Lower Bounds for Eigenvalues of the Stokes Operator

In this paper, we propose a condition that can guarantee the lower bound property of the discrete eigenvalue produced by the finite element method for the Stokes operator. We check and prove this condition for four nonconforming methods and one conforming method. Hence they produce eigenvalues which are smaller than their exact counterparts.     

Explicit A Priori Bounds on Transfer Operator Eigenvalues

We provide explicit bounds on the eigenvalues of transfer operators defined in terms of holomorphic data.     

Unveiling Operator Growth Using Spin Correlation Functions

In this paper, we study non-equilibrium dynamics induced by a sudden quench of strongly correlated Hamiltonians with all-to-all interactions. By relying on a Sachdev-Ye-Kitaev (SYK)-based quench protocol, we show that the time evolution of simple spin-spin correlation functions is highly sensitive to the degree of k-locality of the corresponding operators, once an appropriate set of fundamental fields is identified. By tracking the time-evolution of specific spin-spin correlation functions and their decay, we argue that it is possible to distinguish between operator-hopping and operator growth dynamics; the latter being a hallmark of quantum chaos in many-body quantum systems. Such an observation, in turn, could constitute a promising tool to probe the emergence of chaotic behavior, rather accessible in state-of-the-art quench setups.     

Convergence of Perturbation Expansions in Fermionic Models. Part 1: Nonperturbative Bounds

An estimate on the operator norm of an abstract fermionic renormalization group map is derived. This abstract estimate is applied in another paper to construct the thermodynamic Greens functions of a two dimensional, weakly coupled fermion gas with an asymmetric Fermi curve. The estimate derived here is strong enough to control everything but the sum of all quartic contributions to the Greens functions.Research supported in part by the Natural Sciences and Engineering Research Council of Canada and the Forschungsinstitut für Mathematik, ETH Zürich.     

质量算符和五维概率分布空间的规范场理论(英文)

鉴于量子场论中普遍存在的粒子产生和湮灭,把描述场量的独立变量个数从量子力学波函数的4个常规时空坐标推广到了5个,其中第5个独立变量对应为粒子的内禀固有时,但是粒子运动的背景还是4维的常规时空。在场函数中固有时之所以可以看作为独立于常规时空坐标的变量,不仅是量子物理所特有的概率性描述语言所允许的,而且有可能是描述量子场论中广泛存在的粒子产生和湮灭现象所必需的。与此对应,在量子场论中,引入了质量算符((?)=-i(?))。由此,自由费米场在推广到五维概率分布空间和引入质量算符的基础上,根据相互作用的规范原理,引入了矢量规范相互作用和标量规范相互作用,同时所有的基本粒子的质量项都由质量算符自然地呈现。在此物理图像下,原则上基本粒子的质量应该通过求解相互作用耦合下的质量算符的本征值得到。此外,理论中存在普遍耦合的标量规范场和质量算符天然地联系在一起,有可能和引力作用对应起来。     

Tau Functions for the Dirac Operator on the Cylinder

The goal of the present paper is to calculate the determinant of the Dirac operator with a mass in the cylindrical geometry. The domain of this operator consists of functions that realize a unitary one-dimensional representation of the fundamental group of the cylinder with n marked points. The determinant represents a version of the isomonodromic -function, introduced by M. Sato, T. Miwa and M. Jimbo. It is calculated by comparison of two sections of the det*-bundle over an infinite-dimensional grassmannian. The latter is composed of the spaces of boundary values of some local solutions to the Dirac equation. The principal ingredients of the computation are the formulae for the Green function of the singular Dirac operator and for the so-called canonical basis of global solutions on the 1-punctured cylinder. We also derive a set of deformation equations satisfied by the expansion coefficients of the canonical basis in the general case and find a more explicit expression for the -function in the simplest case n=2.Acknowledgement I would like to thank A. I. Bugrij and V. N. Roubtsov for constant support and numerous stimulating discussions. I am grateful to S. Pakuliak for his lectures on the infinite-dimensional grassmannians and boson-fermion correspondence. I would also like to express my gratitude to J. Palmer, whose clear ideas made this work possible.     

Bounds on the Unstable Eigenvalue for the Asymmetric Renormalization Operator for Period Doubling

We establish rigorous bounds for the unstable eigenvalue of the period-doubling renormalization operator for asymmetric unimodal maps. Herglotz-function techniques and cone invariance ideas are used. Our result generalizes an established result for conventional period doubling.This research is supported by The Leverhulme Trust (grant number F/00144/W).     

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.     

Multiscale Basis Functions for Singular Perturbation on Adaptively Graded Meshes

We apply the multiscale basis functions for the singularly perturbed reaction-diffusion problem on adaptively graded meshes, which can provide a good balance between the numerical accuracy and computational cost. The multiscale space is built through standard finite element basis functions enriched with multiscale basis functions. The multiscale basis functions have abilities to capture originally perturbed information in the local problem, as a result, our method is capable of reducing the boundary layer errors remarkably on graded meshes, where the layer-adapted meshes are generated by a given parameter. Through numerical experiments we demonstrate that the multiscale method can acquire second order convergence in the L² norm and first order convergence in the energy norm on graded meshes, which is independent of ε. In contrast with the conventional methods, our method is much more accurate and effective.     

Well-Posedness and Uniform Bounds for a Nonlocal Third Order Evolution Operator on an Infinite Wedge

We investigate regularity and well-posedness for a fluid evolution model in the presence of a three-phase contact point. We consider a fluid evolution governed by Darcy’s Law. After linearization, we obtain a nonlocal third order operator which contains the Dirichlet-Neumann operator on the wedge with opening angle ${\epsilon > 0}$ . We show well-posedness and regularity for this linear evolution equation. In the limit of vanishing opening angle, we show the convergence of solutions to a fourth order degenerate parabolic operator, related to the thin-film equation. In the course of the analysis, we introduce and characterize a new type of sum of weighted Sobolev spaces which are suitable to capture the singular limit as ${\epsilon \to 0}$ . In particular, the nature of the problem requires the use of techniques that are adapted to the problem in the singular domain as well as the degenerate limit problem.     

Genus Two Partition and Correlation Functions for Fermionic Vertex Operator Superalgebras I

We define the partition and n-point correlation functions for a vertex operator superalgebra on a genus two Riemann surface formed by sewing two tori together. For the free fermion vertex operator superalgebra we obtain a closed formula for the genus two continuous orbifold partition function in terms of an infinite dimensional determinant with entries arising from torus Szegő kernels. We prove that the partition function is holomorphic in the sewing parameters on a given suitable domain and describe its modular properties. Using the bosonized formalism, a new genus two Jacobi product identity is described for the Riemann theta series. We compute and discuss the modular properties of the generating function for all n-point functions in terms of a genus two Szegő kernel determinant. We also show that the Virasoro vector one point function satisfies a genus two Ward identity.     

势阱中粒子能级与波函数微扰计算的代数递推公式

利用超位力定理（HVT）和Hellmann-Feynman定理（HFT）,导出了由有精确解的势阱的能级值用微扰法直接计算一维势阱的各级近似能级的普遍代数公式,并导出由能级近似值计算定态波函数近似表达式的代数公式,给出了代数公式具体应用的几个典型一维势阱实例,此法可推广到二维势阱与三维势阱的情形。     

One-Particle Distribution Functions and Thermodynamics of Crystals with Many-Body Forces. III. Perturbation Theory

The theory of crystalline state described in preceding papers is considered as a zeroth-order approximation. Statistical perturbation theory is used for the purpose of improving this approximation. The second-order corrections to the Helmholtz free energy and thermodynamic properties of strongly anharmonic crystals are derived. The properties of solid Ar, Kr, and Xe are calculated using various pair-potential functions in conjunction with the AXILROD -TELLER -MUTO three-body potential. The first quantum corrections are taken into account in the quasi-classical approximation. The result of computations are compared with the experimental data.     

A Dynamical Uncertainty Principle in von Neumann Algebras by Operator Monotone Functions

Suppose that A ₁,…,A _N are observables (selfadjoint matrices) and ρ is a state (density matrix). In this case the standard uncertainty principle, proved by Robertson, gives a bound for the quantum generalized variance, namely for det {Cov  _ρ (A _j ,A _k )}, using the commutators [A _j ,A _k ]; this bound is trivial when N is odd. Recently a different inequality of Robertson-type has been proved by the authors with the help of the theory of operator monotone functions. In this case the bound makes use of the commutators [ρ,A _j ] and is non-trivial for any N. In the present paper we generalize this new result to the von Neumann algebra case. Nevertheless the proof appears to simplify all the existing ones.     

Wave Functions for Arbitrary Operator Ordering in the de Sitter Minisuperspace Approximation

We derive exact series solutions for the Wheeler–DeWitt equation corresponding to a spatially closed Friedmann–Robertson–Walker universe with cosmological constant for arbitrary operator ordering of the scale factor of the universe. The resulting wave functions are those relevant to the approximation which has been widely used in two-dimensional minisuperspace models with an inflationary scalar field for the purpose of predicting the period of inflation which results from competing boundary condition proposals for the wave function of the universe. The problem that Vilenkin's tunneling wave function is not normalizable for general operator orderings, is shown to persist for other values of the spatial curvature, and when additional matter degrees of freedom such as radiation are included.     

An Operator Method for the Addition of Two Angular Momenta by Symmetrizing the Product of n Spin Functions

A general expression of Clebsch-Gordan coefficients is derived by means of de Broglie-like méthode de fusion: the second angular momentum is added to the first in n steps of half each, the maximum symmetric composition of the n spin functions being taken into account explicitly. The procedure is operatorial. Group-theoretically speaking the irreducible representations of the direct product of two irreducible representations of the rotation group are realized via symmetric group S_n with the aid of projection operators (equivalent to Young's symmetrizers). The decomposition of the representations of S_n therein is performed inductively by utilizing the chain of subgroups: S_n ⊃ S_n−1 … ⊃ S₂ ⊃ S₁.     

On Bosonic Magnetic Flux Operator and Bosonic Faraday Operator Formula

In the literature about mesoscopic Josephson devices the magnetic flux is considered as an operator, the fundamental commutative relation between the magnetic flux operator and the Cooper-pair charge operator is usually preengaged. In this paper we show that such a relation can be deduced from the basic Bose operators＇ commutative relation through the entangled state representation. The Faraday formula in bosonic form is then equivalent to the second Josephson equation. The current operator equation for LC mesoscopic circuit is also derived.     

Operator ordering and supersymmetry

We investigate supersymmetric quantum mechanics in curved space. In particular we discuss, in the framework of the canonical formulation, the role of supersymmetry (SUSY) and of invariance under general coordinate transformations (GCT) in solving the operator ordering ambiguity. It is shown that GCT invariance is instrumental to achieve such result. We also examine the connection with the Weyl ordering of operators and the relevance of our results for the path integral approach.