Discrete symmetries in quantum scattering

Using the discrete symmetries of the Klein—Gordon, Dirac, and Schrödinger wave equations, we obtain from one solution, considered as a function of the quantum numbers and the parameters of the potentials, three other solution. Taken together, these solutions form two complete sets of solutions of the wave equation. The coefficients of the linear relations between the functions of these sets — the connection coefficients — are related in a simple manner to the wave transmission and reflection amplitudes. By virtue of the discrete symmetries of the wave equation, the connection coefficients satisfy certain symmetry relations. We show that in a number of simple cases, the behavior of the wave function near the center of formation of an additional wave determines the amplitude of the wave that is formed at infinity.P. N. Lebedev Physics Institute, Russian Academy of Sciences. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 98, No. 1, pp. 60–79, January, 1994.     

Discrete symmetries and solitons

Landau Institute for Theoretical Physics, Moscow 117940, Russia. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 99, No. 3, pp. 537–544, June, 1994.     

Polynomial supersymmetry and dynamical symmetries in quantum mechanics

A polynomial generalization of supersymmetry in quantum mechanics in one and two dimensions is proposed. Polynomial superalgebras in one dimension are classified. In two dimensions, a detailed analysis is made for supercharges of second order with respect to derivatives and it is shown that in all cases the binomial superalgebra leads to hidden dynamical symmetry generated by the central charge.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 104, No. 3, pp. 463–478, September, 1995.     

On the realization of symmetries in quantum mechanics

The aim of this paper is to give a simple, geometric proof of Wigner’s theorem on the realization of symmetries in quantum mechanics that clarifies its relation to projective geometry. Although several proofs exist already, it seems that the relevance of Wigner’s theorem is not fully appreciated in general. It is Wigner’s theorem which allows the use of linear realizations of symmetries and therefore guarantees that, in the end, quantum theory stays a linear theory. In the present paper, we take a strictly geometrical point of view in order to prove this theorem. It becomes apparent that Wigner’s theorem is nothing else but a corollary of the fundamental theorem of projective geometry. In this sense, the proof presented here is simple, transparent and therefore accessible even to elementary treatments in quantum mechanics.     

On the realization of symmetries in quantum mechanics

The aim of this paper is to give a simple, geometric proof of Wigner’s theorem on the realization of symmetries in quantum mechanics that clarifies its relation to projective geometry. Although several proofs exist already, it seems that the relevance of Wigner’s theorem is not fully appreciated in general. It is Wigner’s theorem which allows the use of linear realizations of symmetries and therefore guarantees that, in the end, quantum theory stays a linear theory. In the present paper, we take a strictly geometrical point of view in order to prove this theorem. It becomes apparent that Wigner’s theorem is nothing else but a corollary of the fundamental theorem of projective geometry. In this sense, the proof presented here is simple, transparent and therefore accessible even to elementary treatments in quantum mechanics.     

Extended supersymmetry and hidden symmetries in one-dimensional matrix quantum mechanics

We study properties of nonlinear supersymmetry algebras realized in the one-dimensional quantum mechanics of matrix systems. Supercharges of these algebras are differential operators of a finite order in derivatives. In special cases, there exist independent supercharges realizing an (extended) supersymmetry of the same super-Hamiltonian. The extended supersymmetry generates hidden symmetries of the super-Hamiltonian. Such symmetries have been found in models with (2×2)-matrix potentials.     

Chu duality theory and coalgebraic representation of quantum symmetries

Building upon Vaughan Pratt's work on applications of Chu space theory to Stone duality, we develop a general theory of categorical dualities on the basis of Chu space theory and closure conditions, which encompasses a variety of dualities for topological spaces, convex spaces, closure spaces, and measurable spaces (some of which are new duality results on their own). It works as a general method to generate analogues of categorical dualities between frames (locales) and topological spaces beyond topology, e.g., for measurable spaces, convex spaces, and closure spaces. After establishing the Chu duality theory, we apply the state-observable duality between quantum lattices and closure spaces to coalgebraic representations of quantum symmetries, showing that the quantum symmetry groupoid fully embeds into a purely coalgebraic category, i.e., the category of Born coalgebras, which refines, through the quantum duality that follows from Chu duality theory, Samson Abramsky's fibred coalgebraic representations of quantum symmetries (which, in turn, builds upon his Chu representations of symmetries).     

Discrete spectrum of cranked quantum and elastic waveguides

The spectrum of quantum and elastic waveguides in the form of a cranked strip is studied. In the Dirichlet spectral problem for the Laplacian (quantum waveguide), in addition to well-known results on the existence of isolated eigenvalues for any angle α at the corner, a priori lower bounds are established for these eigenvalues. It is explained why methods developed in the scalar case are frequently inapplicable to vector problems. For an elastic isotropic waveguide with a clamped boundary, the discrete spectrum is proved to be nonempty only for small or close-to-π angles α. The asymptotics of some eigenvalues are constructed. Elastic waveguides of other shapes are discussed.     

Upper bounds in quantum dynamics

We develop a general method to bound the spreading of an entire wavepacket under Schrödinger dynamics from above. This method derives upper bounds on time-averaged moments of the position operator from lower bounds on norms of transfer matrices at complex energies.

This general result is applied to the Fibonacci operator. We find that at sufficiently large coupling, all transport exponents take values strictly between zero and one. This is the first rigorous result on anomalous transport.

For quasi-periodic potentials associated with trigonometric polynomials, we prove that all lower transport exponents and, under a weak assumption on the frequency, all upper transport exponents vanish for all phases if the Lyapunov exponent is uniformly bounded away from zero. By a well-known result of Herman, this assumption always holds at sufficiently large coupling. For the particular case of the almost Mathieu operator, our result applies for coupling greater than two.

     

Mean field dynamics of a quantum tracer particle interacting with a boson gas

We consider the dynamics of a heavy quantum tracer particle coupled to a non-relativistic boson field in ${\mathbb{R}}^3$. The pair interactions of the bosons are of mean-field type, with coupling strength proportional to $\frac{1}{N}$ where N is the expected particle number. Assuming that the mass of the tracer particle is proportional to N, we derive generalized Hartree equations in the limit $N\to \infty$. Moreover, we prove the global well-posedness of the associated Cauchy problem for sufficiently weak interaction potentials.     

Asymptotic analysis of a non-linear non-local integro-differential equation arising from bosonic quantum field dynamics

We introduce a one parameter family of non-linear, non-local integro-differential equations and its limit equation. These equations originate from a derivation of the linear Boltzmann equation using the framework of bosonic quantum field theory. We show the existence and uniqueness of strong global solutions for these equations, and a result of uniform convergence on every compact interval of the solutions of the one parameter family towards the solution of the limit equation.