Theoretical and Mathematical Physics最新文献

A thermal–electrical ((3+1))-dimensional model of heating a semiconductor in an electric field is considered. For the corresponding Cauchy problem, the existence of a classical solution nonextendable in time is proved and an a priori estimate global in time is obtained.

The Clifford hierarchy plays a crucial role in the stabilizer formalism of quantum error correction and quantum computation. Apart form the zeroth level (the discrete Heisenberg–Weyl group) and the first level (the Clifford group), all other levels of the Clifford hierarchy are not groups. However, the diagonal gates at all levels do form groups, and it is desirable to characterize their generators and structures. In this paper, we study the diagonal gates at the second level of the Clifford hierarchy. For this, we introduce the notion of a (T)-gate in an arbitrary dimension, generalizing the corresponding notion in prime dimensions. By the use of the (T)-gate, we are able to completely characterize the group structures of the diagonal gates at the second level of the Clifford hierarchy in any (not necessarily prime) dimension. It turns out that the classification depends crucially on the number-theoretic nature of the dimension. The results highlight the special role of the first two primes, (2) and (3), in the prime factorization of the dimension. The (T)-gate in an arbitrary dimension, apart from its key role as a generator of the diagonal gates, may have independent interest and further applications in quantum theory.

We study the ((2+1))-dimensional nonlinear Date–Jimbo–Kashiwara–Miwa (DJKM) equation by the CK direct method. In the literature, no one has used the CK direct method to solve the DJKM equation. However, in the process of solving the DJKM equation by the CK direct method, it is almost impossible to solve for all (beta) and (z), and we therefore use a certain method to find the concrete expressions of (beta) and (z) more easily. Finally, some new one-dimensional similarity reductions and new exact solutions of the DJKM equation are obtained via a large amount of complex and tedious calculations; these solutions can contain some arbitrary functions of (t).

We consider one-parameter families of generating quantum channels. Such families are called the generating quantum dynamical mappings or the generating quantum processes. By the generating channels of composite quantum systems, we understand the channels that allow the channels of constituent subsystems, called the induced channels, to be uniquely defined. Using the criterion for generating and induced linear mappings, we study the properties of bijective quantum channels and the properties of quantum processes consisting of such channels. Using a generating quantum dynamical mapping, we naturally construct the induced dynamical mapping. We show that the properties of continuity and completely positive divisibility of generating quantum dynamical mappings are hereditary for induced dynamical mappings. As an application of the obtained results, we construct continuous completely positive evolutions. For generating quantum dynamical mappings taking values in the set of phase-damping channels, we obtain a criterion for the completely positive divisibility.

We develop the formalism of the inverse scattering problem method for the Cauchy problem for the defocusing Hirota equation with a self-consistent source. The specific feature of the considered Cauchy problem is that the solution is assumed to approach nonzero limits as the spatial variable approaches the plus and minus infinities. The purpose of the paper is to present two main steps of the formalism: first, the inverse problem for the associated linear Zakharov–Shabat system and, second, the evolution of the associated scattering data. A theorem is proved on the evolution of scattering data of a self-adjoint Zakharov–Shabat system, with the potential given by a solution of the defocusing Hirota equation with a self-consistent source.

The Hamiltonian mapping of the Dirac equation with a quantum perturbation in (2)D black hole models are investigated. We derive the Hamiltonians of the modified Rabi model for the point matter black hole and noncommutative black hole, and obtain perturbed expressions for the velocity and acceleration of the simulating Dirac particles. The fluctuation equations for the charge–current density in two types of black holes are derived based on the solutions of the Dirac equation. The results indicate that the Hamiltonians contain the correction terms with different powers of the creation and annihilation operators, and the accelerations have some additional Dirac Zitterbewegung terms. In addition, the coordinate operators in the fluctuation equations of the charge–current density have different powers. These characteristics can be used to distinguish different types of black holes in the analogical gravity models.

In the context of the connection discovered in a preceding paper between left-invariant objects (both geometric and dynamical) defined on a Lie group and the algebra of right automorphisms (the dual algebra), we consider the representation of the main geometric characteristics via this algebra and the corresponding metric form. These characteristics are shown to be constant (independent of a point) and defined only by the structure constants of the dual algebra and the coefficients of the metric form. Due to this connection, it is possible to introduce the concept of normal forms of a Lie algebra. Reducing any algebra and any metric to normal form in fact consists in reducing two quadratic forms to canonical form: first, the metric is reduced to the sum of squares of linear differential forms, and then the constant matrix characterizing the Ricci tensor is reduced to diagonal form (with the principal curvatures appearing on the diagonal). It turns out that there are only two different normal forms for three-dimensional Lie algebras, each depending on three parameters associated with three principal curvatures in the general case.

We study the problem of reconstructing a solenoidal vector field from a vortex function with a no-slip condition on the boundary of an external two-dimensional domain. A solvability criterion is obtained as a condition for the orthogonality of the vortex function to harmonic functions. We also obtain some estimates of the solution in the spaces (L_2) and (H_1).

We study a nonlocal abstract Ginzburg–Landau type equation. The equation includes variable coefficients with convolution terms and an abstract linear operator function (A) in a Fourier-type Banach space (E). For sufficiently smooth initial data, assuming growth conditions for the operator (A) and the coefficient (a), the existence and uniqueness of the solution and the (L^p) -regularity properties are established. We obtain the existence and uniqueness of the solution, and the regularity of different classes of nonlocal Ginzburg–Landau-type equations by choosing the space (E) and operator (A) that occur in a wide variety of physical systems.

We present the binary Bargmann symmetry constraint and algebro-geometric solutions for a semidiscrete integrable hierarchy with a bi-Hamiltonian structure. First, we derive a hierarchy associated with a discrete spectral problem by applying the zero-curvature equation and study its bi-Hamiltonian structure. Then, resorting to the binary Bargmann symmetry constraint for the potentials and eigenfunctions, we decompose the hierarchy into an integrable symplectic map and finite-dimensional integrable Hamiltonian systems. Moreover, with the help of the characteristic polynomial of the Lax matrix, we propose a trigonal curve encompassing two infinite points. On this trigonal curve, we introduce a stationary Baker–Akhiezer function and a meromorphic function, and analyze their asymptotic properties and divisors. Based on these preparations, we obtain algebro-geometric solutions for the hierarchy in terms of the Riemann theta function.