We prove that the Hausdorff dimension of the spectrum of a discrete Schr"odinger operator with Sturmian potential of bounded type tends to one as coupling tends to zero. The proof is based on the trace map formalism.
{"title":"On the spectrum of Sturmian Hamiltonians of bounded type in a small coupling regime","authors":"Alexandro Luna","doi":"arxiv-2408.01637","DOIUrl":"https://doi.org/arxiv-2408.01637","url":null,"abstract":"We prove that the Hausdorff dimension of the spectrum of a discrete\u0000Schr\"odinger operator with Sturmian potential of bounded type tends to one as\u0000coupling tends to zero. The proof is based on the trace map formalism.","PeriodicalId":501373,"journal":{"name":"arXiv - MATH - Spectral Theory","volume":"25 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141943889","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Emily B. Dryden, Carolyn Gordon, Javier Moreno, Julie Rowlett, Carlos Villegas-Blas
We explore the Steklov eigenvalue problem on convex polygons, focusing mainly on the inverse Steklov problem. Our primary finding reveals that, for almost all convex polygonal domains, there exist at most finitely many non-congruent domains with the same Steklov spectrum. Moreover, we obtain explicit upper bounds for the maximum number of mutually Steklov isospectral non-congruent polygonal domains. Along the way, we obtain isoperimetric bounds for the Steklov eigenvalues of a convex polygon in terms of the minimal interior angle of the polygon.
{"title":"The Steklov spectrum of convex polygonal domains I: spectral finiteness","authors":"Emily B. Dryden, Carolyn Gordon, Javier Moreno, Julie Rowlett, Carlos Villegas-Blas","doi":"arxiv-2408.01529","DOIUrl":"https://doi.org/arxiv-2408.01529","url":null,"abstract":"We explore the Steklov eigenvalue problem on convex polygons, focusing mainly\u0000on the inverse Steklov problem. Our primary finding reveals that, for almost\u0000all convex polygonal domains, there exist at most finitely many non-congruent\u0000domains with the same Steklov spectrum. Moreover, we obtain explicit upper\u0000bounds for the maximum number of mutually Steklov isospectral non-congruent\u0000polygonal domains. Along the way, we obtain isoperimetric bounds for the\u0000Steklov eigenvalues of a convex polygon in terms of the minimal interior angle\u0000of the polygon.","PeriodicalId":501373,"journal":{"name":"arXiv - MATH - Spectral Theory","volume":"34 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141943891","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Christoph Fischbacher, Fritz Gesztesy, Paul Hagelstein, Lance Littlejohn
We use a model operator approach and the spectral theorem for self-adjoint operators in a Hilbert space to derive the basic results of abstract left-definite theory in a straightforward manner. The theory is amply illustrated with a variety of concrete examples employing scales of Hilbert spaces, fractional Sobolev spaces, and domains of (strictly) positive fractional powers of operators, employing interpolation theory. In particular, we explicitly describe the domains of positive powers of the harmonic oscillator operator in $L^2(mathbb{R})$ $big($and hence that of the Hermite operator in $L^2big(mathbb{R}; e^{-x^2}dx)big)big)$ in terms of fractional Sobolev spaces, certain commutation techniques, and positive powers of (the absolute value of) the operator of multiplication by the independent variable in $L^2(mathbb{R})$.
{"title":"Abstract Left-Definite Theory: A Model Operator Approach, Examples, Fractional Sobolev Spaces, and Interpolation Theory","authors":"Christoph Fischbacher, Fritz Gesztesy, Paul Hagelstein, Lance Littlejohn","doi":"arxiv-2408.01514","DOIUrl":"https://doi.org/arxiv-2408.01514","url":null,"abstract":"We use a model operator approach and the spectral theorem for self-adjoint\u0000operators in a Hilbert space to derive the basic results of abstract\u0000left-definite theory in a straightforward manner. The theory is amply\u0000illustrated with a variety of concrete examples employing scales of Hilbert\u0000spaces, fractional Sobolev spaces, and domains of (strictly) positive\u0000fractional powers of operators, employing interpolation theory. In particular, we explicitly describe the domains of positive powers of the\u0000harmonic oscillator operator in $L^2(mathbb{R})$ $big($and hence that of the\u0000Hermite operator in $L^2big(mathbb{R}; e^{-x^2}dx)big)big)$ in terms of\u0000fractional Sobolev spaces, certain commutation techniques, and positive powers\u0000of (the absolute value of) the operator of multiplication by the independent\u0000variable in $L^2(mathbb{R})$.","PeriodicalId":501373,"journal":{"name":"arXiv - MATH - Spectral Theory","volume":"48 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141943892","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper addresses inverse spectral problems associated with Dirac-type operators with a constant delay, specifically when this delay is less than one-third of the interval length. Our research focuses on eigenvalue behavior and operator recovery from spectra. We find that two spectra alone are insufficient to fully recover the potentials. Additionally, we consider the Ambarzumian-type inverse problem for Dirac-type operators with a delay. Our results have significant implications for the study of inverse problems related to the differential operators with a constant delay and may inform future research directions in this field.
{"title":"Inverse problem for Dirac operators with a small delay","authors":"Nebojša Djurić, Biljana Vojvodić","doi":"arxiv-2408.01229","DOIUrl":"https://doi.org/arxiv-2408.01229","url":null,"abstract":"This paper addresses inverse spectral problems associated with Dirac-type\u0000operators with a constant delay, specifically when this delay is less than\u0000one-third of the interval length. Our research focuses on eigenvalue behavior\u0000and operator recovery from spectra. We find that two spectra alone are\u0000insufficient to fully recover the potentials. Additionally, we consider the\u0000Ambarzumian-type inverse problem for Dirac-type operators with a delay. Our\u0000results have significant implications for the study of inverse problems related\u0000to the differential operators with a constant delay and may inform future\u0000research directions in this field.","PeriodicalId":501373,"journal":{"name":"arXiv - MATH - Spectral Theory","volume":"23 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141943895","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We introduce the notion of p-adic equiangular lines and derive the first fundamental relation between common angle, dimension of the space and the number of lines. More precisely, we show that if ${tau_j}_{j=1}^n$ is p-adic $gamma$-equiangular lines in $mathbb{Q}^d_p$, then begin{align*} (1) quadquad quad quad |n|^2leq |d|max{|n|, gamma^2 }. end{align*} We call Inequality (1) as the p-adic van Lint-Seidel relative bound. We believe that this complements fundamental van Lint-Seidel textit{[Indag. Math., 1966]} relative bound for equiangular lines in the p-adic case.
{"title":"p-adic Equiangular Lines and p-adic van Lint-Seidel Relative Bound","authors":"K. Mahesh Krishna","doi":"arxiv-2408.00810","DOIUrl":"https://doi.org/arxiv-2408.00810","url":null,"abstract":"We introduce the notion of p-adic equiangular lines and derive the first\u0000fundamental relation between common angle, dimension of the space and the\u0000number of lines. More precisely, we show that if ${tau_j}_{j=1}^n$ is p-adic\u0000$gamma$-equiangular lines in $mathbb{Q}^d_p$, then begin{align*} (1)\u0000quadquad quad quad |n|^2leq |d|max{|n|, gamma^2 }. end{align*} We call Inequality (1) as the p-adic van Lint-Seidel relative bound. We\u0000believe that this complements fundamental van Lint-Seidel textit{[Indag.\u0000Math., 1966]} relative bound for equiangular lines in the p-adic case.","PeriodicalId":501373,"journal":{"name":"arXiv - MATH - Spectral Theory","volume":"34 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141943896","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let $S$ be a compact hyperbolic surface of genus $ggeq 2$ and let $I(S) = frac{1}{mathrm{Vol}(S)}int_{S} frac{1}{mathrm{Inj}(x)^2 wedge 1} dx$, where $mathrm{Inj}(x)$ is the injectivity radius at $x$. We prove that for any $kin {1,ldots, 2g-3}$, the $k$-th eigenvalue $lambda_k$ of the Laplacian satisfies begin{equation*} lambda_k geq frac{c k^2}{I(S) g^2} , , end{equation*} where $c>0$ is some universal constant. We use this bound to prove the heat kernel estimate begin{equation*} frac{1}{mathrm{Vol}(S)} int_S Big| p_t(x,x) -frac{1}{mathrm{Vol}(S)} Big | ~dx leq C sqrt{ frac{I(S)}{t}} qquad forall t geq 1 , , end{equation*} where $C
{"title":"A sharp lower bound on the small eigenvalues of surfaces","authors":"Renan Gross, Guy Lachman, Asaf Nachmias","doi":"arxiv-2407.21780","DOIUrl":"https://doi.org/arxiv-2407.21780","url":null,"abstract":"Let $S$ be a compact hyperbolic surface of genus $ggeq 2$ and let $I(S) =\u0000frac{1}{mathrm{Vol}(S)}int_{S} frac{1}{mathrm{Inj}(x)^2 wedge 1} dx$,\u0000where $mathrm{Inj}(x)$ is the injectivity radius at $x$. We prove that for any\u0000$kin {1,ldots, 2g-3}$, the $k$-th eigenvalue $lambda_k$ of the Laplacian\u0000satisfies begin{equation*} lambda_k geq frac{c k^2}{I(S) g^2} , , end{equation*} where $c>0$ is\u0000some universal constant. We use this bound to prove the heat kernel estimate\u0000begin{equation*} frac{1}{mathrm{Vol}(S)} int_S Big| p_t(x,x) -frac{1}{mathrm{Vol}(S)}\u0000Big | ~dx leq C sqrt{ frac{I(S)}{t}} qquad forall t geq 1 , ,\u0000end{equation*} where $C<infty$ is some universal constant. These bounds are\u0000optimal in the sense that for every $ggeq 2$ there exists a compact hyperbolic\u0000surface of genus $g$ satisfying the reverse inequalities with different\u0000constants.","PeriodicalId":501373,"journal":{"name":"arXiv - MATH - Spectral Theory","volume":"49 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141873174","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider the (symmetric) Pascal matrix, in its finite and infinite versions, and prove the existence of symmetric tridiagonal matrices commuting with it by giving explicit expressions for these commuting matrices. This is achieved by studying the associated Fourier algebra, which as a byproduct, allows us to show that all the linear relations of a certain general form for the entries of the Pascal matrix arise from only three basic relations. We also show that pairs of eigenvectors of the tridiagonal matrix define a natural eigenbasis for the binomial transform. Lastly, we show that the commuting tridiagonal matrices provide a numerically stable means of diagonalizing the Pascal matrix.
{"title":"The Pascal Matrix, Commuting Tridiagonal Operators and Fourier Algebras","authors":"W. Riley Casper, Ignacio Zurrian","doi":"arxiv-2407.21680","DOIUrl":"https://doi.org/arxiv-2407.21680","url":null,"abstract":"We consider the (symmetric) Pascal matrix, in its finite and infinite\u0000versions, and prove the existence of symmetric tridiagonal matrices commuting\u0000with it by giving explicit expressions for these commuting matrices. This is\u0000achieved by studying the associated Fourier algebra, which as a byproduct,\u0000allows us to show that all the linear relations of a certain general form for\u0000the entries of the Pascal matrix arise from only three basic relations. We also\u0000show that pairs of eigenvectors of the tridiagonal matrix define a natural\u0000eigenbasis for the binomial transform. Lastly, we show that the commuting\u0000tridiagonal matrices provide a numerically stable means of diagonalizing the\u0000Pascal matrix.","PeriodicalId":501373,"journal":{"name":"arXiv - MATH - Spectral Theory","volume":"74 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141869755","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider an infinite-dimension SIS model introduced by Delmas, Dronnier and Zitt, with a more general incidence rate, and study its equilibria. Unsurprisingly, there exists at least one endemic equilibrium if and only if the basic reproduction number is larger than 1. When the pathogen transmission exhibits one way propagation, it is possible to observe different possible endemic equilibria. We characterize in a general setting all the equilibria, using a decomposition of the space into atoms, given by the transmission operator. We also prove that the proportion of infected individuals converges to an equilibrium, which is uniquely determined by the support of the initial condition.We extend those results to infinite-dimensional SIS models with reservoir or with immigration.
{"title":"Infinite dimensional metapopulation SIS model with generalized incidence rate","authors":"Jean-François DelmasCERMICS, Kacem LefkiLAMA, CERMICS, Pierre-André ZittLAMA","doi":"arxiv-2408.00034","DOIUrl":"https://doi.org/arxiv-2408.00034","url":null,"abstract":"We consider an infinite-dimension SIS model introduced by Delmas, Dronnier\u0000and Zitt, with a more general incidence rate, and study its equilibria.\u0000Unsurprisingly, there exists at least one endemic equilibrium if and only if\u0000the basic reproduction number is larger than 1. When the pathogen transmission\u0000exhibits one way propagation, it is possible to observe different possible\u0000endemic equilibria. We characterize in a general setting all the equilibria,\u0000using a decomposition of the space into atoms, given by the transmission\u0000operator. We also prove that the proportion of infected individuals converges\u0000to an equilibrium, which is uniquely determined by the support of the initial\u0000condition.We extend those results to infinite-dimensional SIS models with\u0000reservoir or with immigration.","PeriodicalId":501373,"journal":{"name":"arXiv - MATH - Spectral Theory","volume":"81 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141885118","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We establish a spectral gap for resonances of the Laplacian of random Schottky surfaces, which is optimal according to a conjecture of Jakobson and Naud.
我们为随机肖特基表面的拉普拉斯基共振建立了一个谱隙，根据雅各布森和诺德的猜想，这个谱隙是最优的。
{"title":"Spectral gap for random Schottky surfaces","authors":"Irving Calderón, Michael Magee, Frédéric Naud","doi":"arxiv-2407.21506","DOIUrl":"https://doi.org/arxiv-2407.21506","url":null,"abstract":"We establish a spectral gap for resonances of the Laplacian of random\u0000Schottky surfaces, which is optimal according to a conjecture of Jakobson and\u0000Naud.","PeriodicalId":501373,"journal":{"name":"arXiv - MATH - Spectral Theory","volume":"181 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141873176","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study Schr"odinger operators on compact finite metric graphs subject to $delta'$-coupling conditions. Based on a novel modified local Weyl law, we derive an explicit expression for the limiting mean eigenvalue distance of two different self-adjoint realisations on a given graph. Furthermore, using this spectral comparison result, we also study the limiting mean eigenvalue distance comparing $delta'$-coupling conditions to so-called anti-Kirchhoff conditions, showing divergence and thereby confirming a numerical observation in [arXiv:2212.12531]. .
{"title":"A modified local Weyl law and spectral comparison results for $δ'$-coupling conditions","authors":"Patrizio Bifulco, Joachim Kerner","doi":"arxiv-2407.21719","DOIUrl":"https://doi.org/arxiv-2407.21719","url":null,"abstract":"We study Schr\"odinger operators on compact finite metric graphs subject to\u0000$delta'$-coupling conditions. Based on a novel modified local Weyl law, we\u0000derive an explicit expression for the limiting mean eigenvalue distance of two\u0000different self-adjoint realisations on a given graph. Furthermore, using this\u0000spectral comparison result, we also study the limiting mean eigenvalue distance\u0000comparing $delta'$-coupling conditions to so-called anti-Kirchhoff conditions,\u0000showing divergence and thereby confirming a numerical observation in\u0000[arXiv:2212.12531]. .","PeriodicalId":501373,"journal":{"name":"arXiv - MATH - Spectral Theory","volume":"212 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141869754","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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