数学物理学术速递[1.10]
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math-ph数学物理，共计12篇
【1】 Classical and quantum walks on paths associated with exceptional Krawtchouk polynomials
标题：例外Krawtchouk多项式路径上的经典和量子行走
链接：https://arxiv.org/abs/2201.02337
备注：22 pages, 4 figures
摘要：Classical and quantum walks on some finite paths are introduced. It is shown
that these walks have explicit solutions given in terms of exceptional
Krawtchouk polynomials and their properties are explored. In particular,
fractional revival is shown to take place in the corresponding quantum walks.
【2】 Quantifying unsharpness of observables in an outcome-independent way
标题：以与结果无关的方式量化可观测对象的不锐度
链接：https://arxiv.org/abs/2201.02578
备注：11 pages, 2 figures
摘要：Recently a very beautiful measure of the unsharpness (fuzziness) of the
observables is discussed in the paper [Phys. Rev. A 104, 052227 (2021)]. The
measure which is defined in this paper is constructed via uncertainty and does
not depend on the values of the outcomes. There exist several properties of a
set of observables (e.g., incompatibility, non-disturbance) that do not depend
on the values of the outcomes. Therefore, the approach in the above-said paper
is consistent with the above-mentioned fact and is able to measure the
intrinsic unsharpness of the observables. In this work, we also quantify the
unsharpness of observables in an outcome-independent way. But our approach is
different than the approach of the above-said paper. In this work, at first, we
construct a Luder's instrument-based unsharpness measure and provide a tight
upper bound of that measure. Then we prove the monotonicity of the above-said
measure under a class of fuzzifying processes (processes that make the
observables more fuzzy). This is consistent with the resource-theoretic
framework. Then we relate our approach to the approach of the above-said paper.
Next, we try to construct an instrument-independent unsharpness measure. In
particular, we define an instrument-independent unsharpness measure and provide
a tight upper bound of that measure and then we derive the condition for the
monotonicity of the measure under a class of fuzzifying processes and prove the
monotonicity for dichotomic qubit observables. Then we show that for an unknown
measurement, the value of these measures can be determined experimentally.
Finally, we present the idea of the resource theory of the sharpness of the
observables.
【3】 The spatial Functional Renormalization Group and Hadamard states on cosmological spacetimes
标题：宇宙时空中的空间泛函重整化群和Hadamard态
链接：https://arxiv.org/abs/2201.02575
备注：59 pages, 1 figure
摘要：A spatial variant of the Functional Renormalization Group (FRG) is introduced
on (Lorentzian signature) globally hyperbolic spacetimes. Through its
perturbative expansion it is argued that such a FRG must inevitably be state
dependent and that it should be based on a Hadamard state. A concrete
implementation is presented for scalar quantum fields on flat
Friedmann-Lema\^{i}tre spacetimes. The universal ultraviolet behavior of
Hadamard states allows the flow to be matched to the one-loop renormalized flow
(where strict removal of the ultraviolet cutoff requires a tower of potentials,
one for each power of the Ricci scalar). The state-dependent infrared behavior
of the flow is investigated for States of Low Energy, which are Hadamard states
deemed to be viable vacua for a pre-inflationary period. A simple
time-dependent infrared fixed point equation (resembling that in Minkowski
space) arises for any scale factor, with analytically computable corrections
coding the non-perturbative ramifications of the Hadamard property in the
infrared.
【4】 Rethinking the ill-posedness of the spectral function reconstruction -- why is it fundamentally hard and how Artificial Neural Networks can help
标题：关于谱函数重构不适定性的再思考--为什么从根本上说是困难的以及人工神经网络是如何提供帮助的
链接：https://arxiv.org/abs/2201.02564
备注：29 pages, 6 figures
摘要：Reconstructing hadron spectral functions through Euclidean correlation
functions are of the important missions in lattice QCD calculations. However,
in a K\"allen--Lehmann(KL) spectral representation, the reconstruction is
observed to be ill-posed in practice. It is usually ascribed to the fewer
observation points compared to the number of points in the spectral function.
In this paper, by solving the eigenvalue problem of continuous KL convolution,
we show analytically that the ill-posedness of the inversion is fundamental and
it exists even for continuous correlation functions. We discussed how to
introduce regulators to alleviate the predicament, in which include the
Artificial Neural Networks(ANNs) representations recently proposed by the
Authors in [arXiv:2111.14760]. The uniqueness of solutions using ANNs
representations is manifested analytically and validated numerically.
Reconstructed spectral functions using different regularization schemes are
also demonstrated, together with their eigen-mode decomposition. We observe
that components with large eigenvalues can be reliably reconstructed by all
methods, whereas those with low eigenvalues need to be constrained by
regulators.
【5】 Solving formally the Auxiliary System of $O(N)$ Non Linear Sigma Model
标题：形式化求解$O(N)$非线性Sigma模型的辅助系统
链接：https://arxiv.org/abs/2201.02493
备注：5 pages
摘要：We show that the integrability of the $SO(N)/SO(N-1)$ Principal Chiral Model
(PCM) originates from the Pohlmeyer reduction of the $O(N)$ Non Linear Sigma
Model (NLSM). In particular, we show that the Lax pair of the PCM is related
upon redefinitions and identification of parameters to the zero curvature
condition, which is a consequence of the flatness of the enhanced space used in
the Pohlmeyer reduction. This identification provides the solution of the
auxiliary system that corresponds to an arbitrary NLSM/PCM solution.
【6】 Hardy inequalities for magnetic $p$-Laplacians
标题：磁性$p$-Laplacian的Hardy不等式
链接：https://arxiv.org/abs/2201.02482
备注：20 pages
摘要：Improved Hardy inequalities for the $p$-Laplacian due to adding magnetic
fields are established, while other expected results are stated as conjectures.
Some general $L^p$ magnetic-free Hardy inequalities in the spirit of Allegretto
and Huang [Nonlinear Anal. 32 (1998)] are also considered.
【7】 Asymptotic shape of isolated magnetic domains
标题：孤立磁畴的渐近形状
链接：https://arxiv.org/abs/2201.02384
备注：27 pages, 2 figures
摘要：We investigate the energy of an isolated magnetized domain $\Omega \subset
\mathbb{R}^n$ for $n=2,3$. In non-dimensionalized variables, the energy given
by $$ \mathcal{E}(\Omega) \ = \ \int_{\mathbb{R}^n} |\nabla \chi_{\Omega}| \ dx
+ \int_{\mathbb{R}^n} |\nabla \Phi|^2 \ dx $$ penalizes the interfacial area of
the domain as well as the energy of the corresponding magnetostatic field.
Here, the magnetostatic potential $\Phi$ is determined by $\Delta \Phi =
\partial_1 \chi_\Omega$, corresponding to uniform magnetization within the
domain. We consider the macroscopic regime $|\Omega| \rightarrow \infty$, in
which we derive compactness and $\Gamma$-limit for the cross-section of the
anisotropically rescaled configuration. The limit energy is local and the
limiting shape of minimizers can be calculated.
【8】 General rogue wave solutions to the discrete nonlinear Schrödinger equation
标题：离散非线性薛定谔方程的一般流氓波解
链接：https://arxiv.org/abs/2201.02359
备注：16 pages, 4 figures
摘要：In the present paper, we attempt to construct both the general rogue wave
solutions to the fully discrete nonlinear Schr\"odinger (fd-NLS) equation via
the KP-Toda reduction method. First, we deduce the general breather solution of
the fd-NLS equation starting from a pair of bilinear equations. We then derive
the general rogue wave solution by taking a limit to the breather solution.
【9】 Projective Embedding of Dynamical Systems: uniform mean field equations
标题：动力系统的射影嵌入：一致平均场方程
链接：https://arxiv.org/abs/2201.02355
备注：45 pages; one column; 10 figures;
摘要：We study embeddings of continuous dynamical systems in larger dimensions via
projector operators. We call this technique PEDS, projective embedding of
dynamical systems, as the stable fixed point of the dynamics are recovered via
projection from the higher dimensional space. In this paper we provide a
general definition and prove that for a particular type of projector operator
of rank-1, the uniform mean field projector, the equations of motion become a
mean field approximation of the dynamical system. While in general the
embedding depends on a specified variable ordering, the same is not true for
the uniform mean field projector. In addition, we prove that the original
stable fixed points remain stable fixed points of the dynamics, saddle points
remain saddle, but unstable fixed points become saddles.
【10】 Schwarzian derivative, Painlevé XXV-Ermakov equation and Bäcklund transformations
标题：Schwarzian导数、PainlevéXXV-Ermakov方程和Bäcklund变换
链接：https://arxiv.org/abs/2201.02267
备注：21 pages
摘要：The role of Schwarzian derivative in the study of nonlinear ordinary
differential equations is revisited. Solutions and invariances admitted by
Painlev\'e XXV-Ermakov equation, Ermakov equation and third order linear
equation in a normal form are shown to be based on solutions of the Schwarzian
equation. Starting from the Riccati equation and the second order element of
the Riccati chian as the simplest examples of linearizable equations, by
introducing a suitable change of variables, it is shown how the Schwarzian
derivative represents a key tool in the construction of solutions. Two families
of B\"acklund transformations which link the linear and nonlinear equations
under investigation are obtained. Some examples with relevant applications are
given and discussed.
【11】 Quantum computing based on complex Clifford algebras
标题：基于复Clifford代数的量子计算
链接：https://arxiv.org/abs/2201.02246
摘要：We propose to represent both $n$--qubits and quantum gates acting on them as
elements in the complex Clifford algebra defined on a complex vector space of
dimension $2n.$ We demonstrate its functionality by performing quantum
computations with several well known examples of quantum gates. We also compare
our approach with representations that use real geometric algebras.
【12】 Persistent oscillations after quantum quenches in $d$ dimensions
标题：$d$维量子猝灭后的持续振荡
链接：https://arxiv.org/abs/2107.13240
备注：None
摘要：We obtain analytical results for the time evolution of local observables in
systems undergoing quantum quenches in $d$ spatial dimensions. For homogeneous
systems we show that oscillations undamped in time occur when the state
produced by the quench includes single-quasiparticle modes and the observable
couples to those modes. In particular, a quench of the transverse field within
the ferromagnetic phase of the Ising model produces undamped oscillations of
the order parameter when $d>1$. For the more general case in which the quench
is performed only in a subregion of the whole $d$-dimensional space occupied by
the system, the time evolution occurs inside a light cone spreading away from
the boundary of the quenched region as time increases. The additional condition
for undamped oscillations is that the volume of the quenched region is
extensive in all dimensions.
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