Dynamical Localization for the Random Dimer Schrödinger Operator

We study the one-dimensional random dimer model, with Hamiltonian H _ω =Δ+V _ω , where for all x∈ $\mathbb{Z}$ , V _ω(2x)=V _ω(2x+1) and where the V _ω(2x) are i.i.d. Bernoulli random variables taking the values ±V, V>0. We show that, for all values of Vand with probability one in ω, the spectrum of His pure point. If V≤1 and V≠1/ $\sqrt 2$ , the Lyapunov exponent vanishes only at the two critical energies given by E=±V. For the particular value V=1/ $\sqrt 2$ , respectively, V= $\sqrt 2$ , we show the existence of new additional critical energies at E=±3/ $\sqrt 2$ , respectively, E=0. On any compact interval Inot containing the critical energies, the eigenfunctions are then shown to be semi-uniformly exponentially localized, and this implies dynamical localization: for all q>0 and for all ψ∈ $\ell$ ²( $\mathbb{Z}$ ) with sufficiently rapid decrease $${\mathop {\sup }\limits_t} r_{\psi ,I}^{\left( q \right)} {\kern 1pt} \left( t \right): = {\mathop {\sup }\limits_t} \left\langle {P_I \left( {H\omega } \right)\psi _t ,\left| X \right|^q P_I \left( {H\omega } \right)\psi _t } \right\rangle < \infty $$ Here $\psi _t = e^{- iH_{\omega ^t}} \psi$ , and P _I(H _ω) is the spectral projector of H _ωonto the interval I. In particular, if V>1 and V≠ $\sqrt 2$ , these results hold on the entire spectrum [so that one can take I=σ(H _ω)].     

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.     

Exponential lower bounds to solutions of the Schrödinger equation: Lower bounds for the spherical average

For a large class of generalizedN-body-Schrödinger operators,H, we show that ifE<Σ=infσ_ess(H) and ψ is an eigenfunction ofH with eigenvalueE, then $$\begin{array}{*{20}c} {\lim } \\ {R \to \infty } \\ \end{array} R^{ - 1} \ln \left( {\int\limits_{S^{n - 1} } {|\psi (R\omega )|} ^2 d\omega } \right)^{1/2} = - \alpha _0 ,$$ with α ₀ ² +E a threshold. Similar results are given forE≧Σ.     

Asymptotic inverse spectral problem for anharmonic oscillators

We study perturbationsL=A+B of the harmonic oscillatorA=1/2(??²+x ²?1) on ?, when potentialB(x) has a prescribed asymptotics at ∞,B(x)～|x|^?α V(x) with a trigonometric even functionV(x)=Σa _mcosω _m x. The eigenvalues ofL are shown to be λ _k =k+μ _k with small μ _k =O(k ^?γ), γ=1/2+1/4. The main result of the paper is an asymptotic formula for spectral fluctuations {μ _k }, $$\mu _k \sim k^{ - \gamma } \tilde V(\sqrt {2k} ) + c/\sqrt {2k} ask \to \infty ,$$ whose leading term \(\tilde V\) represents the so-called “Radon transform” ofV, $$\tilde V(x) = const\sum {\frac{{a_m }}{{\sqrt {\omega _m } }}\cos (\omega _m x - \pi /4)} .$$ as a consequence we are able to solve explicitly the inverse spectral problem, i.e., recover asymptotic part |x ^?α|V(x) ofB from asymptotics of {µ _k }. ₁ ^∞      

Alternative technique for Standard Model estimation of the rare kaon decay branchings \left. {BR(K \to \pi \nu \bar \nu )} \right|_{SM}

We estimate $BR(K \to \pi \nu \bar \nu )$ in the context of the Standard Model by fitting for λ _t≡V _tdV _ts ^* of the “kaon unitarity triangle” relation. To find the vertex of this triangle, we fit data from |? _K|, the CP-violating parameter describing K mixing, and a _ψ,K , the CP-violating asymmetry in B _d ⁰ → J/ψK ⁰ decays, and obtain the values $\left. {BR(K \to \pi \nu \bar \nu )} \right|_{SM} = (7.07 \pm 1.03) \times 10^{ - 11} $ and $\left. {BR(K_L^0 \to \pi ^0 \nu \bar \nu )} \right|_{SM} = (2.60 \pm 0.52) \times 10^{ - 11} $ . Our estimate is independent of the CKM matrix element V _cb and of the ratio of B-mixing frequencies ${{\Delta m_{B_s } } \mathord{\left/ {\vphantom {{\Delta m_{B_s } } {\Delta m_{B_d } }}} \right. \kern-0em} {\Delta m_{B_d } }}$ . We also use the constraint estimation of λ _t with additional data from $\Delta m_{B_d } $ and |V _ub|. This combined analysis slightly increases the precision of the rate estimation of $K^ + \to \pi ^ + \nu \bar \nu $ and $K_L^0 \to \pi ^0 \nu \bar \nu $ (by ?10 and ?20%, respectively). The measured value of $BR(K^ + \to \pi ^ + \nu \bar \nu )$ can be compared both to this estimate and to predictions made from ${{\Delta m_{B_s } } \mathord{\left/ {\vphantom {{\Delta m_{B_s } } {\Delta m_{B_d } }}} \right. \kern-0em} {\Delta m_{B_d } }}$ .     

On the similarity solution of evolution equation u t =H(x,t, u,u x,u xx , ...)

Let $$\begin{gathered} u^* = u + \in \eta (x,{\text{ }}t,{\text{ }}u), \hfill \\ \hfill \\ \hfill \\ x^* = x + \in \xi (x, t, u{\text{),}} \hfill \\ \hfill \\ \hfill \\ {\text{t}}^{\text{*}} = {\text{ }}t + \in \tau {\text{(}}x,{\text{ }}t,{\text{ }}u), \hfill \\ \end{gathered}$$ be an infinitesimal invariant transformation of the evolution equation u _t =H(x,t,u,?u/?x,...,? ⁿ :u/?x ⁿ . In this paper we give an explicit expression for \(\eta ^{X^i }\) in the ‘determining equation’ $$\eta ^T = \sum\limits_{i = 1}^n {{\text{ }}\eta ^{X^i } {\text{ }}\frac{{\partial H}}{{\partial u_i }} + \eta \frac{{\partial H}}{{\partial u_{} }} + \xi \frac{{\partial H}}{{\partial x}} + \tau } \frac{{\partial H}}{{\partial t}},$$ where u _i =? ⁱ u/?x ⁱ . By using this expression we derive a set of equations with η, ξ, τ as unknown functions and discuss in detail the cases of heat and KdV equations.     

A Weighted Dispersive Estimate for Schrödinger Operators in Dimension Two

Let H = ?Δ + V, where V is a real valued potential on ${\mathbb {R}^2}$ satisfying ${\|V(x)|\lesssim \langle x \rangle^{-3-}}$ . We prove that if zero is a regular point of the spectrum of H = ?Δ + V, then $${\| w^{-1} e^{itH}P_{ac}f\|_{L^\infty(\mathbb{R}^2)} \lesssim \frac{1}{|t|\log^2(|t|)} \| w f\|_{L^1(\mathbb{R}^2)},\,\,\,\,\,\,\,\, |t| \geq 2}$$ , with w(x) = (log(2 + |x|))². This decay rate was obtained by Murata in the setting of weighted L ² spaces with polynomially growing weights.     

Collisional damping of transverse waves in a plasma

The effect of collisions on transverse waves in a homogeneous, field free plasma is investigated by means of Gross-Krook collision model. The dispersion relation is calculated by assuming the collision frequency to be small andKλ _D ?1. The damping rate ω _I is obtained as $$\omega _I = \frac{{\nu _{ei} }}{2}\frac{{\omega _p^2 }}{{\omega _0^2 }}\left[ {1 + \frac{{3K^2 \lambda _D^2 \omega _p^2 }}{{\omega _0^2 }} - \frac{{K^2 \lambda _D^2 \omega _p^4 }}{{\omega _0^4 }}} \right] + \frac{{\nu _{ee} }}{2}\frac{{\omega _p^2 }}{{\omega _0^2 }}\left( {\frac{{K^2 \lambda _D^2 \omega _p^2 }}{{\omega _0^2 }}} \right)$$ where ω ₀ ² =c ² K ²+ω ₂ ^p , andv _ei andv _ee are electron-ion and electron-electron collision frequency respectively.     

The local structure of the spectrum of the one-dimensional Schrödinger operator

Let \(H_V = - \frac{{d^{\text{2}} }}{{dt^{\text{2}} }} + q(t,\omega )\) be an one-dimensional random Schrödinger operator in ?²(?V,V) with the classical boundary conditions. The random potentialq(t, ω) has a formq(t, ω)=F(x _t ), wherex _t is a Brownian motion on the compact Riemannian manifoldK andF:K→R ¹ is a smooth Morse function, \(\mathop {\min }\limits_K F = 0\) . Let \(N_V (\Delta ) = \sum\limits_{Ei(V) \in \Delta } 1 \) , where Δ∈(0, ∞),E _i (V) are the eigenvalues ofH _V . The main result (Theorem 1) of this paper is the following. IfV→∞,E ₀>0,k∈Z ₊ anda>0 (a is a fixed constant) then $$P\left\{ {N_V \left( {E_0 - \frac{a}{{2V}},E_0 + \frac{a}{{2V}}} \right) = k} \right\}\xrightarrow[{V \to \infty }]{}e^{ - an(E_0 )} (an(E_0 ))^k |k!,$$ wheren(E ₀) is a limit state density ofH _V ,V→∞. This theorem mean that there is no repulsion between energy levels of the operatorH _V ,V→∞. The second result (Theorem 2) describes the phenomen of the repulsion of the corresponding wave functions.     

The NLS Equation in Dimension One with Spatially Concentrated Nonlinearities: the Pointlike Limit

In the present paper, we study the following scaled nonlinear Schrödinger equation (NLS) in one space dimension: $$ i\frac{\rm d}{{\rm d}t}\psi^{\varepsilon}(t)=-\Delta\psi^{\varepsilon}(t) +\frac{1}{\varepsilon}V\left(\frac{x}{\varepsilon} \right)|\psi^{\varepsilon}(t)|^{2\mu}\psi^{\varepsilon}(t)\quad \varepsilon > 0\,\quad V\in L^1(\mathbb{R},(1+|x|){\rm d}x) \cap L^\infty(\mathbb{R}).$$ This equation represents a nonlinear Schrödinger equation with a spatially concentrated nonlinearity. We show that in the limit \({\varepsilon\to 0}\) the weak (integral) dynamics converges in \({H^1(\mathbb{R})}\) to the weak dynamics of the NLS with point-concentrated nonlinearity: $$ i\frac{{\rm d}}{{\rm d}t} \psi(t) =H_{\alpha} \psi(t) .$$ where H _α is the Laplacian with the nonlinear boundary condition at the origin \({\psi'(t,0+)-\psi'(t,0-)=\alpha|\psi(t,0)|^{2\mu}\psi(t,0)}\) and \({\alpha=\int_{\mathbb{R}}V{\rm d}x}\) . The convergence occurs for every \({\mu\in \mathbb{R}^+}\) if V ≥  0 and for every  \({\mu\in (0,1)}\) otherwise. The same result holds true for a nonlinearity with an arbitrary number N of concentration points.     

Propagation of sound and spectrally resolved rayleigh scattering in weakly ionized plasmas

The scattering mechanisms possible in weakly ionized plasmas are reviewed. The different cases can be discerned by means of the magnitude of three characteristic parameters: 2π/ωτ _c being the ratio of scattering time and mean free collision time, y=1/kλ _c being the scattering parameter defined as the ratio of scattering length and mean free collision path, and χω/c _{s, T} ² being the ratio between the product of thermometric conductivity and scattering frequency to the square of the adiabatic or isothermal sound velocity. For \({{2\pi } \mathord{\left/ {\vphantom {{2\pi } {\omega \tau _c<< 1}}} \right. \kern-0em} {\omega \tau _c<< 1}}\) quantum mechanical formulae have to be used, whereas in the opposite case classical treatments are applicable. The classical methods are Boltzmann equation formalisms ify?1, and fluid dynamics ify?1. In the fluid dynamical case there appear two waves for low frequencies, \({{\chi \omega } \mathord{\left/ {\vphantom {{\chi \omega } {c_{s, T}^2 }}} \right. \kern-0em} {c_{s, T}^2 }}<< 1\) , an adiabatic one which can propagate with weak damping and an isobaric one which cannot propagate, both wave types yielding together three scattered lines. In the opposite case of high frequencies, \({{\chi \omega } \mathord{\left/ {\vphantom {{\chi \omega } {c_{s, T}^2 }}} \right. \kern-0em} {c_{s, T}^2 }} > > 1\) , the scattering behavior is different from ordinary hydrodynamics. Here also do exist two types of waves, isochoric and isothermal, but none of them can propagate. Since the intensity of the scattered isochoric wave can be neglected, there is only one scattered line. Local temperatures and particle densities can be determined from the scattered spectrum. On the other hand, transport coefficients like shear and bulk viscosity as well as thermometric conductivity can be derived from sound absorption or Rayleigh scattering experiments if an independent temperature measurement is performed at the same time. The general formalism is applied to a weakly ionized hydrogen arc plasma.     

On Lévy (or stable) distributions and the Williams-Watts model of dielectric relaxation

This paper is concerned with the Lévy, or stable distribution function defined by the Fourier transform $$Q_\alpha \left( z \right) = \frac{1}{{2\pi }}\int {_{ - \infty }^\infty \exp \left( { - izu - \left| u \right|^\alpha } \right)du} with 0< \alpha \leqslant 2$$ Whenα=2 it becomes the Gauss distribution function and whenα=1, the Cauchy distribution. Whenα≠2 the distribution has a long inverse power tail $$Q_\alpha \left( z \right) \sim \frac{{\Gamma \left( {1 + \alpha } \right)\sin \tfrac{1}{2}\pi \alpha }}{{\pi \left| z \right|^{1 + \alpha } }}$$ In the regime of smallα, ifα¦logz¦?1, the distribution is mimicked by a log normal distribution. We have derived rapidly converging algorithms for the numerical calculation ofQ _α (z) for variousα in the range 0<α<1. The functionQ _α (z) appears naturally in the Williams-Watts model of dielectric relaxation. In that model one expresses the normalized dielectric parameter as $$ \in _n \left( \omega \right) \equiv \in '_n \left( \omega \right) - i \in ''_n \left( \omega \right) = - \int {_0^\infty e^{ - i\omega t} \left[ {{{d\phi \left( t \right)} \mathord{\left/ {\vphantom {{d\phi \left( t \right)} {dt}}} \right. \kern-\nulldelimiterspace} {dt}}} \right]} dt$$ with $$\phi \left( t \right) = \exp - \left( {{t \mathord{\left/ {\vphantom {t \tau }} \right. \kern-\nulldelimiterspace} \tau }} \right)^\alpha $$ It has been found empirically by various authors that observed dielectric parameters of a wide variety of materials of a broad range of frequencies are fitted remarkably accurately by using this form ofφ(t).ε″ _n (ω) is shown to be directly related toQ _α (z). It is also shown that if the Williams-Watts exponential is expressed as a weighted average of exponential relaxation functions $$\exp - \left( {{t \mathord{\left/ {\vphantom {t \tau }} \right. \kern-\nulldelimiterspace} \tau }} \right)^\alpha = \int {_0^\infty } g\left( {\lambda , \alpha } \right)e^{ - \lambda t} dt$$ the weight functiong(λ, α) is expressible as a stable distribution. Some suggestions are made about physical models that might lead to the Williams-Watts form ofφ(t).     

Electromagnetic moment investigation of two short-lived isomeric states in118Sb

Two short-lived isomeric states in¹¹⁸Sb have been investigated by the¹¹⁸Sn(p, n),¹¹⁸Sn(d, 2n) and¹¹⁵In(α, n) reactions. The TDPAD method on solid and liquid metallic targets was used to measure the electromagnetic moments of these states. The results of the experiments are: $$\begin{gathered} T_{1/2} = 13.4{\text{ }}(3){\text{ }}ns I^\pi = 3^ - {\text{ }}g = - 1.254(31){\text{ }}|Q| = 0.25{\text{ }}(5){\text{ }}b, \hfill \\ T_{1/2} = 22.8{\text{ }}(4){\text{ }}ns I^\pi = 7^ + {\text{ }}g = + 0.680(18){\text{ }}|Q| > 1.4{\text{ }}b. \hfill \\ \end{gathered}$$ Pure \([\pi 2d_{5/2} \otimes v1h_{1{\text{ }}1/2} ]_{3 - }\) and \([\pi 1g_{9/2}^{ - 1} \otimes v2d_{5/2}^{ - 1} ]_{7 + }\) configurations have been established for the two isomeric states. An experimental evidence concerning the participation of the 1g ₉ ^2/?1 proton shell-model intruder excitation into the positive parity low-lying level structure of the odd-odd¹¹⁸Sb nucleus was obtained.     

Universal lower bounds on eigenvalue splittings for one dimensional Schrödinger operators

We provide lower bounds on the eigenvalue splitting for ?d ²/dx ²+V(x) depending only on qualitative properties ofV. For example, ifV is C^∝ on [a, b] andE _n ,E _n?1 are two successive eigenvalues of ?d ²/dx ²+V withu(a)=u(b)=0 boundary conditions, and if \(\lambda = \mathop {\max }\limits_{E \in (E_{n - 1} ,E_n );x \in (a,b)} |E - V(x)|^{1/2} \) , then $$E_n - E_{n - 1} \geqq \pi \lambda ^2 \exp \left[ { - \lambda (b - a)} \right]$$ . The exponential factor in such bounds are saturated precisely in tunneling examples. Our results arenot restricted toV's of compact support, but only require \(E_n< \mathop {\lim }\limits_{\overline {x \to \infty } } V(x)\) .     

Light quark masses in quantum chromodynamics and chiral symmetry breaking

We derive model independent lower bounds for the sums of effective quark masses \(\bar m_u + \bar m_d \) and \(\bar m_u + \bar m_s \) . The bounds follow from the combination of the spectral representation properties of the hadronic axial currents two-point functions and their behavior in the deep euclidean region (known from a perturbative QCD calculation to two loops and the leading non-perturbative contribution). The bounds incorporate PCAC in the Nambu-Goldstone version. If we define the invariant masses \(\hat m\) by $$\bar m_i = \hat m_i \left( {{{\frac{1}{2}\log Q^2 } \mathord{\left/ {\vphantom {{\frac{1}{2}\log Q^2 } {\Lambda ^2 }}} \right. \kern-\nulldelimiterspace} {\Lambda ^2 }}} \right)^{{{\gamma _1 } \mathord{\left/ {\vphantom {{\gamma _1 } {\beta _1 }}} \right. \kern-\nulldelimiterspace} {\beta _1 }}} $$ and <F ²> is the vacuum expectation value of $$F^2 = \Sigma _a F_{(a)}^{\mu v} F_{\mu v(a)} $$ , we find, e.g., $$\hat m_u + \hat m_d \geqq \sqrt {\frac{{2\pi }}{3} \cdot \frac{{8f_\pi m_\pi ^2 }}{{3\left\langle {\alpha _s F^2 } \right\rangle ^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} }}} $$ ; with the value <α _u F ²?0.04GeV⁴, recently suggested by various analysis, this gives $$\hat m_u + \hat m_d \geqq 35MeV$$ . The corresponding bounds on \(\bar m_u + \bar m_s \) are obtained replacingm _π ² f _π bym _K ² f _K . The PCAC relation can be inverted, and we get upper bounds on the spontaneous masses, \(\hat \mu \) : $$\hat \mu \leqq 170MeV$$ where \(\hat \mu \) is defined by $$\left\langle {\bar \psi \psi } \right\rangle \left( {Q^2 } \right) = \left( {{{\frac{1}{2}\log Q^2 } \mathord{\left/ {\vphantom {{\frac{1}{2}\log Q^2 } {\Lambda ^2 }}} \right. \kern-\nulldelimiterspace} {\Lambda ^2 }}} \right)^d \hat \mu ^3 ,d = {{12} \mathord{\left/ {\vphantom {{12} {\left( {33 - 2n_f } \right)}}} \right. \kern-\nulldelimiterspace} {\left( {33 - 2n_f } \right)}}$$ .     

The Polyanalytic Ginibre Ensembles

For integers n,q=1,2,3,…?, let Pol _n,q denote the ${\mathbb{C}}$ -linear space of polynomials in z and $\bar{z}$ , of degree ≤n?1 in z and of degree ≤q?1 in $\bar{z}$ . We supply Pol _n,q with the inner product structure of $$\begin{aligned} L^2 \bigl({\mathbb{C}},\mathrm{e}^{-m|z|^2} {\mathrm{d}}A \bigr),\quad \mbox {where } {\mathrm{d}}A(z)=\pi^{-1}{\mathrm{d}}x {\mathrm{d}}y,\ z= x+ {\mathrm{i}}y; \end{aligned}$$ the resulting Hilbert space is denoted by Pol _m,n,q . Here, it is assumed that m is a positive real. We let K _m,n,q denote the reproducing kernel of Pol _m,n,q , and study the associated determinantal process, in the limit as m,n→+∞ while n=m+O(1); the number q, the degree of polyanalyticity, is kept fixed. We call these processes polyanalytic Ginibre ensembles, because they generalize the Ginibre ensemble—the eigenvalue process of random (normal) matrices with Gaussian weight. There is a physical interpretation in terms of a system of free fermions in a uniform magnetic field so that a fixed number of the first Landau levels have been filled. We consider local blow-ups of the polyanalytic Ginibre ensembles around points in the spectral droplet, which is here the closed unit disk $\bar{\mathbb{D}}:=\{z\in{\mathbb{C}}:|z|\le1\}$ . We obtain asymptotics for the blow-up process, using a blow-up to characteristic distance m ^?1/2; the typical distance is the same both for interior and for boundary points of $\bar{\mathbb{D}}$ . This amounts to obtaining the asymptotical behavior of the generating kernel K _m,n,q . Following (Ameur et al. in Commun. Pure Appl. Math. 63(12):1533–1584, 2010), the asymptotics of the K _m,n,q are rather conveniently expressed in terms of the Berezin measure (and density) For interior points |z|<1, we obtain that ${\mathrm{d}}B^{\langle z\rangle}_{m,n,q}(w)\to{\mathrm{d}}\delta_{z} $ in the weak-star sense, where δ _z denotes the unit point mass at z. Moreover, if we blow up to the scale of m ^?1/2 around z, we get convergence to a measure which is Gaussian for q=1, but exhibits more complicated Fresnel zone behavior for q>1. In contrast, for exterior points |z|>1, we have instead that ${\mathrm{d}}B^{\langle z\rangle}_{m,n,q}(w) \to{\mathrm{d}}\omega(w,z, {\mathbb{D}}^{e}) $ , where ${\mathrm{d}}\omega(w,z,{\mathbb{D}}^{e})$ is the harmonic measure at z with respect to the exterior disk ${\mathbb{D}}^{e}:= \{w\in{\mathbb{C}}:\, |w|>1\}$ . For boundary points, |z|=1, the Berezin measure ${\mathrm{d}}B^{\langle z\rangle}_{m,n,q}$ converges to the unit point mass at z, as with interior points, but the blow-up to the scale m ^?1/2 exhibits quite different behavior at boundary points compared with interior points. We also obtain the asymptotic boundary behavior of the 1-point function at the coarser local scale q ^1/2 m ^?1/2.     

One electron molecules with relativistic kinetic energy: Properties of the discrete spectrum

We discuss the discrete spectrum of the operator $$H_K (c) = \left[ { - \hbar ^2 c^2 \Delta + m^2 c^4 } \right]^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} - \sum\limits_{k = 1}^K {Z_k e^2 \left| {x - R_k } \right|^{ - 1} } $$ . More specifically, we study 1) the behaviour of the eigenvalues when the internuclear distances contract, 2) the existence of ac-independent lower bound forH _K (c)?mc ², 3) the nonrelativistic limit of the eigenvalues ofH _K (c)?mc ².     

A remark on tensor representations

The identity $$\sum\limits_{v = 0} {\left( {\begin{array}{*{20}c} {n + 1} \\ v \\ \end{array} } \right)\left[ {\left( {\begin{array}{*{20}c} {n - v} \\ v \\ \end{array} } \right) - \left( {\begin{array}{*{20}c} {n - v} \\ {v - 1} \\ \end{array} } \right)} \right] = ( - 1)^n } $$ is proved and, by means of it, the coefficients of the decomposition ofD ₁ ⁿ into irreducible representations are found. It holds: ifD ₁ ⁿ \(\mathop {\sum ^n }\limits_{m = 0} A_{nm} D_m \) , then $$A_{nm} = \mathop \sum \limits_{\lambda = 0} \left( {\begin{array}{*{20}c} n \\ \lambda \\ \end{array} } \right)\left[ {\left( {\begin{array}{*{20}c} \lambda \\ {n - m - \lambda } \\ \end{array} } \right) - \left( {\begin{array}{*{20}c} \lambda \\ {n - m - \lambda - 1} \\ \end{array} } \right)} \right].$$