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Integrability, non-integrability, chaos and control in classical and quantum mechanics

About The Project

The main aim of the project is a systematic analysis of solvability, integrability and control in quantum mechanics. The fundamental problem of solvability/integrability was raised from the beginning of quantum mechanics and now many solvable systems have been found by various techniques, usually based on specific properties of a given system, e.g., its symmetries suggesting a certain clever Ansatz for general solutions.

**For classical systems solvability (in quadratures) is related directly to such notions as**

1. integrability, understood as the existence of an appropriate number of conserved quantities

(first integrals, symmetries, etc.),

2. control problems for systems, or, more precisely, for Hamiltonians governing them,

3. the absence of chaos.

The quantum counterparts of these notions cause some fundamental problems already at the level of their definitions, but solvability itself seems to be the most amenable to definition for the quantum case and further to become a solid basis for the manifold concepts listed above. For this reason, it is very important to make solvability analysis as systematic as possible and perform it for wide classes of quantum systems.

We aim at obtaining general results and gaining a better understanding of the general mechanisms related to quantum solvability. To achieve this goal, we propose a novel approach to the solvability problem. Equations of quantum mechanics are linear differential equations with variable coefficients, possibly in an infinte-dimensional Hilbert space. To analyze solvability of such equations the powerful tools of differential algebra seem to be the most natural, but they have not been used widely enough in previous investigations of this topic.

The project consists of researchers and students from various institutions: The Center for Theoretical Physics of the Polish Acedemy of Science (host institution,** head Prof. dr hab. M. Kuś**): Faculty of Physics and Astronomy of University of Zielona Góra; Institute of Physics of N. Copernicus University and Institut National des Sciences Appliquées de Rouen. At every of these institutions some Postdoctoral Research Posts and/or PhD positions are available.

Integrability, non-integrability, chaos and control in classical and quantum mechanics

About The Project

The main aim of the project is a systematic analysis of solvability, integrability and control in quantum mechanics. The fundamental problem of solvability/integrability was raised from the beginning of quantum mechanics and now many solvable systems have been found by various techniques, usually based on specific properties of a given system, e.g., its symmetries suggesting a certain clever Ansatz for general solutions.

**For classical systems solvability (in quadratures) is related directly to such notions as**

1. integrability, understood as the existence of an appropriate number of conserved quantities

(first integrals, symmetries, etc.),

2. control problems for systems, or, more precisely, for Hamiltonians governing them,

3. the absence of chaos.

The quantum counterparts of these notions cause some fundamental problems already at the level of their definitions, but solvability itself seems to be the most amenable to definition for the quantum case and further to become a solid basis for the manifold concepts listed above. For this reason, it is very important to make solvability analysis as systematic as possible and perform it for wide classes of quantum systems.

We aim at obtaining general results and gaining a better understanding of the general mechanisms related to quantum solvability. To achieve this goal, we propose a novel approach to the solvability problem. Equations of quantum mechanics are linear differential equations with variable coefficients, possibly in an infinte-dimensional Hilbert space. To analyze solvability of such equations the powerful tools of differential algebra seem to be the most natural, but they have not been used widely enough in previous investigations of this topic.

The project consists of researchers and students from various institutions: The Center for Theoretical Physics of the Polish Acedemy of Science (host institution,** head Prof. dr hab. M. Kuś**): Faculty of Physics and Astronomy of University of Zielona Góra; Institute of Physics of N. Copernicus University and Institut National des Sciences Appliquées de Rouen. At every of these institutions some Postdoctoral Research Posts and/or PhD positions are available.