Quantum dynamics of wave packets in a non-stationary parabolic potential and the Kramers escape rate theory

At sufficiently low temperatures, the reaction rates in solids are controlled by quantum rather than by thermal fluctuations. We solve the Schrödinger equation for a Gaussian wave packet in a non-stationary harmonic oscillator and derive simple analytical expressions for the increase of its mean energy with time induced by the time-periodic modulation. Applying these expressions to the modified Kramers theory, we demonstrate a strong increase of the rate of escape out of a potential well under the time-periodic driving, when the driving frequency of the well position equals its eigenfrequency, or when the driving frequency of the well width exceeds its eigenfrequency by a factor of ~2~2. Such regimes can be realized near localized anharmonic vibrations (LAVs), in which the amplitude of atomic oscillations greatly exceeds that of harmonic oscillations (phonons) that determine the system temperature. LAVs can be excited either thermally or by external triggering, which can result in strong catalytic effects due to amplification of the Kramers rate

Critical Temperature and Energy Gap for the BCS Equation

We derive upper and lower bounds on the critical temperature $T_c$ and the energy gap $\Xi$ (at zero temperature) for the BCS gap equation, describing spin 1/2 fermions interacting via a local two-body interaction potential $\lambda V(x)$. At weak coupling $\lambda \ll 1$ and under appropriate assumptions on $V(x)$, our bounds show that $T_c \sim A \exp(-B/\lambda)$ and $\Xi \sim C \exp(-B/\lambda)$ for some explicit coefficients $A$, $B$ and $C$ depending on the interaction $V(x)$ and the chemical potential $\mu$. The ratio $A/C$ turns out to be a universal constant, independent of both $V(x)$ and $\mu$. Our analysis is valid for any $\mu$; for small $\mu$, or low density, our formulas reduce to well-known expressions involving the scattering length of $V(x)$.Comment: RevTeX4, 23 pages. Revised version, to appear in Phys. Rev.

On the AC spectrum of one-dimensional random Schroedinger operators with matrix-valued potentials

We consider discrete one-dimensional random Schroedinger operators with decaying matrix-valued, independent potentials. We show that if the l^2-norm of this potential has finite expectation value with respect to the product measure then almost surely the Schroedinger operator has an interval of purely absolutely continuous (ac) spectrum. We apply this result to Schroedinger operators on a strip. This work provides a new proof and generalizes a result obtained by Delyon, Simon, and Souillard.Comment: (1 figure

Цифровая трансформация инструментов управления современными корпорациями: состояние и пути развития

The subject. Digital technologies have been integrated into all aspects of public life, including politics, law, finance, business, education, science, and society. Yet, digitalization exerts an even greater impact on the economy, which should prompt the State, represented by its legislative and executive bodies, to take timely action to ensure the legal regulation of diverse aspects of the digital economy. Digital transformation of the economy has redefined the approaches to the issues of legal capacity, corporate governance and management of business processes. Traditional management mechanisms are no longer competitive, unless used in conjunction with dynamically developing digital technologies.This article explores the issues related to digital legal personality of a corporation (online registration (e-residency) of corporations and the digital footprint that companies leave in public registers), digital corporate governance, and discuss the operation of digital corporations, including networked and decentralized autonomous organizations. The authors distinguish three types of digital corporate governance: remote management (exercised by human individuals), smart management (based on algorithms designed by human engineers), and artificial intelligence (AI) management (that does not require human involvement). Some tools of digital corporation management are illustrated, replacing traditional forms of management of the human cognitive system. Finally, we provide an overview of the operational characteristics of decentralized autonomous organizations.Purpose of the research. This article is devoted to the transformation of management tools for modern corporations in the digital economy. In order to comply with the Russian corporate legislation of the existing digital reality, it is necessary to develop a comprehensive scientific and legal concept of corporate governance, ensuring the balance and protection of the rights and legitimate interests of all participants in corporate relations and others related to corporate relations, as well as increasing the transparency and efficiency of corporations.Methodology. The methodology of this study was based on the following methods of scientific knowledge: general scientific empirical methods (observation (over the course of development of the use of digital technologies in corporate law), comparison (of the effectiveness of the use of digital technologies in corporate law of different countries)); methods of theoretical knowledge (analysis (of advantages and disadvantages of digitalization of individual institutions of corporate law), formal legal method (in the formulation and research of various concepts, determination of their characteristics and classification), theoretical modeling (of the prospects and areas of possible application of digital technologies in corporate law).The main results. Digitalization of corporate management is bound to increase business profitability and improve competitiveness on the market. We believe that in the coming years science will have to tackle the issues of assessing the implications of the introduction of digital technologies, determining technical, economic and legal prerequisites for their implementation, and identifying their limits. In addition, issues related to professional training / retraining of personnel capable of working with modern technologies are of importance.Conclusions. The authors came to the conclusion that the main direction of improving corporate legislation in the context of digitalization is currently the creation and provision of conditions for effective interaction between corporate actors and persons directly associated with them in the digital environment.Развитие и активное внедрение информационных технологий в современную экономику потребовало оценки трансформации сознания цифрового общества. Цифровая трансформация экономики изменила подходы к вопросу правоспособности, управления корпорациями и бизнес-процессами. Традиционные механизмы управления утрачивают свою конкурентоспособность при их использовании в отрыве от динамично развивающихся цифровых технологий.В статье исследуются вопросы, связанные с цифровой правосубъектностью корпорации (онлайн-регистрация корпораций (электронное (цифровое) резидентство корпораций), а также «цифровой отпечаток» компании в публичных реестрах), цифровым корпоративным управлением, а также деятельностью непосредственно цифровых (сетевых или децентрализованных автономных организаций) корпораций. Выделяются три типа цифрового корпоративного управления: дистанционное управление (с участием человека), смарт-управление (по алгоритмам, установленным человеком) и управление искусственным интеллектом (без участия человека). Иллюстрируются отдельные инструменты цифрового управления корпорацией, вытесняющие традиционные формы управления когнитивной системой человека. Раскрываются особенности функционирования децентрализованных автономных организаций

Lieb-Thirring inequalities for geometrically induced bound states

We prove new inequalities of the Lieb-Thirring type on the eigenvalues of Schr\"odinger operators in wave guides with local perturbations. The estimates are optimal in the weak-coupling case. To illustrate their applications, we consider, in particular, a straight strip and a straight circular tube with either mixed boundary conditions or boundary deformations.Comment: LaTeX2e, 14 page

Geometric extensions of many-particle Hardy inequalities

Certain many-particle Hardy inequalities are derived in a simple and systematic way using the so-called ground state representation for the Laplacian on a subdomain of $\mathbb{R}^n$. This includes geometric extensions of the standard Hardy inequalities to involve volumes of simplices spanned by a subset of points. Clifford/multilinear algebra is employed to simplify geometric computations. These results and the techniques involved are relevant for classes of exactly solvable quantum systems such as the Calogero-Sutherland models and their higher-dimensional generalizations, as well as for membrane matrix models, and models of more complicated particle interactions of geometric character.Comment: Revised version. 28 page

Cross-View Action Recognition from Temporal Self-Similarities

This paper concerns recognition of human actions under view changes. We explore self-similarities of action sequences over time and observe the striking stability of such measures across views. Building upon this key observation we develop an action descriptor that captures the structure of temporal similarities and dissimilarities within an action sequence. Despite this descriptor not being strictly view-invariant, we provide intuition and experimental validation demonstrating the high stability of self-similarities under view changes. Self-similarity descriptors are also shown stable under action variations within a class as well as discriminative for action recognition. Interestingly, self-similarities computed from different image features possess similar properties and can be used in a complementary fashion. Our method is simple and requires neither structure recovery nor multi-view correspondence estimation. Instead, it relies on weak geometric cues captured by self-similarities and combines them with machine learning for efficient cross-view action recognition. The method is validated on three public datasets, it has similar or superior performance compared to related methods and it performs well even in extreme conditions such as when recognizing actions from top views while using side views for training only