Mathematical Physics
New submissions
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New submissions for Fri, 29 Mar 24
- [1] arXiv:2403.18931 [pdf, other]
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Title: Local topology for periodic Hamiltonians and fuzzy toriSubjects: Mathematical Physics (math-ph); Disordered Systems and Neural Networks (cond-mat.dis-nn)
A variety of local index formulas is constructed for quantum Hamiltonians with periodic boundary conditions. All dimensions of physical space as well as many symmetry constraints are covered, notably one-dimensional systems in Class DIII as well as two- and three-dimensional systems in Class AII. The constructions are based on several periodic variations of the spectral localizer and are rooted in the existence of underlying fuzzy tori. For these latter, a general invariant theory is developed.
- [2] arXiv:2403.18942 [pdf, other]
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Title: Transfer matrix analysis of non-hermitian Hamiltonians: asymptotic spectra and topological eigenvaluesSubjects: Mathematical Physics (math-ph); Disordered Systems and Neural Networks (cond-mat.dis-nn)
Transfer matrix techniques are used to {provide} a new proof of Widom's results on the asymptotic spectral theory of finite block Toeplitz matrices. Furthermore, a rigorous treatment of {the} skin effect, spectral outliers, the generalized Brillouin zone and the bulk-boundary correspondence in such systems is given. This covers chiral Hamiltonians with topological eigenvalues close to zero, but no line-gap.
- [3] arXiv:2403.18944 [pdf, ps, other]
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Title: The generators of the K-groups of the sphereJournal-ref: Expositiones Mathematicae 41, 125519 (2023)Subjects: Mathematical Physics (math-ph)
This note presents an elementary iterative construction of the generators for the complex $K$-groups $K_i(C(\SM^d))$ of the $d$-dimensional spheres. These generators are explicitly given as the restrictions of Dirac or Weyl Hamiltonians to the unit sphere. Connections to solid state physics are briefly elaborated on.
- [4] arXiv:2403.18948 [pdf, ps, other]
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Title: Topological indices in condensed matterAuthors: Hermann Schulz-BaldesComments: to appear in Encyclopedia of Mathematical PhysicsSubjects: Mathematical Physics (math-ph)
This contribution describes the mathematical theory of topological indices in solid state systems composed of non-interacting Fermions. In particular, this covers the spectral localizer and the bulk-boundary correspondence.
- [5] arXiv:2403.19023 [pdf, ps, other]
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Title: Two-sided Lieb-Thirring boundsComments: 29 pages. Comments are welcome!Subjects: Mathematical Physics (math-ph); Spectral Theory (math.SP)
We prove upper and lower bounds for the number of eigenvalues of semi-bounded Schr\"odinger operators in all spatial dimensions. As a corollary, we obtain two-sided estimates for the sum of the negative eigenvalues of atomic Hamiltonians with Kato potentials. Instead of being in terms of the potential itself, as in the usual Lieb-Thirring result, the bounds are in terms of the landscape function, also known as the torsion function, which is a solution of $(-\Delta + V +M)u_M =1$ in $\mathbb{R}^d$; here $M\in\mathbb{R}$ is chosen so that the operator is positive. We further prove that the infimum of $(u_M^{-1} - M)$ is a lower bound for the ground state energy $E_0$ and derive a simple iteration scheme converging to $E_0$.
- [6] arXiv:2403.19045 [pdf, ps, other]
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Title: Sister Celine's polynomials in the quantum theory of angular momentumAuthors: Jean-Christophe PainSubjects: Mathematical Physics (math-ph); Quantum Physics (quant-ph)
The polynomials introduced by Sister Celine cover different usual orthogonal polynomials as special cases. Among them, the Jacobi and discrete Hahn polynomials are of particular interest for the quantum theory of angular momentum. In this note, we show that characters of irreducible representations of the rotation group as well as Wigner rotation "d" matrices, can be expressed as Sister Celine's polynomials. Since many relations were proposed for the latter polynomials, such connections could lead to new identities for quantities important in quantum mechanics and atomic physics.
- [7] arXiv:2403.19618 [pdf, ps, other]
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Title: What is Ballistic Transport?Subjects: Mathematical Physics (math-ph); Spectral Theory (math.SP)
In this article, we review some notions of ballistic transport from the mathematics and physics literature, describe their basic interrelations, and contrast them with other commonly studied notions of wave packet spread.
Cross-lists for Fri, 29 Mar 24
- [8] arXiv:2403.18854 (cross-list from math.AP) [pdf, ps, other]
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Title: Homogenization and continuum limit of mechanical metamaterialsSubjects: Analysis of PDEs (math.AP); Mathematical Physics (math-ph)
When used in bulk applications, mechanical metamaterials set forth a multiscale problem with many orders of magnitude in scale separation between the micro and macro scales. However, mechanical metamaterials fall outside conventional homogenization theory on account of the flexural, or bending, response of their members, including torsion. We show that homogenization theory, based on calculus of variations and notions of Gamma-convergence, can be extended to account for bending. The resulting homogenized metamaterials exhibit intrinsic generalized elasticity in the continuum limit. We illustrate these properties in specific examples including two-dimensional honeycomb and three-dimensional octet-truss metamaterials.
- [9] arXiv:2403.18927 (cross-list from quant-ph) [pdf, other]
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Title: Optimal Coherent Quantum Phase Estimation via TaperingComments: 23 pages, 6 figuresSubjects: Quantum Physics (quant-ph); Mathematical Physics (math-ph)
Quantum phase estimation is one of the fundamental primitives that underpins many quantum algorithms, including quantum amplitude estimation, the HHL algorithm for solving linear systems of equations, and quantum principal component analysis. Due to its significance as a subroutine, in this work, we study the coherent version of the phase estimation problem, where given an arbitrary input state and black-box access to unitaries $U$ and controlled-$U$, the goal is to estimate the phases of $U$ in superposition. Unlike most existing phase estimation algorithms, which employ intermediary measurements steps that inevitably destroy coherence, only a couple of algorithms, including the well-known standard quantum phase estimation algorithm, consider this coherent setting. In this work, we propose an improved version of this standard algorithm that utilizes tapering/window functions. Our algorithm, which we call tapered quantum phase estimation algorithm, achieves the optimal query complexity (total number of calls to $U$ and controlled-$U$) without requiring the use of a computationally expensive quantum sorting network for median computation, which the standard algorithm uses to boost the success probability arbitrarily close to one. We also show that the tapering functions that we use are optimal by formulating optimization problems with different optimization criteria. Beyond the asymptotic regime, we also provide non-asymptotic query complexity of our algorithm, as it is crucial for practical implementation. Finally, we also propose an efficient algorithm that prepares the quantum state corresponding to the optimal tapering function.
- [10] arXiv:2403.18977 (cross-list from math.AP) [pdf, ps, other]
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Title: Semiclassical wave-packets for weakly nonlinear Schrödinger equations with rotationComments: 13 pagesSubjects: Analysis of PDEs (math.AP); Mathematical Physics (math-ph)
We consider semiclassically scaled, weakly nonlinear Schr\"odinger equations with external confining potentials and additional angular-momentum rotation term. This type of model arises in the Gross-Pitaevskii theory of trapped, rotating quantum gases. We construct asymptotic solutions in the form of semiclassical wave-packets, which are concentrated in both space and in frequency around an classical Hamiltonian phase-space flow. The rotation term is thereby seen to alter this flow, but not the corresponding classical action.
- [11] arXiv:2403.19030 (cross-list from physics.class-ph) [pdf, other]
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Title: Symmetry criteria for the equality of interior and exterior shape factorsSubjects: Classical Physics (physics.class-ph); Mathematical Physics (math-ph); Classical Analysis and ODEs (math.CA)
Lienhard (2019) reported that the shape factor of the interior of a simply-connected region ($\Omega$) is equal to that of its exterior ($\mathbb{R}^2\backslash\Omega$) under the same boundary conditions. In that study, numerical examples supported the claim in particular cases; for example, it was shown that for certain boundary conditions on circles and squares, the conjecture holds. In the present paper, we show that the conjecture is not generally true, unless some additional condition is met. We proceed by elucidating why the conjecture does in fact hold in all of the examples analysed by Lienhard. We thus deduce a simple criterion which, when satisfied, ensures the equality of interior and exterior shape factors in general. Our criterion notably relies on a beautiful and little-known symmetry method due to Hersch (1982) which we introduce in a tutorial manner.
- [12] arXiv:2403.19033 (cross-list from math.AP) [pdf, ps, other]
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Title: Carleman factorization of layer potentials on smooth domainsSubjects: Analysis of PDEs (math.AP); Mathematical Physics (math-ph); Functional Analysis (math.FA)
One of the unexplored benefits of studying layer potentials on smooth, closed hypersurfaces of Euclidean space is the factorization of the Neumann-Poincar\'e operator into a product of two self-adjoint transforms. Resurrecting some pertinent indications of Carleman and M. G. Krein, we exploit this grossly overlooked structure by confining the spectral analysis of the Neumann-Poincar\'e operator to the amenable $L^2$-space setting, rather than bouncing back and forth the computations between Sobolev spaces of negative or positive fractional order. An enhanced, fresh new look at symmetrizable linear transforms enters into the picture in the company of geometric-microlocal analysis techniques. The outcome is manyfold, complementing recent advances on the theory of layer potentials, in the smooth boundary setting.
- [13] arXiv:2403.19048 (cross-list from cond-mat.mtrl-sci) [pdf, other]
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Title: Phase-space representation of coherent states generated through SUSY QM for tilted anisotropic Dirac materialsComments: 14 pages, 8 figuresSubjects: Materials Science (cond-mat.mtrl-sci); Mathematical Physics (math-ph); Quantum Physics (quant-ph)
In this paper, we examine the electron interaction within tilted anisotropic Dirac materials when subjected to external electric and magnetic fields possessing translational symmetry. Specifically, we focus on a distinct non-zero electric field magnitude, enabling the separation of the differential equation system inherent in the eigenvalue problem. Subsequently, employing supersymmetric quantum mechanics facilitates the determination of eigenstates and eigenvalues corresponding to the Hamiltonian operator. To delve into a semi-classical analysis of the system, we identify a set of coherent states. Finally, we assess the characteristics of these states using fidelity and the phase-space representation through the Wigner function.
- [14] arXiv:2403.19055 (cross-list from math.SP) [pdf, other]
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Title: Computing the spectrum and pseudospectrum of infinite-volume operators from local patchesSubjects: Spectral Theory (math.SP); Mathematical Physics (math-ph); Numerical Analysis (math.NA)
We show how the spectrum of normal discrete short-range infinite-volume operators can be approximated with two-sided error control using only data from finite-sized local patches. As a corollary, we prove the computability of the spectrum of such infinite-volume operators with the additional property of finite local complexity and provide an explicit algorithm. Such operators appear in many applications, e.g. as discretizations of differential operators on unbounded domains or as so-called tight-binding Hamiltonians in solid state physics. For a large class of such operators, our result allows for the first time to establish computationally also the absence of spectrum, i.e. the existence and the size of spectral gaps. We extend our results to the $\varepsilon$-pseudospectrum of non-normal operators, proving that also the pseudospectrum of such operators is computable.
- [15] arXiv:2403.19090 (cross-list from math.NA) [pdf, other]
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Title: A Stabilized Physics Informed Neural Networks Method for Wave EquationsSubjects: Numerical Analysis (math.NA); Mathematical Physics (math-ph)
In this article, we propose a novel Stabilized Physics Informed Neural Networks method (SPINNs) for solving wave equations. In general, this method not only demonstrates theoretical convergence but also exhibits higher efficiency compared to the original PINNs. By replacing the $L^2$ norm with $H^1$ norm in the learning of initial condition and boundary condition, we theoretically proved that the error of solution can be upper bounded by the risk in SPINNs. Based on this, we decompose the error of SPINNs into approximation error, statistical error and optimization error. Furthermore, by applying the approximating theory of $ReLU^3$ networks and the learning theory on Rademacher complexity, covering number and pseudo-dimension of neural networks, we present a systematical non-asymptotic convergence analysis on our method, which shows that the error of SPINNs can be well controlled if the number of training samples, depth and width of the deep neural networks have been appropriately chosen. Two illustrative numerical examples on 1-dimensional and 2-dimensional wave equations demonstrate that SPINNs can achieve a faster and better convergence than classical PINNs method.
- [16] arXiv:2403.19145 (cross-list from math.RT) [pdf, ps, other]
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Title: The Cartan-Helgason theorem for supersymmetric spaces: spherical weights for Kac-Moody superalgebrasAuthors: Alexander ShermanComments: 21 pages; comments welcome!Subjects: Representation Theory (math.RT); Mathematical Physics (math-ph)
Let $(\mathfrak{g},\mathfrak{k})$ be a supersymmetric pair arising from a finite-dimensional, symmetrizable Kac-Moody superalgebra $\mathfrak{g}$. An important branching problem is to determine the finite-dimensional highest-weight $\mathfrak{g}$-modules which admit a $\mathfrak{k}$-coinvariant, and thus appear as functions in a corresponding supersymmetric space $\mathcal{G}/\mathcal{K}$. This is the super-analogue of the Cartan-Helgason theorem. We solve this problem via a rank one reduction and an understanding of reflections in singular roots, which generalize odd reflections in the theory of Kac-Moody superalgebras. An explicit presentation of spherical weights is provided for every pair when $\mathfrak{g}$ is indecomposable.
- [17] arXiv:2403.19157 (cross-list from math.PR) [pdf, other]
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Title: Correlation functions between singular values and eigenvaluesSubjects: Probability (math.PR); Mathematical Physics (math-ph); Statistics Theory (math.ST)
Exploiting the explicit bijection between the density of singular values and the density of eigenvalues for bi-unitarily invariant complex random matrix ensembles of finite matrix size we aim at finding the induced probability measure on $j$ eigenvalues and $k$ singular values that we coin $j,k$-point correlation measure. We fully derive all $j,k$-point correlation measures in the simplest cases for one- and two-dimensional matrices. For $n>2$, we find a general formula for the $1,1$-point correlation measure. This formula reduces drastically when assuming the singular values are drawn from a polynomial ensemble, yielding an explicit formula in terms of the kernel corresponding to the singular value statistics. These expressions simplify even further when the singular values are drawn from a P\'{o}lya ensemble and extend known results between their eigenvalue and singular value statistics.
- [18] arXiv:2403.19538 (cross-list from hep-th) [pdf, other]
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Title: Superintegrability of the monomial Uglov matrix modelComments: 42 pagesSubjects: High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph)
In this paper we propose a resolution to the problem of $\beta$-deforming the non-Gaussian monomial matrix models. The naive guess of substituting Schur polynomials with Jack polynomials does not work in that case, therefore, we are forced to look for another basis for superintegrability. We find that the relevant symmetric functions are given by Uglov polynomials, and that the integration measure should also be deformed. The measure appears to be related to the Uglov limit as well, when the quantum parameters $(q,t)$ go to a root of unity. The degree of the root must be equal to the degree of the potential. One cannot derive these results directly, for example, by studying Virasoro constraints. Instead, we use the recently developed techniques of $W$-operators to arrive at the root of unity limit. From the perspective of matrix models this new example demonstrates that even with a rather nontrivial integration measure one can find a superintegrability basis by studying the hidden symmetry of the moduli space of deformations.
- [19] arXiv:2403.19566 (cross-list from math.DS) [pdf, ps, other]
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Title: Level-2 IFS Thermodynamic Formalism: Gibbs probabilities in the space of probabilities and the push-forward mapSubjects: Dynamical Systems (math.DS); Statistical Mechanics (cond-mat.stat-mech); Mathematical Physics (math-ph); Probability (math.PR)
We will denote by $\mathcal{M}$ the space of Borel probabilities on the symbolic space $\Omega=\{1,2...,m\}^\mathbb{N}$. $\mathcal{M}$ is equipped Monge-Kantorovich metric. We consider here the push-forward map $\mathfrak{T}:\mathcal{M} \to \mathcal{M}$ as a dynamical system. The space of Borel probabilities on $\mathcal{M}$ is denoted by $\mathfrak{M}$. Given a continuous function $A: \mathcal{M}\to \mathbb{R}$, an {\it a priori} probability $\Pi_0$ on $\mathcal{M}$, and a certain convolution operation acting on pairs of probabilities on $\mathcal{M}$, we define an associated Level-2 IFS Ruelle operator. We show the existence of an eigenfunction and an eigenprobability $\hat{\Pi}\in\mathfrak{M}$ for such an operator. Under a normalization condition for $A$, we show the existence of some $\mathfrak{T}$-invariant probabilities $\hat{\Pi}\in\mathfrak{M}.$ We are able to define the variational entropy of such $\hat{\Pi}$ and a related maximization pressure problem associated to $A$. In some particular examples, we show how to get eigenprobabilities solutions on $\mathfrak{M}$ for the Level-2 Thermodynamic Formalism problem from eigenprobabilities on $\mathcal{M}$ for the classical (Level-1) Thermodynamic Formalism. These examples highlight the fact that our approach is a natural generalization of the classic case.
- [20] arXiv:2403.19644 (cross-list from math.PR) [pdf, ps, other]
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Title: Gaussian statistics for left and right eigenvectors of complex non-Hermitian matricesComments: 46 pagesSubjects: Probability (math.PR); Mathematical Physics (math-ph)
We consider a constant-size subset of left and right eigenvectors of an $N\times N$ i.i.d. complex non-Hermitian matrix associated with the eigenvalues with pairwise distances at least $N^{-\frac12+\epsilon}$. We show that arbitrary constant rank projections of these eigenvectors are Gaussian and jointly independent.
Replacements for Fri, 29 Mar 24
- [21] arXiv:2005.14445 (replaced) [pdf, ps, other]
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Title: Higher Complex Structures and Flat ConnectionsAuthors: Alexander ThomasComments: The previous version of the paper has been completely revisedSubjects: Differential Geometry (math.DG); Mathematical Physics (math-ph)
- [22] arXiv:2012.09767 (replaced) [pdf, ps, other]
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Title: On microlocalisation and the construction of Feynman Propagators for normally hyperbolic operatorsComments: 48 pages; typos and inaccuracies have been corrected and some details in two instances have been provided. To be published in the Communications in Analysis and GeometrySubjects: Analysis of PDEs (math.AP); Mathematical Physics (math-ph); Differential Geometry (math.DG)
- [23] arXiv:2205.09141 (replaced) [pdf, other]
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Title: Topological phases of unitary dynamics: Classification in Clifford categoryAuthors: Jeongwan HaahComments: 48 pages (v2) minor revision in Sec.5.7Subjects: Mathematical Physics (math-ph); Strongly Correlated Electrons (cond-mat.str-el); Quantum Physics (quant-ph)
- [24] arXiv:2211.13076 (replaced) [pdf, ps, other]
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Title: Birkhoff normal form in low regularity for the nonlinear quantum harmonic oscillatorAuthors: Charbella Abou Khalil (LMJL)Subjects: Analysis of PDEs (math.AP); Mathematical Physics (math-ph)
- [25] arXiv:2305.11799 (replaced) [pdf, ps, other]
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Title: Estimates for the lowest Neumann eigenvalues of parallelograms and domains of constant widthComments: 18 pages, 1 figure. Minor edits and updated list of references. To appear in "Analysis and Mathematical Physics"Subjects: Spectral Theory (math.SP); Mathematical Physics (math-ph); Analysis of PDEs (math.AP)
- [26] arXiv:2307.12583 (replaced) [pdf, ps, other]
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Title: Maximum of the Gaussian interface model in random external fieldsAuthors: Hironobu SakagawaSubjects: Probability (math.PR); Mathematical Physics (math-ph)
- [27] arXiv:2310.11064 (replaced) [pdf, other]
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Title: On currents in the $O(n)$ loop modelComments: 47 pages, v2: clarifications on a few points and improved presentationSubjects: High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph)
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