength. For any temperature the equilibrium magnetization, \(m_n\), tends to zero in the thermodynamic limit, and the true order parameter is the renormalized magnetization \(r_n=n^{3/2}m_n\), where n is the number of generations in the Cayley tree. Below \(T_\mathrm{c}\), the equilibrium values of the order parameter are given by \(\pm \rho ^*\), where

$$\begin{aligned} \rho ^*=\frac{2\pi }{(\sqrt{2}-1)^2}\sqrt{1-\frac{T}{T_\mathrm{c}}}. \end{aligned}$$

One more notable temperature in the model is the penetration temperature

$$\begin{aligned} T_\mathrm{p}=\frac{J}{W_\mathrm{Cayley}(3/2)}\left( 1-\frac{1}{\sqrt{2}}\left( \frac{h}{2J}\right) ^2\right) . \end{aligned}$$

Below \(T_\mathrm{p}\) the influence of homogeneous boundary field of magnitude h penetrates throughout the tree. The main new technical result of the paper is a complete set of orthonormal eigenvectors for the discrete Laplace operator on a Cayley tree.