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We extend projectively equivariant quantization and symbol calculus to symbols of pseudo-differential operators. An explicit expression in terms of hypergeometric functions with noncommutative arguments is given. Some examples are worked…
Considered is quantum tunnelling in anisotropic spin systems in a magnetic field perpendicular to the anisotropy axis. In the domain of small field the problem of calculating tunnelling splitting of energy levels is reduced to constructing…
Variations in distinct restricted spaces of wave functions generate distinct density functionals. In particular, angular momentum projected Slater determinants define a new density functional, compatible simultaneously with angular momentum…
In the deformed quantum mechanics with a minimal length, one WKB connection formula through a turning point is derived. We then use it to calculate tunnelling rates through potential barriers under the WKB approximation. Finally, the…
The characterization of quantum processes is a key tool in quantum information processing tasks for several reasons: on one hand, it allows to acknowledge errors in the implementations of quantum algorithms; on the other, it allows to…
We describe how geometrical methods can be applied to a system with explicitly time-dependent second-class constraints so as to cast it in Hamiltonian form on its physical phase space. Examples of particular interest are systems which…
The operator and the functional formulations of the dynamics of constrained systems are explored for determining unambiguously the quantum Hamiltonian of a nonrelativistic particle in a curved space.
It is shown that quantization of the dynamical systems with second class constraints actually can be reduced to quantization of the systems with first class constraints. The motion of the non-relativistic particle along the plane curve and…
In this paper, we perform a rigourous version of shape and topological derivatives for optimizations problems under constraint Helmoltz problems. A shape and topological optimization problem is formulated by introducing cost functional. We…
Following the guidelines of classical differential geometry the `building material' for the tensor calculus in non-commutative geometry is suggested. The algebraic account of moduli of vectors and covectors is carried out.
For decades, mathematical physicists have searched for a coordinate independent quantization procedure to replace the ad hoc process of canonical quantization. This effort has largely coalesced into two distinct research programs: geometric…
Invariant manifolds are important constructs for the quantitative and qualitative understanding of nonlinear phenomena in dynamical systems. In nonlinear damped mechanical systems, for instance, spectral submanifolds have emerged as useful…
We consider in what sense quantum tunnelling is associated with non-classical probabilistic behaviour. We use the Wigner function quasi-probability description of quantum states. We give a definition of tunnelling that allows us to say…
Quantum hamiltonian reduction is a fundamental tool of conformal field theory and vertex algebra representation theory. It has traditionally been applied to study highest-weight modules. On the other hand, inverse quantum hamiltonian…
This article deals with a particular class of shape and topology optimization problems: the optimized design is a region $G$ of the boundary $\partial \Omega$ of a given domain $\Omega$, which supports a particular type of boundary…
We prove quantitative bounds on the eigenvalues of non-selfadjoint bounded and unbounded operators. We use the perturbation determinant to reduce the problem to one of studying the zeroes of a holomorphic function.
In this paper, the degenerate ground states of Z2 topological order on a plane with holes (the so-called surface codes) are used as the protected code subspace to build a topological quantum computer by tuning their quantum tunneling…
The scalar functional determinants on sectors of the two-dimensional disc and spherical cap are determined for arbitrary angles (rational factors of $\pi$). The wholesphere and hemisphere expressions are also given, in low dimensions, for…
We consider shape functionals obtained as minima on Sobolev spaces of classical integrals having smooth and convex densities, under mixed Dirichlet-Neumann boundary conditions. We propose a new approach for the computation of the second…
We discuss the pairs of quadratic integrals of motion belonging to the $n$-dimensional space of independent integrals of motion in involution, that provide integrability of the corresponding Hamiltonian equations of motion by quadratures.…