On the Spectral Theory of Operator Pencils in a Hilbert Space

Abstract

Consider the operator pencil L _λ = A ? λB ? λ ² C, where A, B, and C are linear, in general unbounded and nonsymmetric, operators densely defined in a Hilbert space H. Sufficient conditions for the existence of the eigenvalues of L _λ are investigated in the case when A, B and C are K-positive and K-symmetric operators in H, and a method to bracket the eigenvalues of L _λ is developed by using a variational characterization of the problem (i) L _λ u = 0. The method generates a sequence of lower and upper bounds converging to the eigenvalues of L _λ and can be considered an extension of the Temple-Lehman method to quadratic eigenvalue problems (i).     

The N-mode squeezed state with enhanced squeezing

It is known that exp [iλ (Q1P1i/2)] is a unitary single-mode squeezing operator,where Q1,P1 are the coordinate and momentum operators,respectively.In this paper we employ Dirac’s coordinate representation to prove that the exponential operator S n ≡ exp [iλ sum((QiPi+1+Qi+1Pi))) from i=1 to n ],(Qn+1=Q1,Pn+1=P1),is an n-mode squeezing operator which enhances the standard squeezing.By virtue of the technique of integration within an ordered product of operators we derive S n ’s normally ordered expansion and obtain new n-mode squeezed vacuum states,its Wigner function is calculated by using the Weyl ordering invariance under similar transformations.     

确定等价电子杨盘基的等概率比对方法

给出了等价电子正则杨盘T^［λ］_ig的基本对称算子、完全对称算子概念,同时给出了这些对称算子作用于任一Slater函数_i所产生的根态、生成态概念.由正交归一化杨盘T^［λ］_ie的纵置换算子A^［λ］_ie的构造规则,给出了A^［λ］_ie中存在的对称算子和确定T^［λ］_ie的等概率比对方法,从而基本避免了牵涉到许多算子的极其复杂的代数,给出了求解N值较大的电子系统杨盘基问题的新方法. 关键词： 正则杨盘 对称算子 根态 等概率比对方法     

A new theoretical development of the limiting electric conductivity of ions in solution

The physical interpretation of the limiting equivalent conductivity of binary electrolytes in solution, Λ⁰ (=Σ_i?=?+,– λ⁰_i), has been a challenge for almost a century. Here, as a follow-up on a preliminary communication [D. Fraenkel, Mol. Phys. 115, 2944 (2017)], a quantitative theoretical treatment is offered and fully developed for the limiting ionic equivalent conductivity, λ⁰_i, based on assuming that around any singled-out ion i in solution, there is an electrostatic atmosphere that includes oriented solvent dipoles and the counterion as ‘charge density’. This leads to a simple mathematical expression for λ⁰_i as the result of the tandem operation of the dipole viscosity and dipole relaxation forces on the drifting ion. Contrary to literature teaching, λ⁰_i is found proportional to the effective ionic radius, r_i. An ‘electric ion-radius scale’ derived from experimental λ⁰_i’s for aqueous solutions, compares reasonably well with known ion-size scales. A newly defined solvent-specific dipole random-walk correction factor, ξ, is used for nonaqueous solvents as an adjustable parameter, to attain a parallel electric ion-size scale. The new theory provides overall a comprehensive view of the limiting ion mobility in solution, based on the ionic properties, solvent variables and universal constants.     

A Particle in a Magnetic Field of an Infinite Rectilinear Current

We consider the Schrödinger operator H=(i+A)² in the space L ₂(R ³) with a magnetic potential A created by an infinite rectilinear current. We show that the operator H is absolutely continuous, its spectrum has infinite multiplicity and coincides with the positive half-axis. Then we find the large-time behavior of solutions exp(–i H t)f of the time dependent Schrödinger equation. Our main observation is that a quantum particle has always a preferable (depending on its charge) direction of propagation along the current. Similar result is true in classical mechanics.     

New transformation of Wigner operator in phase space quantum mechanics for the two-mode entangled case

As a natural and important extension of Fan's paper (Fan Hong-Yi 2010 Chin. Phys. B 19 040305) by employing the formula of operators' Weyl ordering expansion and the bipartite entangled state representation this paper finds a new two-fold complex integration transformation about the Wigner operator Δ ( μ,v ) (in its entangled form) in phase space quantum mechanics, and its inverse transformation. In this way, some operator ordering problems regarding to ( a₁⁺-a₂) and (a₁+a₂⁺) can be solved and the contents of phase space quantum mechanics can be enriched, where a_i,a_i⁺ are bosonic creation and annihilation operators, respectively.     

Nonformal recursion operators and nonlinear integrable evolution equations

A new property involving the recursion operator L and the Hamiltonian operator J of the nonlinear evolution equations integrable by the inverse scattering transform method is derived. It follows that these equations are completely determined in terms of the L and J operators.Unité Associée au CNRS No. 040768. Recherche Coopérative sur Programme No. 080264.     

Initial-boundary value problem for the two-component Gerdjikov-Ivanov equation on the interval

In this paper, we apply Fokas unified method to study initial-boundary value problems for the two-component Gerdjikov-Ivanov equation formulated on the finite interval with 3×3 Lax pairs. The solution can be expressed in terms of the solution of a 3×3 Riemann-Hilbert problem. The relevant jump matrices are explicitly given in terms of three matrix-value spectral functions s (λ), S (λ) and S_L(λ), which arising from the initial values at t = 0, boundary values at x = 0 and boundary values at x = L, respectively. Moreover, The associated Dirichlet to Neumann map is analyzed via the global relation. The relevant formulae for boundary value problems on the finite interval can reduce to ones on the half-line as the length of the interval tends to infinity.     

Symmetry Classification of the One–Dimensional Second Order Equation of a Hydrodynamic Type

Abstract

The paper contains a symmetry classification of the one–dimensional second order equation of a hydrodynamical type L(Lu)+λLu=F (u), where L ≡ ? _t+u? _x. Some classes of exact solutions of this equation are given.     

Polaronic origin of the isotope effect on the London penetration depth in high-temperature superconducting oxides

Direct measurements of the in-plane London penetration depth λ _L have recently been performed on high-temperature superconducting copper oxides by a new low-energy muon spin rotation technique. The results show that λ _L is isotope dependent, evidencing unconventional electron–phonon interactions as its source. The data are interpreted here in terms of polaronic effects on the single-particle energies, which leads to level shifts and exponential band narrowing. Good agreement with the experimental data is obtained.     

A<Subscript>4</Subscript> symmetry and lepton masses and mixing

Stimulated by Ma’s idea, which explains the tribimaximal neutrino mixing by assuming an A₄ flavor symmetry, a lepton mass matrix model is investigated. A Frogatt–Nielsen-type model is assumed, and the flavor structures of the masses and mixing are caused by the VEVs of SU(2)_L singlet scalars φ_i ^u and φ_i ^d (i=1,2,3), which are assigned to 3 and (1 ,1 ’,1 ”) of A₄, respectively. Possible charged lepton and neutrino mass spectra and mixing are investigated.     

Quantum Motion on 2D Surfaces of Spherical Topology

When the motion of a particle is constrained on the two-dimensional surface, excess terms exist in usual kinetic energy 1/(2μ) ∑ p _i ² with hermitian form of Cartesian momentum p _i (i = 1,2,3), and the operator ordering should be taken into account in the kinetic energy which turns out to be 1/(2μ) ∑ (1/f _i )p _i f _i p _i where the functions f _i are dummy factors in classical mechanics and nontrivial in quantum mechanics. In this article, the explicit forms of the dummy functions f _i for quantum motion on some 2D surfaces of revolution of spherical topology are given. PACS numbers: 03.65.-w Quantum mechanics, 04.60.Ds Canonical quantization.     

Universality of Operator Ordering in Kinetic Energy Operator for Particles Moving on two Dimensional Surfaces

When the motion of a particle is constrained on the two-dimensional surface, excess terms exist in usual kinetic energy 1/(2m)∑ p _i ² with hermitian form of Cartesian momentum p _i (i = 1,2,3), and the operator ordering should be taken into account in the kinetic energy which turns out to be 1/(2m)∑ (1/f _i )p _i f _i p _i where the functions f _i are dummy factors in classical mechanics and nontrivial in quantum mechanics. The existence of non-trivial f _i shows the universality of this constraint induced operator ordering in quantum kinetic energy operator for the constraint systems.     

Repeated application of the recursion operator for a new hierarchy of negative-order integrable KdV equations

ABSTRACT

In this work we use the repeated application of the recursion operator to establish a new hierarchy of negative-order integrable KdV equations of higher orders. The concept of the inverse recursion operator allows us to develop this new hierarchy. The complete integrability of each equation is guaranteed via the use of the recursion operator. We show that the dispersion relations of this hierarchy follow an infinite geometric series. Multiple soliton solutions for each equation of the hierarchy are obtained.     

The Study on the Arc Plasma Temperature Measurement by Optical Emission Spectroscopy with Fiber Optical Transmission

Abstract

A method for transmitting radiation of the arc plasma with multimode fused quartz fiber onto the spectrograph has been studied. The plot of the Boltzmann function in emission spectral analysis is used for measuring temperature of the arc plasma. The measured temperature of the arc plasma is 5946.6K from least square linear regression of ln[λI/(gA)] and E_i for a number of the emission line intensities of the excited copper atom. Its regression coefficient and measured precision are ?0.97% and 1.7%, respectively. The advantages of the method of the diagnostic temperature for the arc plasma are absolute measurements of the temperature, remote sensing, precision and suitable for mal-environment, such as high temperature, toxic, explosion, strong magnetic or/and electrical fields.

In addition, we have discussed the effect of the spectroscopic constants, such as transition probability, A , the statistical weight of the upper level, g , and the energy of the upper level, E_i , of copper lines on calculating temperature with a plot of the Boltzmann function in detail. The results show that the accurate measurement of the temperature for the arc plasma is obtained only when the spectroscopic constants are selected correctly.     

Solutions and applications of tridiagonal vector recurrence relations

The applications of infinite systems of linear first order differential equations with 2L+1-term recursion formulas are discussed. It is shown that such systems can be written as a system of linear tridiagonal vector equations of dimensionL. A general method is presented by which the initial value problem can be solved by iteration. For special but physically important initial conditions the solution is given by a matrix continued fraction. The eigenvalues of the tridiagonal vector recurrence relations are obtained as the roots of aL×L determinant the elements of which are determined by a matrix continued fraction. The applicability of the method is demonstrated by calculating the eigenvalues of the laser Fokker-Planck operator.     

Pauli Operator and Aharonov–Casher Theorem¶ for Measure Valued Magnetic Fields

We define the two dimensional Pauli operator and identify its core for magnetic fields that are regular Borel measures. The magnetic field is generated by a scalar potential hence we bypass the usual A∈L ² _loc condition on the vector potential, which does not allow to consider such singular fields. We extend the Aharonov–Casher theorem for magnetic fields that are measures with finite total variation and we present a counterexample in case of infinite total variation. One of the key technical tools is a weighted L ² estimate on a singular integral operator. Received: 14 May 2001 / Accepted: 5 September 2001     

Response to the Comment: “On the computation of molecular auxiliary functions<Emphasis Type="Italic">A</Emphasis><Subscript>n</Subscript>and B<Subscript>n</Subscript>”

The Comment ‘on the computation of auxiliary functionsA _n(p) and B_n(pt)’ (FEHarris, Pramana - J. Phys. 61, C779 (2003)) is analysed in the arbitrary range of parametersn,p andpt. It is shown that our downward recursion approach forB _n(pt) in the range (n/pt) > 1 is more efficient than the well-known upward recursion method, and the upward recursion procedure forA _n(p) does not have merit for smaller non-zero values ofp (p < 001).     

Thickness and angular dependence of the L‐edge X‐ray absorption of nickel thin films

We report on the near‐edge X‐ray absorption fine structure spectroscopy of the L₃ (2p_3/2) and L₂ (2p_1/2) edges for ferromagnetic pure nickel transition metal and show that the L_2,3 edge peak intensity and satellite feature at ~6 eV above the L₃ edge in nickel increase with increasing nickel film thickness both in the total electron yield and transmission modes. The absorption spectra of nickel metal, however, exhibit strong angular‐dependent effects when measured in total electron yield mode. In addition, we calculated the mean electron escape depth of the emitted electrons (λ_e), which was found for pure nickel metal to be λ_e=25 ± 2 Å. We point out the advantages of the total electron yield technique for the study of the L‐edge of 3d transition metals. Copyright © 2011 John Wiley & Sons, Ltd.     

Andronov-Hopf bifurcations in planar, piecewise-smooth, continuous flows

An equilibrium of a planar, piecewise-C¹, continuous system of differential equations that crosses a curve of discontinuity of the Jacobian of its vector field can undergo a number of discontinuous or border-crossing bifurcations. Here we prove that if the eigenvalues of the Jacobian limit to λL±iωL on one side of the discontinuity and −λR±iωR on the other, with λL,λR>0, and the quantity Λ=λL/ωL−λR/ωR is nonzero, then a periodic orbit is created or destroyed as the equilibrium crosses the discontinuity. This bifurcation is analogous to the classical Andronov-Hopf bifurcation, and is supercritical if Λ<0 and subcritical if Λ>0.