Searcharxiv⌕ Search

arXiv subjects

Raphael Loewy

Publications and source records attributed to Raphael Loewy.

14 recordsLinked to original sources
arXiv2026

Complete hierarchical structure of the spectral bands in the Kohmoto model

We study the Kohmoto model, a family of discrete Schr\"odinger operators with Sturmian potentials depending on a frequency and a coupling constant. We prove that, for all non-vanishing coupling constants, all spectral bands admit a hierarchical structure. This structure offers a variety of applications, including a detailed description of the Kohmoto butterfly and a central step towards the resolution of the dry ten Martini problem for Sturmian Hamiltonians, which we carry out in a subsequent work.

math-ph↗
arXiv2024

The Dry Ten Martini Problem for Sturmian Hamiltonians

The dry ten Martini problem for Sturmian Hamiltonians is solved. Concretely, we prove that all the predicted spectral gaps "are there" for all the Schr\"odinger operators with Sturmian potentials and non-vanishing coupling constant. A key approach towards the solution is a representation of the spectrum as the boundary of an infinite tree. This tree is constructed using periodic approximations and encodes substantial spectral characteristics.

math-ph↗
arXiv2023

MFO Report: The dry ten Martini problem for Sturmian dynamical systems

This extended Oberwolfach report (to appear in the proceedings of the MFO Workshop 2335: Aspects of Aperiodic Order) announces the full solution to the Dry Ten Martini Problem for Sturmian Hamiltonians. Specifically, we show that all spectral gaps of Sturmian Hamiltonians (as predicted by the gap labeling theorem) are open for all nonzero couplings and all irrational rotations. We present here the proof strategy.

math-ph↗
arXiv2022

Proof of a conjecture on polynomials preserving nonnegative matrices

We consider polynomials in R[x] which map the set of nonnegative (element-wise) matrices of a given order into itself. Let n be a positive integer and define P(n)= {p in R[x] : p(A) is nonnegative (element-wise), for all A, A an n-by-n nonnegative (element-wise) matrix}. This set plays a role in the Nonnegative Inverse Eigenvalue Problem. Clark and Paparella conjectured that P(n+1) is strictly contained in P(n). We prove this conjecture.

math.RA↗
arXiv2016

On Faces of the set of Quantum Channels

A linear map $L$ from ${\mathbb C}^{n \times n}$ into ${\mathbb C}^{n \times n}$ is called a quantum channel if it is completely positive and trace preserving. The set ${\cal L}_n$ of all such quantum channels is known to be a compact convex set. While the extreme points of ${\cal L}_n$ can be characterized, not much is known about the structure of its higher dimensional faces. Using the so called Choi matrix $Z(L)$ associated with the quantum channel $L$, we compute the maximum dimension of a proper face of ${\cal L}_n$, and in addition the possible dimensions of faces generated by $L$ when $rank \ Z(L)=2 $.

quant-ph↗
arXiv2016

A Necessary Condition for the Spectrum of Nonnegative Symmetric $ 5 \times 5 $ Matrices

Let $A$ be a nonnegative symmetric $ 5 \times 5 $ matrix with eigenvalues $ λ_1 \geq λ_2 \geq λ_3 \geq λ_4 \geq λ_5 $. We show that if $ \sum_{i=1}^{5} λ_{i} \geq \frac{1}{2} λ_1 $ then $ λ_3 \leq \sum_{i=1}^{5} λ_{i} $. McDonald and Neumann showed that $ λ_1 + λ_3 + λ_4 \geq 0 $. Let $ σ= \left( λ_1, λ_2, λ_3, λ_4, λ_5 \right) $ be a list of decreasing real numbers satisfying: 1. $ \sum_{i=1}^{5} λ_{i} \geq \frac{1}{2} λ_1 $, 2. $ λ_3 \leq \sum_{i=1}^{5} λ_{i} $, 3. $ λ_1 + λ_3 + λ_4 \geq 0 $, 4. the Perron property, that is $ λ_1 = \max_{λ\in σ} \left| λ\right| $. We show that $ σ$ is the spectrum of a nonnegative symmetric $ 5 \times 5 $ matrix. Thus, we solve the symmetric nonnegative inverse eigenvalue problem for $ n = 5 $ in a region for which a solution has not been known before.

math.RA↗
arXiv2014

On the extreme points of quantum channels

Let L(m,n) denote the convex set of completely positive trace preserving operators from C^{m x m} to C^{n x n}$, i.e quantum channels. We give a necessary condition for L in L(m,n) to be an extreme point. We show that generically, this condition is also sufficient. We characterize completely the extreme points of L_(2,2) and L(3,2), i.e. quantum channels from qubits to qubits and from qutrits to qubits.

math-ph↗
arXiv2011

Asymptotic behavior of the smallest eigenvalue of matrices associated with completely even functions (mod r)

In this paper we present systematically analysis on the smallest eigenvalue of matrices associated with completely even functions (mod $r$). We obtain several theorems on the asymptotic behavior of the smallest eigenvalue of matrices associated with completely even functions (mod $r$). In particular, we get information on the asymptotic behavior of the smallest eigenvalue of the famous Smith matrices. Finally some examples are given to demonstrate the main results.

math.NT↗
arXiv2010

On the minimum rank of a graph over finite fields

In this paper we deal with two aspects of the minimum rank of a simple undirected graph $G$ on $n$ vertices over a finite field $\FF_q$ with $q$ elements, which is denoted by $\mr(\FF_q,G)$. In the first part of this paper we show that the average minimum rank of simple undirected labeled graphs on $n$ vertices over $\FF_2$ is $(1-\varepsilon_n)n$, were $\lim_{n\to\infty} \varepsilon_n=0$. In the second part of this paper we assume that $G$ contains a clique $K_k$ on $k$-vertices. We show that if $q$ is not a prime then $\mr(\FF_q,G)\le n-k+1$ for $4\le k\le n-1$ and $n\ge 5$. It is known that $\mr(\FF_q,G)\le 3$ for $k=n-2$, $n\ge 4$ and $q\ge 4$. We show that for $k=n-2$ and each $n\ge 10$ there exists a graph $G$ such that $\mr(\FF_3,G)>3$. For $k=n-3$, $n\ge 5$ and $q\ge 4$ we show that $\mr(\FF_q,G)\le 4$.

math.CO↗
arXiv2009

Maximal Exponents of K-Primitive Matrices: The Polyhedral Cone Case

Let $K$ be a proper (i.e., closed, pointed, full convex) cone in ${\Bbb R}^n$. An $n\times n$ matrix $A$ is said to be $K$-primitive if there exists a positive integer $k$ such that $A^k(K \setminus \{0 \}) \subseteq$ int $K$; the least such $k$ is referred to as the exponent of $A$ and is denoted by $γ(A)$. For a polyhedral cone $K$, the maximum value of $γ(A)$, taken over all $K$-primitive matrices $A$, is denoted by $γ(K)$. It is proved that for any positive integers $m,n, 3 \le n \le m$, the maximum value of $γ(K)$, as $K$ runs through all $n$-dimensional polyhedral cones with $m$ extreme rays, equals $(n-1)(m-1)+1$ when $m$ is even or $m$ and $n$ are both odd, and is at least $(n-1)(m-1)$ and at most $(n-1)(m-1)+1$ when $m$ is odd and $n$ is even. For the cases when $m = n, m = n+1$ or $n = 3$, the cones $K$ and the corresponding $K$-primitive matrices $A$ such that $γ(K)$ and $γ(A)$ attain the maximum value are identified up to respectively linear isomorphism and cone-equivalence modulo positive scalar multiplication.

math.DS↗
arXiv2008

The minimum rank problem over the finite field of order 2: minimum rank 3

Our main result is a sharp bound for the number of vertices in a minimal forbidden subgraph for the graphs having minimum rank at most 3 over the finite field of order 2. We also list all 62 such minimal forbidden subgraphs. We conclude by exploring how some of these results over the finite field of order 2 extend to arbitrary fields and demonstrate that at least one third of the 62 are minimal forbidden subgraphs over an arbitrary field for the class of graphs having minimum rank at most 3 in that field.

math.CO↗
arXiv2007

The inverse inertia problem for graphs

Let G be an undirected graph on n vertices and let S(G) be the set of all real symmetric n x n matrices whose nonzero off-diagonal entries occur in exactly the positions corresponding to the edges of G. The inverse inertia problem for G asks which inertias can be attained by a matrix in S(G). We give a complete answer to this question for trees in terms of a new family of graph parameters, the maximal disconnection numbers of a graph. We also give a formula for the inertia set of a graph with a cut vertex in terms of inertia sets of proper subgraphs. Finally, we give an example of a graph that is not inertia-balanced, and investigate restrictions on the inertia set of any graph.

math.CO↗
arXiv2007

The Graphs for which the Maximum Multiplicity of an Eigenvalue is Two

Characterized are all simple undirected graphs $G$ such that any real symmetric matrix that has graph $G$ has no eigenvalues of multiplicity more than 2. All such graphs are partial 2-trees (and this follows from a result for rather general fields), but only certain partial 2-trees guarantee maximum multiplicity 2. Among partial linear 2-trees, they are only those whose vertices can be covered by two "parallel" induced paths. The remaining graphs that guarantee maximum multiplicity 2 are comprised by certain identified families of "exceptional" partial 2-trees that are not linear.

math.CO↗