We prove that mean decay of the coefficient of Krein system is equivalent to the mean decay of the Fourier transform of its SzegH{o} function.
我们证明，Krein 系统系数的平均衰减等同于其 SzegH{o} 函数傅里叶变换的平均衰减。
{"title":"Krein systems with oscillating potentials","authors":"Pavel Gubkin","doi":"arxiv-2409.08614","DOIUrl":"https://doi.org/arxiv-2409.08614","url":null,"abstract":"We prove that mean decay of the coefficient of Krein system is equivalent to\u0000the mean decay of the Fourier transform of its SzegH{o} function.","PeriodicalId":501373,"journal":{"name":"arXiv - MATH - Spectral Theory","volume":"4 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142248714","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}
The longstanding emph{nonnegative inverse eigenvalue problem} (NIEP) is to determine which multisets of complex numbers occur as the spectrum of an entry-wise nonnegative matrix. Although there are some well-known necessary conditions, a solution to the NIEP is far from known. An invertible matrix is called a emph{Perron similarity} if it diagonalizes an irreducible, nonnegative matrix. Johnson and Paparella developed the theory of real Perron similarities. Here, we fully develop the theory of complex Perron similarities. Each Perron similarity gives a nontrivial polyhedral cone and polytope of realizable spectra (thought of as vectors in complex Euclidean space). The extremals of these convex sets are finite in number, and their determination for each Perron similarity would solve the diagonalizable NIEP, a major portion of the entire problem. By considering Perron similarities of certain realizing matrices of Type I Karpelevich arcs, large portions of realizable spectra are generated for a given positive integer. This is demonstrated by producing a nearly complete geometrical representation of the spectra of $4 times 4$ stochastic matrices. Similar to the Karpelevich region, it is shown that the subset of complex Euclidean space comprising the spectra of stochastic matrices is compact and star-shaped. emph{Extremal} elements of the set are defined and shown to be on the boundary. It is shown that the polyhedral cone and convex polytope of the emph{discrete Fourier transform (DFT) matrix} corresponds to the conical hull and convex hull of its rows, respectively. Similar results are established for multifold Kronecker products of DFT matrices and multifold Kronecker products of DFT matrices and Walsh matrices. These polytopes are of great significance with respect to the NIEP because they are extremal in the region comprising the spectra of stochastic matrices.
{"title":"Perron similarities and the nonnegative inverse eigenvalue problem","authors":"Charles R. Johnson, Pietro Paparella","doi":"arxiv-2409.07682","DOIUrl":"https://doi.org/arxiv-2409.07682","url":null,"abstract":"The longstanding emph{nonnegative inverse eigenvalue problem} (NIEP) is to\u0000determine which multisets of complex numbers occur as the spectrum of an\u0000entry-wise nonnegative matrix. Although there are some well-known necessary\u0000conditions, a solution to the NIEP is far from known. An invertible matrix is called a emph{Perron similarity} if it diagonalizes\u0000an irreducible, nonnegative matrix. Johnson and Paparella developed the theory\u0000of real Perron similarities. Here, we fully develop the theory of complex\u0000Perron similarities. Each Perron similarity gives a nontrivial polyhedral cone and polytope of\u0000realizable spectra (thought of as vectors in complex Euclidean space). The\u0000extremals of these convex sets are finite in number, and their determination\u0000for each Perron similarity would solve the diagonalizable NIEP, a major portion\u0000of the entire problem. By considering Perron similarities of certain realizing\u0000matrices of Type I Karpelevich arcs, large portions of realizable spectra are\u0000generated for a given positive integer. This is demonstrated by producing a\u0000nearly complete geometrical representation of the spectra of $4 times 4$\u0000stochastic matrices. Similar to the Karpelevich region, it is shown that the subset of complex\u0000Euclidean space comprising the spectra of stochastic matrices is compact and\u0000star-shaped. emph{Extremal} elements of the set are defined and shown to be on\u0000the boundary. It is shown that the polyhedral cone and convex polytope of the\u0000emph{discrete Fourier transform (DFT) matrix} corresponds to the conical hull\u0000and convex hull of its rows, respectively. Similar results are established for\u0000multifold Kronecker products of DFT matrices and multifold Kronecker products\u0000of DFT matrices and Walsh matrices. These polytopes are of great significance\u0000with respect to the NIEP because they are extremal in the region comprising the\u0000spectra of stochastic matrices.","PeriodicalId":501373,"journal":{"name":"arXiv - MATH - Spectral Theory","volume":"15 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142179333","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}
In this article, we consider the Dirac operator with constant magnetic field in $mathbb R^2$. Its spectrum consists of eigenvalues of infinite multiplicities, known as the Landau-Dirac levels. Under compactly supported perturbations, we study the distribution of the discrete eigenvalues near each Landau-Dirac level. Similarly to the Landau (Schr"odinger) operator, we demonstrate that a three-terms asymptotic formula holds for the eigenvalue counting function. One of the main novelties of this work is the treatment of some perturbations of variable sign. In this context we explore some remarkable phenomena related to the finiteness or infiniteness of the discrete eigenvalues, which depend on the interplay of the different terms in the matrix perturbation.
{"title":"Spectrum of the perturbed Landau-Dirac operator","authors":"Vincent Bruneau, Pablo Miranda","doi":"arxiv-2409.08218","DOIUrl":"https://doi.org/arxiv-2409.08218","url":null,"abstract":"In this article, we consider the Dirac operator with constant magnetic field\u0000in $mathbb R^2$. Its spectrum consists of eigenvalues of infinite\u0000multiplicities, known as the Landau-Dirac levels. Under compactly supported\u0000perturbations, we study the distribution of the discrete eigenvalues near each\u0000Landau-Dirac level. Similarly to the Landau (Schr\"odinger) operator, we\u0000demonstrate that a three-terms asymptotic formula holds for the eigenvalue\u0000counting function. One of the main novelties of this work is the treatment of\u0000some perturbations of variable sign. In this context we explore some remarkable\u0000phenomena related to the finiteness or infiniteness of the discrete\u0000eigenvalues, which depend on the interplay of the different terms in the matrix\u0000perturbation.","PeriodicalId":501373,"journal":{"name":"arXiv - MATH - Spectral Theory","volume":"14 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142179332","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}
Guglielmo Fucci, Mateusz Piorkowski, Jonathan Stanfill
In this work we analyze the spectral $zeta$-function associated with the self-adjoint extensions, $T_{A,B}$, of quasi-regular Sturm--Liouville operators that are bounded from below. By utilizing the Green's function formalism, we find the characteristic function which implicitly provides the eigenvalues associated with a given self-adjoint extension $T_{A,B}$. The characteristic function is then employed to construct a contour integral representation for the spectral $zeta$-function of $T_{A,B}$. By assuming a general form for the asymptotic expansion of the characteristic function, we describe the analytic continuation of the $zeta$-function to a larger region of the complex plane. We also present a method for computing the value of the spectral $zeta$-function of $T_{A,B}$ at all positive integers. We provide two examples to illustrate the methods developed in the paper: the generalized Bessel and Legendre operators. We show that in the case of the generalized Bessel operator, the spectral $zeta$-function develops a branch point at the origin, while in the case of the Legendre operator it presents, more remarkably, branch points at every nonpositive integer value of $s$.
{"title":"The spectral $ζ$-function for quasi-regular Sturm--Liouville operators","authors":"Guglielmo Fucci, Mateusz Piorkowski, Jonathan Stanfill","doi":"arxiv-2409.06922","DOIUrl":"https://doi.org/arxiv-2409.06922","url":null,"abstract":"In this work we analyze the spectral $zeta$-function associated with the\u0000self-adjoint extensions, $T_{A,B}$, of quasi-regular Sturm--Liouville operators\u0000that are bounded from below. By utilizing the Green's function formalism, we\u0000find the characteristic function which implicitly provides the eigenvalues\u0000associated with a given self-adjoint extension $T_{A,B}$. The characteristic\u0000function is then employed to construct a contour integral representation for\u0000the spectral $zeta$-function of $T_{A,B}$. By assuming a general form for the\u0000asymptotic expansion of the characteristic function, we describe the analytic\u0000continuation of the $zeta$-function to a larger region of the complex plane.\u0000We also present a method for computing the value of the spectral\u0000$zeta$-function of $T_{A,B}$ at all positive integers. We provide two examples\u0000to illustrate the methods developed in the paper: the generalized Bessel and\u0000Legendre operators. We show that in the case of the generalized Bessel\u0000operator, the spectral $zeta$-function develops a branch point at the origin,\u0000while in the case of the Legendre operator it presents, more remarkably, branch\u0000points at every nonpositive integer value of $s$.","PeriodicalId":501373,"journal":{"name":"arXiv - MATH - Spectral Theory","volume":"6 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142179337","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}
Marco Carfagnini, Maria Gordina, Alexander Teplyaev
In the context of irreducible ultracontractive Dirichlet metric measure spaces, we demonstrate the discreteness of the Laplacian spectrum and the corresponding diffusion's irreducibility in connected open sets, without assuming regularity of the boundary. This general result can be applied to study various questions, including those related to small deviations of the diffusion and generalized heat content. Our examples include Riemannian and sub-Riemannian manifolds, as well as non-smooth and fractal spaces.
{"title":"Dirichlet metric measure spaces: spectrum, irreducibility, and small deviations","authors":"Marco Carfagnini, Maria Gordina, Alexander Teplyaev","doi":"arxiv-2409.07425","DOIUrl":"https://doi.org/arxiv-2409.07425","url":null,"abstract":"In the context of irreducible ultracontractive Dirichlet metric measure\u0000spaces, we demonstrate the discreteness of the Laplacian spectrum and the\u0000corresponding diffusion's irreducibility in connected open sets, without\u0000assuming regularity of the boundary. This general result can be applied to\u0000study various questions, including those related to small deviations of the\u0000diffusion and generalized heat content. Our examples include Riemannian and\u0000sub-Riemannian manifolds, as well as non-smooth and fractal spaces.","PeriodicalId":501373,"journal":{"name":"arXiv - MATH - Spectral Theory","volume":"10 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142179334","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}
David Krejcirik, Pier Domenico Lamberti, Michele Zaccaron
We consider a natural eigenvalue problem for the vector Laplacian related to stationary Maxwell's equations in a cavity and we prove that an analog of the celebrated Faber-Krahn inequality doesn't hold.
{"title":"A note on the failure of the Faber-Krahn inequality for the vector Laplacian","authors":"David Krejcirik, Pier Domenico Lamberti, Michele Zaccaron","doi":"arxiv-2409.07206","DOIUrl":"https://doi.org/arxiv-2409.07206","url":null,"abstract":"We consider a natural eigenvalue problem for the vector Laplacian related to\u0000stationary Maxwell's equations in a cavity and we prove that an analog of the\u0000celebrated Faber-Krahn inequality doesn't hold.","PeriodicalId":501373,"journal":{"name":"arXiv - MATH - Spectral Theory","volume":"30 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142179335","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}
As a non-trivial extension of the celebrated Cheeger inequality, the higher-order Cheeger inequalities for graphs due to Lee, Oveis Gharan and Trevisan provide for each $k$ an upper bound for the $k$-way Cheeger constant in forms of $C(k)sqrt{lambda_k(G)}$, where $lambda_k(G)$ is the $k$-th eigenvalue of the graph Laplacian and $C(k)$ is a constant depending only on $k$. In this article, we prove some new bounds for multi-way Cheeger constants. By shifting the index of the eigenvalue via cyclomatic number, we establish upper bound estimates with an absolute constant instead of $C(k)$. This, in particular, gives a more direct proof of Miclo's higher order Cheeger inequalities on trees. We also show a new lower bound of the multi-way Cheeger constants in terms of the spectral radius of the graph. The proofs involve the concept of discrete nodal domains and a probability argument showing generic properties of eigenfunctions.
{"title":"Spectral bounds of multi-way Cheeger constants via cyclomatic number","authors":"Chuanyuan Ge","doi":"arxiv-2409.07097","DOIUrl":"https://doi.org/arxiv-2409.07097","url":null,"abstract":"As a non-trivial extension of the celebrated Cheeger inequality, the\u0000higher-order Cheeger inequalities for graphs due to Lee, Oveis Gharan and\u0000Trevisan provide for each $k$ an upper bound for the $k$-way Cheeger constant\u0000in forms of $C(k)sqrt{lambda_k(G)}$, where $lambda_k(G)$ is the $k$-th\u0000eigenvalue of the graph Laplacian and $C(k)$ is a constant depending only on\u0000$k$. In this article, we prove some new bounds for multi-way Cheeger constants.\u0000By shifting the index of the eigenvalue via cyclomatic number, we establish\u0000upper bound estimates with an absolute constant instead of $C(k)$. This, in\u0000particular, gives a more direct proof of Miclo's higher order Cheeger\u0000inequalities on trees. We also show a new lower bound of the multi-way Cheeger\u0000constants in terms of the spectral radius of the graph. The proofs involve the\u0000concept of discrete nodal domains and a probability argument showing generic\u0000properties of eigenfunctions.","PeriodicalId":501373,"journal":{"name":"arXiv - MATH - Spectral Theory","volume":"82 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142179336","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}
In this paper we present a complete spectral analysis of Dirac operators with non-Hermitian matrix potentials of the form $ioperatorname{sgn}(x)+V(x)$ where $Vin L^1$. For $V=0$ we compute explicitly the matrix Green function. This allows us to determine the spectrum, which is purely essential, and its different types. It also allows us to find sharp enclosures for the pseudospectrum and its complement, in all parts of the complex plane. Notably, this includes the instability region, corresponding to the interior of the band that forms the numerical range. Then, with the help of a Birman-Schwinger principle, we establish in precise manner how the spectrum and pseudospectrum change when $Vnot=0$, assuming the hypotheses $|V|_{L^1}<1$ or $Vin L^1cap L^p$ where $p>1$. We show that the essential spectra remain unchanged and that the $varepsilon$-pseudospectrum stays close to the instability region for small $varepsilon$. We determine sharp asymptotic for the discrete spectrum, whenever $V$ satisfies further conditions of decay at infinity. Finally, in one of our main findings, we give a complete description of the weakly-coupled model.
{"title":"Spectral analysis of Dirac operators for dislocated potentials with a purely imaginary jump","authors":"Lyonell Boulton, David Krejcirik, Tho Nguyen Duc","doi":"arxiv-2409.06480","DOIUrl":"https://doi.org/arxiv-2409.06480","url":null,"abstract":"In this paper we present a complete spectral analysis of Dirac operators with\u0000non-Hermitian matrix potentials of the form $ioperatorname{sgn}(x)+V(x)$ where\u0000$Vin L^1$. For $V=0$ we compute explicitly the matrix Green function. This\u0000allows us to determine the spectrum, which is purely essential, and its\u0000different types. It also allows us to find sharp enclosures for the\u0000pseudospectrum and its complement, in all parts of the complex plane. Notably,\u0000this includes the instability region, corresponding to the interior of the band\u0000that forms the numerical range. Then, with the help of a Birman-Schwinger\u0000principle, we establish in precise manner how the spectrum and pseudospectrum\u0000change when $Vnot=0$, assuming the hypotheses $|V|_{L^1}<1$ or $Vin L^1cap\u0000L^p$ where $p>1$. We show that the essential spectra remain unchanged and that\u0000the $varepsilon$-pseudospectrum stays close to the instability region for\u0000small $varepsilon$. We determine sharp asymptotic for the discrete spectrum,\u0000whenever $V$ satisfies further conditions of decay at infinity. Finally, in one\u0000of our main findings, we give a complete description of the weakly-coupled\u0000model.","PeriodicalId":501373,"journal":{"name":"arXiv - MATH - Spectral Theory","volume":"48 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142179338","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 magnetic Dirac operator on a curved strip whose boundary carries the infinite mass boundary condition. When the magnetic field is large, we provide the reader with accurate estimates of the essential and discrete spectra. In particular, we give sufficient conditions ensuring that the discrete spectrum is non-empty.
{"title":"Magnetic Dirac operator in strips submitted to strong magnetic fields","authors":"Loïc Le Treust, Julien Royer, Nicolas Raymond","doi":"arxiv-2409.06284","DOIUrl":"https://doi.org/arxiv-2409.06284","url":null,"abstract":"We consider the magnetic Dirac operator on a curved strip whose boundary\u0000carries the infinite mass boundary condition. When the magnetic field is large,\u0000we provide the reader with accurate estimates of the essential and discrete\u0000spectra. In particular, we give sufficient conditions ensuring that the\u0000discrete spectrum is non-empty.","PeriodicalId":501373,"journal":{"name":"arXiv - MATH - Spectral Theory","volume":"39 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142179339","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 the behavior of the spectrum of the Dirac operator on degenerating families of compact Riemannian surfaces, when the length $t$ of a simple closed geodesic shrinks to zero, under the hypothesis that the spin structure along the pinched geodesic is non-trivial. The difficulty of the problem stems from the non-compactness of the limit surface, which has finite area and two cusps. The main idea in this investigation is to construct an adapted pseudodifferential calculus, in the spirit of the celebrated b-algebra of Melrose, which includes both the family of Dirac operators on the family of compact surfaces and the Dirac operator on the limit non-compact surface, together with their resolvents. We obtain smoothness of the spectral projectors, and $t^2 log t$ regularity for the cusp-surgery trace of the relative resolvent in the degeneracy process as $t searrow 0$.
{"title":"On the Dirac spectrum on degenerating Riemannian surfaces","authors":"Cipriana Anghel","doi":"arxiv-2409.05616","DOIUrl":"https://doi.org/arxiv-2409.05616","url":null,"abstract":"We study the behavior of the spectrum of the Dirac operator on degenerating\u0000families of compact Riemannian surfaces, when the length $t$ of a simple closed\u0000geodesic shrinks to zero, under the hypothesis that the spin structure along\u0000the pinched geodesic is non-trivial. The difficulty of the problem stems from\u0000the non-compactness of the limit surface, which has finite area and two cusps.\u0000The main idea in this investigation is to construct an adapted\u0000pseudodifferential calculus, in the spirit of the celebrated b-algebra of\u0000Melrose, which includes both the family of Dirac operators on the family of\u0000compact surfaces and the Dirac operator on the limit non-compact surface,\u0000together with their resolvents. We obtain smoothness of the spectral\u0000projectors, and $t^2 log t$ regularity for the cusp-surgery trace of the\u0000relative resolvent in the degeneracy process as $t searrow 0$.","PeriodicalId":501373,"journal":{"name":"arXiv - MATH - Spectral Theory","volume":"12 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142179343","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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