Degeneracy Theorem Research Articles

Kramers' degeneracy for open systems in thermal equilibrium

Kramers' degeneracy theorem underpins many interesting effects in quantum systems with time-reversal symmetry. We show that the generator of dynamics for Markovian open fermionic systems can exhibit an analogous degeneracy, protected by a combination of time-reversal symmetry and the microreversibility (detailed balance) property of systems at thermal equilibrium -- the degeneracy is lifted if either condition is not met. We provide simple examples of this phenomenon and show that the degeneracy is reflected in the single-particle Green's functions. Furthermore, we show that certain experimental signatures of topological edge modes in open many-body systems can be protected by microreversibility in the same way. Our results highlight the importance of detailed balance in characterizing open topological matter.

On the degeneracy of integral points and entire curves in the complement of nef effective divisors

As a consequence of the divisorial case of our recently established generalization of Schmidt's subspace theorem, we prove a degeneracy theorem for integral points on the complement of a union of nef effective divisors. A novel aspect of our result is the attainment of a strong degeneracy conclusion (arithmetic quasi-hyperbolicity) under weak positivity assumptions on the divisors. The proof hinges on applying our recent theorem with a well-situated ample divisor realizing a certain lexicographical minimax. We also explore the connections with earlier work by other authors and make a conjecture regarding bounds for the numbers of divisors necessary, including consideration of the question of arithmetic hyperbolicity. Under the standard correspondence between statements in Diophantine approximation and Nevanlinna theory, one obtains analogous degeneration statements for entire curves.

Geometry of logarithmic forms and deformations of complex structures

We present a new method to solve certain ∂ ¯ \bar \partial -equations for logarithmic differential forms by using harmonic integral theory for currents on Kähler manifolds. The result can be considered as a ∂ ∂ ¯ \partial \bar \partial -lemma for logarithmic forms. As applications, we generalize the result of Deligne about closedness of logarithmic forms, give geometric and simpler proofs of Deligne’s degeneracy theorem for the logarithmic Hodge to de Rham spectral sequences at E 1 E_1 -level, as well as a certain injectivity theorem on compact Kähler manifolds. Furthermore, for a family of logarithmic deformations of complex structures on Kähler manifolds, we construct the extension for any logarithmic ( n , q ) (n,q) -form on the central fiber and thus deduce the local stability of log Calabi-Yau structure by extending an iteration method to the logarithmic forms. Finally we prove the unobstructedness of the deformations of a log Calabi-Yau pair and a pair on a Calabi-Yau manifold by the differential geometric method.

Uniqueness problem for meromorphic mappings in several complex variables with few hyperplanes

The purpose of this article is to deal with uniqueness problem with truncated multiplicities for meromorphic mappings in several complex variables. We obtain a degeneracy theorem of meromorphic mappings without taking account of multiplicities of order > k in counting functions and a uniqueness theorem for meromorphic mappings sharing 2n + 2(n ≥ 2) hyperplanes in general position, which improve and extend some earlier work.

A degeneracy theorem for meromorphic mappings with truncated multiplicities

In this article, we prove a degeneracy theorem for three linearly non-degenerate meromorphic mappings from {mathbb{ C}} n into {mathbb{ P} N({mathbb{ C}}), sharing 2 N + 2 hyperplanes in general position, counted with multiplicities truncated by 2.

Kramers Degeneracy Theorem in Nonrelativistic QED

Degeneracy of the eigenvalues of the Pauli-Fierz Hamiltonian with spin 1/2 is proven by the Kramers degeneracy theorem. The Pauli-Fierz Hamiltonian at fixed total momentum is also investigated.

A degeneracy theorem for meromorphic mappings with few hyperplanes and low truncation level of multiplicities

A degeneracy theorem for meromorphic mappings with few hyperplanes and low truncation level of multiplicities

Three meromorphic mappings sharing some common hyperplanes

In this paper, using techniques of Nevanlinna theory, we give a degeneracy theorem of meromorphic mappings of C m into C P n with few hyperplanes and truncated multiplicities.

A DEGENERACY THEOREM FOR MEROMORPHIC MAPPINGS WITH MOVING TARGETS

The purpose of this article is to prove a degeneracy theorem for meromorphic mappings of ℂm into ℂPn with (2n + 2) moving targets.

The one-dimensional hydrogen atom revisited

The one-dimensional Schrödinger hydrogen atom is an interesting mathematical and physical problem for the study of bound states, eigenfunctions, and quantum-degeneracy issues. This one-dimensional physical system has given rise to some intriguing controversy for more than four decades. Presently, still no definite consensus seems to have been reached. We reanalyzed this apparently controversial problem, approaching it from a Fourier-transform representation method combined with some fundamental (basic) ideas found in self-adjoint extensions of symmetric operators. In disagreement with some previous claims, we found that the complete Balmer energy spectrum is obtained together with an odd-parity set of eigenfunctions. Closed-form solutions in both coordinate and momentum spaces were obtained. No twofold degeneracy was observed as predicted by the degeneracy theorem in one dimension, though it does not necessarily have to hold for potentials with singularities. No ground state with infinite energy exists since the corresponding eigenfunction does not satisfy the Schrödinger equation at the origin.PACS Nos.: 03.65.Ge, 03.65.–w

Monotonicity inequalities for ther-area and a degeneracy theorem forr-minimal graphs

We establish monotonicity inequalities for the r-area of a complete oriented properly immersed r-minimal hypersurface in Euclidean space under appropriate quasi-positivity assumptions on certain invariants of the immersion. The proofs are based on the corresponding first variational formula. As an application, we derive a degeneracy theorem for an entire r-minimal graph whose defining function ƒ has first and second derivatives decaying fast enough at infinity: Its Hessian operator D2 ƒ has at least n − r null eigenvalues everywhere.

Theory of fractional quantum hall effect

A theory is constructed by introducing 3-particle interaction breaking the symmetry for ν = 1 3 according to a degeneracy theorem proved here. Order parameter is introduced and a gap in the single particle spectrum is found. Critical temperature, critical filling number and critical behaviour are determined as well as Ginsburg-Landau equation coefficients. First principles calculation of the Hall current is given. 3, 5, 7 electron tunneling and Josephson interference effects are predicted.

Algebraic degeneracy theorem for holomorphic mappings into smooth projective algebraic varieties

The famous Picard theorem states that a holomorphic mapping f: C → P1(C) omitting distinct three points must be constant. Borel [1] showed that a non-degenerate holomorphic curve can miss at most n + 1 hyperplanes in Pn(C) in general position, thus extending Picard’s theorem (n = 1). Recently, Fujimoto [3], Green [4] and [5] obtained many Picard type theorems using Borel’s methods for holomorphic mappings.

Infinite-momentum limit of factorizable current algebras

A simple method of studying factorizable current algebra at infinite momentum is described, starting from the corresponding finitemomentum representations. It is shown that, for these algebras, the infinite-momentum limit of the current commutator is well defined. Other points discussed include the explicit construction of all operators at infinite momentum as functions of their finite-momentum counterparts, the derivation of the decoupled angilar contributions and the mass degeneracy theorem.