构造阶梯算符的定量及其应用

给出一个基本定理，提供了构造二阶线性常微分算符相应的阶梯算符的普遍适用的方法，利用这个定理得到量子力学中常见的本征问题的解。     

构造阶梯算符的定理及其应用

给出一个基本定理，提供了构造二阶线性常微分算符相应的阶梯算符的普遍适用的方法.利用这个定理得到量子力学中常见的本征问题的解.     

A Quantum Analogue of the First Fundamental Theorem of Classical Invariant Theory

We establish a noncommutative analogue of the first fundamental theorem of classical invariant theory. For each quantum group associated with a classical Lie algebra, we construct a noncommutative associative algebra whose underlying vector space forms a module for the quantum group and whose algebraic structure is preserved by the quantum group action. The subspace of invariants is shown to form a subalgebra, which is finitely generated. We determine generators of this subalgebra of invariants and determine their commutation relations. In each case considered, the noncommutative modules we construct are flat deformations of their classical commutative analogues. Our results are therefore noncommutative generalisations of the first fundamental theorem of classical invariant theory, which follows from our results by taking the limit as q → 1. Our method similarly leads to a definition of quantum spheres, which is a noncommutative generalisation of the classical case with orthogonal quantum group symmetry.     

Implications of Local Friendliness Violation for Quantum Causality

We provide a new formulation of the Local Friendliness no-go theorem of Bong et al. [Nat. Phys. 16, 1199 (2020)] from fundamental causal principles, providing another perspective on how it puts strictly stronger bounds on quantum reality than Bell’s theorem. In particular, quantum causal models have been proposed as a way to maintain a peaceful coexistence between quantum mechanics and relativistic causality while respecting Leibniz’s methodological principle. This works for Bell’s theorem but does not work for the Local Friendliness no-go theorem, which considers an extended Wigner’s Friend scenario. More radical conceptual renewal is required; we suggest that cleaving to Leibniz’s principle requires extending relativity to events themselves.     

Quantum Hypothesis Testing for Gaussian States: Quantum Analogues of χ2, t-, and F-Tests

We consider quantum counterparts of testing problems for which the optimal tests are the χ², t-, and F-tests. These quantum counterparts are formulated as quantum hypothesis testing problems concerning Gaussian state families, and they contain nuisance parameters, which have group symmetry. The quantum Hunt-Stein theorem removes some of these nuisance parameters, but other difficulties remain. In order to remove them, we combine the quantum Hunt-Stein theorem and other reduction methods to establish a general reduction theorem that reduces a complicated quantum hypothesis testing problem to a fundamental quantum hypothesis testing problem. Using these methods, we derive quantum counterparts of the χ², t-, and F-tests as optimal tests in the respective settings.     

静电场环路定理的非抵消法证明

在普通物理学中,"静电场环路定理"是一个重要的基本定理.与一般教科书中的证明方法--"抵消法"不同,提出了另外两种巧妙而更为简单的非抵消法的证法.     

Abel’s theorem in the noncommutative case

We define noncommutative binary forms. Using the typical representation of Hermite we prove the fundamental theorem of algebra and we derive a noncommutative Cardano formula for cubic forms. We define quantized elliptic and hyperelliptic differentials of the first kind. Following Abel we prove Abel’s theorem.     

No-broadcasting theorem and its classical counterpart

Although it is widely accepted that "no-broadcasting"-the nonclonability of quantum information-is a fundamental principle of quantum mechanics, an impossibility theorem for the broadcasting of general density matrices has not yet been formulated. In this Letter, we present a general proof for the no-broadcasting theorem, which applies to arbitrary density matrices. The proof relies on entropic considerations, and as such can also be directly linked to its classical counterpart, which applies to probabilistic distributions of statistical ensembles.     

Mathematical properties of the Navier–Stokes dynamical system for incompressible Newtonian fluids

The connection between fluid dynamics and classical statistical mechanics has motivated in the past mathematical investigations of the incompressible Navier–Stokes (NS) equations (INSE) by means of an asymptotic kinetic theory. This feature has suggested the search for possible alternative exact approaches, based on the construction of a suitable inverse kinetic theory (IKT), which can avoid the asymptotic character and the intrinsic mathematical difficulty of direct kinetic theories. In this paper the fundamental mathematical properties of the NS phase-space dynamical system underlying INSE and determined by IKT are investigated. In particular, an equivalence theorem with the INSE problem and a global existence theorem are proved to hold for the NS dynamical system.     

线性方程、叠加原理与复指数表示法

对线性微分方程作了简要介绍,并指出线性方程是叠加原理成立并得以广泛应用的根源,最后指出复指数表示法得以适用的根本原因也是线性方程．     

Generalization of the Khinchin theorem to Lévy flights

One of the most fundamental theorems in statistical mechanics is the Khinchin ergodic theorem, which links the ergodicity of a physical system with the irreversibility of the corresponding autocorrelation function. However, the Khinchin theorem cannot be successfully applied to processes with infinite second moment, in particular, to the relevant class of Lévy flights. Here, we solve this challenging problem. Namely, using the recently developed measure of dependence called Lévy correlation cascade, we derive a version of the Khinchin theorem for Lévy flights. This result allows us to verify the Boltzmann hypothesis for systems displaying Lévy-flight-type dynamics.     

About the nonlinear fluctuation-dissipation theorem for plasmas in two and three dimensions

The authors establish for the first time the fundamental nonlinear fluctuation-dissipation theorem for classical electron films trapped on liquid helium surfaces. When formulated in a reduced polarizability language, this theorem becomes identical to its three dimensional counterpart.     

Generalized Cayley-Hamilton-Newton identities

The q-generalizations of the two fundamental statements of matrix algebra — the Cayley-Hamilton theorem and the Newton relations — to the cases of quantum matrix algebras of RTT and Reflection-equation types have been obtained. We construct a family of matrix identities which we call Cayley-Hamilton-Newton identities and which underlie the characteristic identity as well as the Newton relations for the RTT and Reflection equation algebras, in the sense that both the characteristic identity and the Newton relations are direct consequences of the Cayley-Hamilton-Newton identities.     

Imaging of electron potential landscapes on Au(111)

The Hohenberg-Kohn theorem states that the ground state electron density completely determines the external potential acting on an electron system. Inspired by this fundamental theorem, we developed a novel approach to map directly the electron potential in surface systems: linear response theory applied to the total electron density as measured with scanning tunneling microscopy determines the external potential. Potential imaging is demonstrated for the s-p derived surface state on Au(111), where the "herringbone" reconstruction induces a periodic potential modulation, the details of which are revealed by our technique.     

Intersection properties of invariant manifolds in certain twist maps

We consider the spaceN ofC ² twist maps that satisfy the following requirements. The action is the sum of a purely quadratic term and a periodic potential times a constantk (hereafter called the nonlinearity). The potential restricted to the unit circle is bimodal, i.e. has one local minimum and one local maximum. The following statements are proven for maps inN with nonlinearityk large enough. The intersection of the unstable and stable invariant manifolds to the hyperbolic minimizing periodic points contains minimizing homoclinic points. Consider two finite pieces of these manifolds that connect two adjacent homoclinic minimizing points (hereafter called fundamental domains). We prove that all such fundamental domains have precisely one point in their intersection (the Single Intersection theorem). In addition, we show that limit points of minimizing points are recurrent, which implies that Aubry Mather sets (with irrational rotation number) are contained in diamonds formed by local stable and unstable manifolds of nearby minimizing periodic orbits (the Diamond Configuration theorem). Another corollary concerns the intersection of the minimax orbits with certain symmetry lines of the map.     

A remark on the energy conditions for Hawking’s area theorem

Hawking’s area theorem is a fundamental result in black hole theory that is universally associated with the null energy condition. That this condition can be weakened is illustrated by the formulation of a strengthened version of the theorem based on an energy condition that allows for violations of the null energy condition. With the semi-classical context in mind, some brief remarks pertaining to the suitability of the area theorem and its energy condition are made.     

Anomaly matching conditions and the moduli space of supersymmetric gauge theories

The structure of the moduli space of N = 1 supersymmetric gauge theories is analyzed from an algebraic geometric viewpoint. The connection between the fundamental fields of the ultraviolet theory, and the gauge-invariant composite fields of the infrared theory is explained in detail. The results are then used to prove an anomaly matching theorem. The theorem is used to study anomaly matching for supersymmetric QCD, and can explain all the known anomaly matching results for this case.     

A note on symmetric connections

In this paper we analyze a reciprocal of the fundamental theorem of Riemannian geometry. We give a condition for a symmetric connection to be locally the Levi-Civita connection of a metric. We also construct a couple of natural examples of connections on the n-dimensional torus and investigate the global problem.     

Stochastic calculus of variations

The fundamental theorem of stochastic calculus of variations is presented.     

The insolubility proof of the quantum measurement problem

Modern insolubility proofs of the measurement problem in quantum mechanics not only differ in their complexity and degree of generality, but also reveal a lack of agreement concerning the fundamental question of what constitutes such a proof. A systematic reworking of the (incomplete) 1970 Fine theorem is presented, which is intended to go some way toward clarifying the issue.