# Mathematical Physics

## New submissions

[ total of 19 entries: 1-19 ]
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### New submissions for Thu, 22 Feb 18

[1]
Title: Relative Phase Shifts for Metaplectic Isotopies Acting on Mixed Gaussian States
Subjects: Mathematical Physics (math-ph); Functional Analysis (math.FA); Symplectic Geometry (math.SG); Quantum Physics (quant-ph)

We address in this paper the notion of relative phase shift for mixed quantum systems. We study the Pancharatnam-Sjoeqvist phase shift for metaplectic isotopies acting on Gaussian mixed states. We complete and generalize previous results obtained by one of us while giving rigorous proofs. This gives us the opportunity to review and complement the theory of the Conley-Zehnder index which plays an essential role in the determination of phase shifts.

[2]
Title: On a family of a linear maps from $M_{n}(\mathbb{C})$ to $M_{n^{2}}(\mathbb{C})$
Subjects: Mathematical Physics (math-ph); Quantum Physics (quant-ph)

Bhat characterizes the family of linear maps defined on $B(\mathcal{H})$ which preserve unitary conjugation. We generalize this idea and study the maps with a similar equivariance property on finite-dimensional matrix algebras. We show that the maps with equivariance property are significant to study $k$-positivity of linear maps defined on finite-dimensional matrix algebras. Choi showed that $n$-positivity is different from $(n-1)$-positivity for the linear maps defined on $n$ by $n$ matrix algebras. In this paper, we present a parametric family of linear maps $\Phi_{\alpha, \beta,n} : M_{n}(\mathbb{C}) \rightarrow M_{n^{2}}(\mathbb{C})$ and study the properties of positivity, completely positivity, decomposability etc. We determine values of parameters $\alpha$ and $\beta$ for which the family of maps $\Phi_{\alpha, \beta,n}$ is positive for any natural number $n \geq 3$. We focus on the case of $n=3,$ that is, $\Phi_{\alpha, \beta,3}$ and study the properties of $2$-positivity, completely positivity and decomposability. In particular, we give values of parameters $\alpha$ and $\beta$ for which the family of maps $\Phi_{\alpha, \beta,3}$ is $2$-positive and not completely positive.

[3]
Title: A note on $\mathfrak{gl}_2$-invariant Bethe vectors
Subjects: Mathematical Physics (math-ph); High Energy Physics - Theory (hep-th)

We consider $\mathfrak{gl}_2$-invariant quantum integrable models solvable by the algebraic Bethe ansatz. We show that the form of on-shell Bethe vectors is preserved under certain twist transformations of the monodromy matrix. We also derive the actions of the twisted monodromy matrix entries onto twisted off-shell Bethe vectors.

[4]
Title: From Hamiltonian to zero curvature formulation for classical integrable boundary conditions
Subjects: Mathematical Physics (math-ph); High Energy Physics - Theory (hep-th); Exactly Solvable and Integrable Systems (nlin.SI)

We reconcile the Hamiltonian formalism and the zero curvature representation in the approach to integrable boundary conditions for a classical integrable system in 1+1 space-time dimensions. We start from an ultralocal Poisson algebra involving a Lax matrix and two (dynamical) boundary matrices. Sklyanin's formula for the double-row transfer matrix is used to derive Hamilton's equations of motion for both the Lax matrix {\bf and} the boundary matrices in the form of zero curvature equations. A key ingredient of the method is a boundary version of the Semenov-Tian-Shansky formula for the generating function of the time-part of a Lax pair. The procedure is illustrated on the finite Toda chain for which we derive Lax pairs of size $2\times 2$ for previously known Hamiltonians of type $BC_N$ and $D_N$ corresponding to constant and dynamical boundary matrices respectively.

[5]
Title: Relativistic Entropy Inequality
Authors: Hans Wilhelm Alt
Journal-ref: Advances in Mathematical Sciences and Applications, Vol. 26, No. 1 (2017), pp. 243-272
Subjects: Mathematical Physics (math-ph)

In this paper we apply the entropy principle to the relativistic version of the differential equations describing a standard fluid flow, that is, the equations for mass, momentum, and a system for the energy matrix. These are the second order equations which have been introduced in [3]. Since the principle also says that the entropy equation is a scalar equation, this implies, as we show, that one has to take a trace in the energy part of the system. Thus one arrives at the relativistic mass-momentum-energy system for the fluid. In the procedure we use the well-known Liu-M\"uller sum [10] in order to deduce the Gibbs relation and the residual entropy inequality.

### Cross-lists for Thu, 22 Feb 18

[6]  arXiv:1802.07271 (cross-list from hep-th) [pdf, ps, other]
Title: Self-Dual Skyrmions on the Spheres $S^{2N+1}$
Subjects: High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph); Exactly Solvable and Integrable Systems (nlin.SI); Nuclear Theory (nucl-th)

We construct self-dual sectors for scalar field theories on a $(2N+2)$-dimensional Minkowski space-time with target space being the $2N+1$-dimensional sphere $S^{2N+1}$. The construction of such self-dual sectors is made possible by the introduction of an extra functional on the action that renders the static energy and the self-duality equations conformally invariant on the $(2N+1)$-dimensional spatial submanifold. The conformal and target space symmetries are used to build an ansatz that leads to an infinite number of exact self-dual solutions with arbitrary values of the topological charge. The five dimensional case is discussed in detail where it is shown that two types of theories admit self dual sectors. Our work generalizes the known results in the three-dimensional case that leads to an infinite set of self-dual Skyrmion solutions.

[7]  arXiv:1802.07291 (cross-list from math.PR) [pdf, ps, other]
Title: On Spin Distributions for Generic $p$-spin models
Comments: An early version of these results appeared in arXiv:1612.06359v1
Subjects: Probability (math.PR); Mathematical Physics (math-ph)

We provide an alternative formula for spin distributions of generic $p$-spin glass models. As a main application of this expression, we write spin statistics as solutions of partial differential equations and we show that the generic $p$-spin models satisfy multiscale Thouless--Anderson--Palmer equations as originally predicted in the work of M\'ezard--Virasoro [15].

[8]  arXiv:1802.07522 (cross-list from math.SP) [pdf, ps, other]
Title: Gap control by singular Schrödinger operators in a periodically structured metamaterial
Subjects: Spectral Theory (math.SP); Mathematical Physics (math-ph); Analysis of PDEs (math.AP)

We consider a family $\{\mathcal{H}_\varepsilon\}_{\varepsilon}$ of $\varepsilon\mathbb{Z}^n$-periodic Schr\"odinger operators with $\delta'$-interactions supported on a lattice of closed compact surfaces; within a minimal period cell one has $m\in\mathbb{N}$ surfaces. We show that in the limit when $\varepsilon\to 0$ and the interactions strengths are appropriately scaled, $\mathcal{H}_\varepsilon$ has at most $m$ gaps within finite intervals, and moreover, the limiting behavior of the first $m$ gaps can be completely controlled through a suitable choice of those surfaces and of the interactions strengths.

[9]  arXiv:1802.07587 (cross-list from quant-ph) [pdf, other]
Title: Attaining the ultimate precision limit in quantum state estimation
Subjects: Quantum Physics (quant-ph); Mathematical Physics (math-ph)

We derive an attainable bound on the precision of quantum state estimation for finite dimensional systems, providing a construction for the asymptotically optimal measurement. Our results hold under an assumption called local asymptotic covariance, which is weaker than unbiasedness or local unbiasedness. The derivation is based on an analysis of the limiting distribution of the estimator's deviation from the true value of the parameter, and takes advantage of quantum local asymptotic normality, a duality between sequences of identically prepared states and Gaussian states of continuous variable systems. We first prove our results for the mean square error of a special class of models, called D-invariant, and then extend the results to arbitrary models, generic cost functions, and global state estimation, where the unknown parameter is not restricted to a local neighbourhood of the true value. The extension includes a treatment of nuisance parameters, namely parameters that are not of interest to the experimenter but nevertheless affect the estimation. As an illustration of the general approach, we provide the optimal estimation strategies for the joint measurement of two qubit observables, for the estimation of qubit states in the presence of amplitude damping noise, and for noisy multiphase estimation.

[10]  arXiv:1802.07666 (cross-list from math.PR) [pdf, ps, other]
Title: Classical large deviations theorems on complete Riemannian manifolds
Subjects: Probability (math.PR); Mathematical Physics (math-ph); Differential Geometry (math.DG)

We generalize classical large deviations theorems to the setting of complete Riemannian manifolds. We prove the analogue of Mogulskii's theorem for geodesic random walks via a general approach using visocity solutions for Hamilton-Jacobi equations. As a corollary, we also obtain the analogue of Cram\'er's theorem. The approach also provides a new proof of Schilder's theorem. Additionally, we provide a proof of Schilder's theorem by using an embedding into Euclidean space, together with Freidlin-Wentzell theory.

[11]  arXiv:1802.07705 (cross-list from math.AP) [pdf, ps, other]
Title: Regularity results for a class of generalized surface quasi-geostrophic equations
Subjects: Analysis of PDEs (math.AP); Mathematical Physics (math-ph)

We show a global existence result of weak solutions for a class of generalized Surface Quasi-Geostrophic equation in the inviscid case. We also prove the global regularity of such solutions for the equation with slightly supercritical dissipation, which turns out to correspond to a logarithmically supercritical diffusion due to the singular nature of the velocity. Our last result is the eventual regularity in the supercritical cases for such weak solutions. The main idea in the proof of the existence part is based on suitable commutator estimates along with a careful cutting into low/high frequencies and inner/outer spatial scales to pass to the limit; while the proof of both the global regularity result and the eventual regularity for the supercritical diffusion are essentially based on the use of the so-called modulus of continuity method.

### Replacements for Thu, 22 Feb 18

[12]  arXiv:1511.05935 (replaced) [pdf, other]
Title: The Bogoliubov free energy functional I. Existence of minimizers and phase diagram
Comments: Published version, 54 pages, 1 figure
Subjects: Mathematical Physics (math-ph)
[13]  arXiv:1612.06359 (replaced) [pdf, ps, other]
Title: Thouless-Anderson-Palmer equations for the generic p-spin glass model
Comments: New title, introduction revised. Major changes to Section 1 and 7
Subjects: Probability (math.PR); Mathematical Physics (math-ph)
[14]  arXiv:1701.05173 (replaced) [pdf, other]
Title: Convergence of the free Boltzmann quadrangulation with simple boundary to the Brownian disk
Comments: 41 pages, 6 figures; revised
Subjects: Probability (math.PR); Mathematical Physics (math-ph); Combinatorics (math.CO)
[15]  arXiv:1708.06660 (replaced) [pdf, other]
Title: Regularized maximum pure-state input-output fidelity of a quantum channel
Journal-ref: Phys. Rev. A 96, 062319 (2017)
Subjects: Quantum Physics (quant-ph); Mathematical Physics (math-ph)
[16]  arXiv:1710.04834 (replaced) [pdf, other]
Title: Morse index for figure-eight choreographies of the planar equal mass three-body problem
Subjects: Mathematical Physics (math-ph)
[17]  arXiv:1710.09427 (replaced) [pdf, ps, other]
Title: On the nonintegrability of equations for long- and short-wave interactions
Comments: 9 pages, presented as a poster at The Tenth IMACS International Conference on Nonlinear Evolution Equations and Wave Phenomena: Computation and Theory
Subjects: Mathematical Physics (math-ph); Exactly Solvable and Integrable Systems (nlin.SI)
[18]  arXiv:1712.06093 (replaced) [pdf, ps, other]
Title: On the relation between Staruszkiewicz's quantum theory of the Coulomb field and the causal perturbative approach to QED