# Mathematical Physics

## New submissions

[ total of 35 entries: 1-35 ]
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### New submissions for Tue, 19 Sep 17

[1]
Title: Some trace inequalities for exponential and logarithmic functions
Subjects: Mathematical Physics (math-ph)

Consider a function $F(X,Y)$ of pairs of positive matrices with values in the positive matrices such that whenever $X$ and $Y$ commute $F(X,Y)= X^pY^q.$ Our first main result gives conditions on $F$ such that ${\rm Tr}[ X \log (F(Z,Y))] \leq {\rm Tr}[X(p\log X + q \log Y)]$ for all $X,Y,Z$ such that ${\rm Tr} Z = {\rm Tr} X$. (Note that $Z$ is absent from the right side of the inequality.) We give several examples of functions $F$ to which the theorem applies.
Our theorem allows us to give simple proofs of the well known logarithmic inequalities of Hiai and Petz and several new generalizations of them which involve three variables $X,Y,Z$ instead of just $X,Y$ alone. The investigation of these logarithmic inequalities is closely connected with three quantum relative entropy functionals: The standard Umegaki quantum relative entropy $D(X||Y) = {\rm Tr} [X(\log X-\log Y])$, and two others, the Donald relative entropy $D_D(X||Y)$, and the Belavkin-Stasewski relative entropy $D_{BS}(X||Y)$. They are known to satisfy $D_D(X||Y) \leq D(X||Y)\leq D_{BS}(X||Y)$. We prove that the Donald relative entropy provides the sharp upper bound, independent of $Z$, on ${\rm Tr}[ X \log (F(Z,Y))]$ in a number of cases in which $(Z,Y)$ is homogeneous of degree $1$ in $Z$ and $-1$ in $Y$. We also investigate the Legendre transforms in $X$ of $D_D(X||Y)$ and $D_{BS}(X||Y)$, and show how our results for these lead to new refinements of the Golden-Thompson inequality.

[2]
Title: Defects in the supersymmetric mKdV hierarchy via Bäcklund transformations
Subjects: Mathematical Physics (math-ph); High Energy Physics - Theory (hep-th); Exactly Solvable and Integrable Systems (nlin.SI)

The integrability of the ${\cal N}=1$ supersymmetric modified Korteweg de-Vries (smKdV) hierarchy in the presence of defects is investigated through the construction of its super B\"acklund transformation. The construction of such transformation is performed by using essentially two methods: the B\"acklund-defect matrix approach and the superfield approach. Firstly, we employ the defect matrix associated to the hierarchy which turns out to be the same for the supersymmetric sinh-Gordon (sshG) model. The method is general for all flows and as an example we derive explicitly the B\"acklund equations in components for the first few flows of the hierarchy, namely $t_3$ and $t_5$. Secondly, the supersymmetric extension of the B\"acklund transformation in the superspace formalism is constructed for those flows. Finally, this super B\"acklund transformation is employed to introduce type I defects for the supersymmetric mKdV hierarchy. Further integrability aspects by considering modified conserved quantities are derived from the defect matrix.

[3]
Title: Wavepackets in inhomogeneous periodic media: propagation through a one-dimensional band crossing
Subjects: Mathematical Physics (math-ph); Mesoscale and Nanoscale Physics (cond-mat.mes-hall); Analysis of PDEs (math.AP); Quantum Physics (quant-ph)

We consider a model of an electron in a crystal moving under the influence of an external electric field: Schroedinger's equation in one spatial dimension with a potential which is the sum of a periodic function $V$ and a smooth function $W$. We assume that the period of $V$ is much shorter than the scale of variation of $W$ and denote the ratio of these scales by $\epsilon$. We consider the dynamics of $\textit{semiclassical wavepacket}$ asymptotic (in the limit $\epsilon \downarrow 0$) solutions which are spectrally localized near to a $\textit{crossing}$ of two Bloch band dispersion functions of the periodic operator $- \frac{1}{2} \partial_z^2 + V(z)$. We show that the dynamics is qualitatively different from the case where bands are well-separated: at the time the wavepacket is incident on the band crossing, a second wavepacket is excited' which has $\textit{opposite}$ group velocity to the incident wavepacket. We then show that our result is consistent with the solution of a Landau-Zener'-type model.

[4]
Title: Sharp bound on the largest positive eigenvalue for one-dimensional Schrödinger operators
Authors: Wencai Liu
Comments: After we finished this paper, we found that Remling [The absolutely continuous spectrum of one-dimensional Schr\"odinger operators with decaying potentials, CMP 1998] has already addressed the problem and obtained a stronger result by a similar approach. So this paper is not intended for publication
Subjects: Mathematical Physics (math-ph)

Let $H=-D^2+V$ be a Schr\"odinger operator on $L^2(\mathbb{R})$, or on $L^2(0,\infty)$. Suppose the potential satisfies $\limsup_{x\to \infty}|xV(x)|=a<\infty$. We prove that $H$ admits no eigenvalue larger than $\frac{4a^2}{\pi^2}$. For any positive $a$ and $\lambda$ with $0<\lambda< \frac{4a^2}{\pi^2}$, we construct potentials $V$ such that $\limsup_{x\to \infty}|xV(x)|=a$ and the associated Sch\"rodinger operator $H=-D^2+V$ has eigenvalue $\lambda$.

[5]
Title: Continuous quasiperiodic Schrödinger operators with Gordon type potentials
Authors: Wencai Liu
Subjects: Mathematical Physics (math-ph)

Let us concern the quasi-periodic Schr\"odinger operator in the continuous case, \begin{equation*}
(Hy)(x)=-y^{\prime\prime}(x)+q(x,\omega x)y(x), \end{equation*} where $q:(\R/\Z)^2\to \R$ is piecewisely $\gamma$-H\"older continuous with respect to the second variable. Let $L(E)$ be the Lyapunov exponent of $Hy=Ey$. Define $\beta(\omega)$ as \begin{equation*}%\label{equ2}
\beta(\omega)= \limsup_{k\to \infty}\frac{-\ln ||k\omega||}{k}. \end{equation*} We prove that $H$ admits no eigenvalue in regime $\{E\in\R:L(E)<\gamma\beta(\omega)\}$.

[6]
Title: From the Adler-Moser polynomials to the polynomial tau functions of KdV
Subjects: Mathematical Physics (math-ph)

In 1978, M. Adler and J. Moser proved that there exists a unique change of variables that transforms the Adler-Moser polynomials into the polynomial tau functions of the KdV hierarchy. In this paper we exhibit this change of variables.

[7]
Title: On the instability of the essential spectrum for block Jacobi matrices
Subjects: Mathematical Physics (math-ph)

We are interested in the phenomenon of the essential spectrum instability for a class of unbounded (block) Jacobi matrices. We give a series of sufficient conditions for the matrices from certain classes to have a discrete spectrum on a half-axis of a real line. An extensive list of examples showing the sharpness of obtained results is provided.

[8]
Title: Hopf Algebras and Topological Recursion
Authors: João N. Esteves
Comments: Submitted on November 16, 2016, to the proceedings of the 2016 von Neumann Symposium on Topological Recursion and its influence in Analysis, Geometry and Topology, July 4-8 2016, Hilton Charlotte University Place, Charlotte NC USA
Subjects: Mathematical Physics (math-ph)

We first review our previous work arxiv:1503.02993 [math-ph] where we considered a model for topological recursion based on the Hopf Algebra of planar binary trees of Loday and Ronco and showed that extending this Hopf Algebra by identifying pairs of nearest neighbor leaves and thus producing graphs with loops we obtain the full recursion formula of Eynard and Orantin. Then we discuss the algebraic structure of the spaces of correlation functions in g = 0 and in g > 0. By taking a classical and a quantum product respectively we endow both spaces with a ring structure. This is an extended version of the contributed talk given at the 2016 von Neumann Symposium on Topological Recursion and its Influence in Analysis, Geometry and Topology, from 4 to 8 July 2016 at Hilton Charlotte University Place, USA.

[9]
Title: New representations for square-integrable spheroidal functions
Authors: V. N. Kovalenko (1), A. M. Puchkov (1), ((1) Saint-Petersburg State University, Russia)
Comments: 5 pages, 3 figures, proc. Days on Diffraction 2017
Subjects: Mathematical Physics (math-ph)

We discuss the solution of boundary value problems that arise after the separation of variables in the Schr\"odinger equation in oblate spheroidal coordinates. The specificity of these boundary value problems is that the singular points of the differential equation are outside the region in which the eigenfunctions are considered. This prevents the construction of eigenfunctions as a convergent series. To solve this problem, we generalized and applied the Jaffe transformation. We found the solution of the problem as trigonometric and power series in the particular case when the charge parameter is zero. The application of the obtained results to the spectral problem for the model of a quantum ring in the form of a potential well of a spheroidal shape is discussed with introducing a potential well of finite depth.

[10]
Title: A systematic method for constructing discrete Painlevé equations in the degeneration cascade of the E$_8$ group
Subjects: Mathematical Physics (math-ph); Exactly Solvable and Integrable Systems (nlin.SI)

We present a systematic and quite elementary method for constructing discrete Painlev\'e equations in the degeneration cascade for E$_8^{(1)}$. Starting from the invariant for the autonomous limit of the E$_8^{(1)}$ equation one wishes to study, the method relies on choosing simple homographies that will cast this invariant into certain judiciously chosen canonical forms. These new invariants lead to mappings the deautonomisations of which allow us to build up the entire degeneration cascade of the original mapping. We explain the method on three examples, two symmetric mappings and an asymmetric one, and we discuss the link between our results and the known geometric structure of these mappings.

[11]
Title: Kac-Ward formula and its extension to order-disorder correlators through a graph zeta function
Subjects: Mathematical Physics (math-ph)

A streamlined derivation of the Kac-Ward formula for the planar Ising model's partition function is presented and applied in relating the kernel of the Kac-Ward matrices' inverse with the correlation functions of the Ising model's order-disorder correlation functions. Used in the analysis is the formula's minor extension beyond planarity. A shortcut for both is enabled through the Bowen-Lanford graph zeta function relation.

### Cross-lists for Tue, 19 Sep 17

[12]  arXiv:1708.01727 (cross-list from gr-qc) [pdf, ps, other]
Title: SU(2) graph invariants, Regge actions and polytopes
Comments: 34 pages, many figures and many footnotes
Subjects: General Relativity and Quantum Cosmology (gr-qc); High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph)

We revisit the the large spin asymptotics of 15j symbols in terms of cosines of the 4d Euclidean Regge action, as derived by Barrett and collaborators using a saddle point approximation. We bring it closer to the perspective of area-angle Regge calculus and twisted geometries, and compute explicitly the Hessian and phase offsets. We then extend it to more general SU(2) graph invariants, showing that saddle points still exist and have a similar structure. For graphs dual to 4d polytopes we find again two distinct saddle points leading to a cosine asymptotic formula, however a conformal shape-mismatch is allowed by these configurations, and the asymptotic action is thus a generalisation of the Regge action. The allowed mismatch correspond to angle-matched twisted geometries, 3d polyhedral tessellations with adjacent faces matching areas and 2d angles, but not their diagonals. We study these geometries, identify the relevant subsets corresponding to 3d Regge data and flat polytope data, and discuss the corresponding Regge actions emerging in the asymptotics. Finally, we also provide the first numerical confirmation of the large spin asymptotics of the 15j symbol. We show that the agreement is accurate to the per cent level already at spins of order 10, and the next-to-leading order oscillates with the same frequency and same global phase.

[13]  arXiv:1709.05008 (cross-list from cond-mat.str-el) [pdf, other]
Title: Spin Hall insulators beyond the Helical Luttinger model
Subjects: Strongly Correlated Electrons (cond-mat.str-el); Mathematical Physics (math-ph)

We consider the interacting, spin conserving, extended Kane-Mele-Hubbard model, and we rigorously establish the exact quantization of the edge spin conductance and the validity of the Helical Luttinger relations for Drude weights and susceptibilities. Our analysis fully takes into account lattice effects, typically neglected in the Helical Luttinger liquid approximation, which play an essential role for universality. The proof is based on exact renormalization group methods and on a combination of lattice and emergent Ward identities, which allow to relate the emergent chiral anomaly with the finite renormalizations due to lattice corrections.

[14]  arXiv:1709.05347 (cross-list from hep-th) [pdf, ps, other]
Title: Casimir recursion relations for general conformal blocks
Authors: Petr Kravchuk
Subjects: High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph)

We study the structure of series expansions of general spinning conformal blocks. We find that the terms in these expansions are naturally expressed by means of special functions related to matrix elements of Spin(d) representations in Gelfand-Tsetlin basis, of which the Gegenbauer polynomials are a special case. We study the properties of these functions and explain how they can be computed in practice. We show how the Casimir equation in Dolan-Osborn coordinates leads to a simple one-step recursion relation for the coefficients of the series expansion of general spinning conformal block. The form of this recursion relation is determined by 6j symbols of Spin(d-1). In particular, it can be written down in closed form in d=3, d=4, for seed blocks in general dimensions, or in any other situation when the required 6j symbols can be computed. We work out several explicit examples and briefly discuss how our recursion relation can be used for efficient numerical computation of general conformal blocks.

[15]  arXiv:1709.05606 (cross-list from math.AP) [pdf, ps, other]
Title: Monotonicity of principal eigenvalue for elliptic operators with incompressible flow: A functional approach
Authors: Shuang Liu, Yuan Lou
Subjects: Analysis of PDEs (math.AP); Mathematical Physics (math-ph)

We establish the monotonicity of the principal eigenvalue $\lambda_1(A)$, as a function of the advection amplitude $A$, for the elliptic operator $L_{A}=-\mathrm{div}(a(x)\nabla)+A\mathbf{V}\cdot\nabla +c(x)$ with incompressible flow $\mathbf{V}$, subject to Dirichlet, Robin and Neumann boundary conditions. As a consequence, the limit of $\lambda_1(A)$ as $A\to \infty$ always exists and is finite for Robin boundary conditions. These results answer some open questions raised by [Berestycki, H., Hamel, F., Nadirashvili, N.: The speed of propagation for KPP-type problems. I. Periodic framework. J. Eur. Math. Soc. 7, 173-213 (2005)]. Our method relies upon some functional which is associated with principal eigenfuntions for operator $L_A$ and its adjoint operator. As a byproduct of the approach, a new min-max characterization of $\lambda_1(A)$ is given.

[16]  arXiv:1709.05653 (cross-list from cond-mat.stat-mech) [pdf, ps, other]
Title: Large deviation principles and fluctuation theorems for currents in semi-Markov processes
Authors: A. Faggionato
Subjects: Statistical Mechanics (cond-mat.stat-mech); Mathematical Physics (math-ph); Probability (math.PR)

In this short note we consider semi-Markov processes satisfying the condition of direction-time independence (Markov renewal processes). We derive large deviation principles and fluctuation theorems for the empirical current and the empirical currents along cycles. Our derivation is based on the joint LDP for the empirical measure and flow recently proved in \cite{MZ}.

[17]  arXiv:1709.05714 (cross-list from math.RT) [pdf, ps, other]
Title: The varieties of semi-conformal vectors of affine vertex operator algebras
Subjects: Representation Theory (math.RT); Mathematical Physics (math-ph)

This is a continuation of our work to understand vertex operator algebras using the geometric properties of varieties attached to vertex operator algebras. For a class of vertex operator algebras including affine vertex operator algebras associated to a finite dimensional simple Lie algebra $\mathfrak{g}$, we describe their varieties of semi-conformal vectors by some matrix equations. These matrix equations are too complicated to be solved for us. However, for affine vertex operator algebras associated to the simple Lie algebra $\mathfrak{g}$, we find the adjoint group $G$ of $\mathfrak{g}$ acts on the corresponding varieties by a natural way, which implies that such varieties should be described more clearly by studying the corresponding $G$-orbit structures. Based on above methods for general cases, as an example, considering affine vertex operator algebras associated to the Lie algebra $\mathfrak{sl}_2(\mathbb{C})$, we shall give the decompositions of $G$-orbits of varieties of their semi-conformal vectors according to different levels. Our results imply that such orbit structures depends on the levels of affine vertex operator algebras associated to a finite dimensional simple Lie algebra $\mathfrak{g}$

[18]  arXiv:1709.05724 (cross-list from math.AG) [pdf, other]
Title: A lax monoidal Topological Quantum Field Theory for representation varieties
Subjects: Algebraic Geometry (math.AG); Mathematical Physics (math-ph); Commutative Algebra (math.AC); Category Theory (math.CT)

We construct a lax monoidal Topological Quantum Field Theory that computes Deligne-Hodge polynomials of representation varieties of the fundamental group of any closed manifold into any complex algebraic group $G$. As byproduct, we obtain formulas for these polynomials in terms of homomorphisms between the space of mixed Hodge modules on $G$. The construction is developed in a categorical-theoretic framework allowing its application to other situations.

### Replacements for Tue, 19 Sep 17

[19]  arXiv:1411.6720 (replaced) [pdf, ps, other]
Title: Microformal geometry and homotopy algebras
Comments: LaTeX 2e. 45 p. Version 4: the paper was completely reworked and new results (and two new sections) included. Version 5: minor editorial changes
Subjects: Differential Geometry (math.DG); Mathematical Physics (math-ph); Symplectic Geometry (math.SG)
[20]  arXiv:1604.06639 (replaced) [pdf, other]
Title: Polychromatic Arm Exponents for the Critical Planar FK-Ising model
Authors: Hao Wu
Subjects: Probability (math.PR); Mathematical Physics (math-ph)
[21]  arXiv:1607.07282 (replaced) [pdf, ps, other]
Title: On the convergence of minimizers of singular perturbation functionals
Subjects: Analysis of PDEs (math.AP); Mathematical Physics (math-ph)
[22]  arXiv:1608.02269 (replaced) [pdf, ps, other]
Title: Combinatorial properties of symmetric polynomials from integrable vertex models in finite lattice
Authors: Kohei Motegi
Journal-ref: Journal of Mathematical Physics 58, 091703 (2017)
Subjects: Quantum Algebra (math.QA); Mathematical Physics (math-ph); Combinatorics (math.CO)
[23]  arXiv:1608.08194 (replaced) [pdf, ps, other]
Title: Homogenization of Dissipative, Noisy, Hamiltonian Dynamics
Subjects: Mathematical Physics (math-ph)
[24]  arXiv:1610.01218 (replaced) [pdf, other]
Title: Reconstruction phases in the planar three- and four-vortex problems
Comments: 45 pages, 4 figures, 1 table
Subjects: Mathematical Physics (math-ph)
[25]  arXiv:1610.06502 (replaced) [pdf, ps, other]
Title: On concentration inequalities and their applications for Gibbs measures in lattice systems
Comments: 58 pages. Some typos corrected. Section 7.4 improved (better bound in the lemma). A point in Section 8.3 was clarified about epsilon-entropy. To appear in J. Stat. Phys
Subjects: Probability (math.PR); Mathematical Physics (math-ph)
[26]  arXiv:1610.10052 (replaced) [pdf, other]
Title: Microscopic densities and Fock-Sobolev spaces
Comments: We have improved Section 4, particularly around Corollary 4.7. This is useful for the applications we have in mind
Subjects: Complex Variables (math.CV); Mathematical Physics (math-ph)
[27]  arXiv:1701.00534 (replaced) [pdf, ps, other]
Title: The Theoretical Proof for GLHUA EM Invisible Double Layer Cloak By Using GL No Scattering Modeling and Inversion
Comments: 16 pages and 144 formulas in this paper, this paper is theoretical proof of paper arXiv:1612.02857. It is academical publication for open review. Please colleagues send comments and question to Jianhua Li by email glhua@glgeo.com, or give coments in arXiv. If some colleague refer our paper, please cite it as reference
Subjects: Classical Physics (physics.class-ph); Mathematical Physics (math-ph); Optics (physics.optics)
[28]  arXiv:1701.03289 (replaced) [pdf, ps, other]
Title: Random Hermitian Matrices and Gaussian Multiplicative Chaos
Comments: Version 3: analysis of the differential identities simplified slightly. Version 4: some errors fixed
Subjects: Probability (math.PR); Mathematical Physics (math-ph)
[29]  arXiv:1704.02119 (replaced) [pdf, other]
Title: Underdamped stochastic harmonic oscillator
Subjects: Statistical Mechanics (cond-mat.stat-mech); Mathematical Physics (math-ph)
[30]  arXiv:1705.01211 (replaced) [pdf, other]
Title: On the n-body problem on surfaces of revolution
Authors: Cristina Stoica
Subjects: Exactly Solvable and Integrable Systems (nlin.SI); Mathematical Physics (math-ph)
[31]  arXiv:1705.01878 (replaced) [pdf, ps, other]
Title: Positive Hamiltonians can give purely exponential decay
Journal-ref: Phys. Rev. A 96, 010103 (2017)
Subjects: Quantum Physics (quant-ph); Mathematical Physics (math-ph)
[32]  arXiv:1705.07769 (replaced) [pdf, ps, other]
Title: On the correspondence between boundary and bulk lattice models and (logarithmic) conformal field theories
Comments: v2: 63 pp, few typos fixed, the final version in a special issue of J Phys A
Subjects: High Energy Physics - Theory (hep-th); Statistical Mechanics (cond-mat.stat-mech); Mathematical Physics (math-ph); Representation Theory (math.RT)
[33]  arXiv:1706.06361 (replaced) [pdf, other]
Title: Relativistic Collisions as Yang-Baxter maps