{"title":"On the spectral properties of the Hilbert transform operator on multi-intervals","authors":"M. Bertola, A. Katsevich, A. Tovbis","doi":"10.4171/jst/403","DOIUrl":"https://doi.org/10.4171/jst/403","url":null,"abstract":"Let $J,Esubsetmathbb R$ be two multi-intervals with non-intersecting interiors. Consider the following operator $$A:, L^2( J )to L^2(E), (Af)(x) = frac 1piint_{ J } frac {f(y)text{d} y}{x-y},$$ and let $A^dagger$ be its adjoint. We introduce a self-adjoint operator $mathscr K$ acting on $L^2(E)oplus L^2(J)$, whose off-diagonal blocks consist of $A$ and $A^dagger$. In this paper we study the spectral properties of $mathscr K$ and the operators $A^dagger A$ and $A A^dagger$. Our main tool is to obtain the resolvent of $mathscr K$, which is denoted by $mathscr R$, using an appropriate Riemann-Hilbert problem, and then compute the jump and poles of $mathscr R$ in the spectral parameter $lambda$. We show that the spectrum of $mathscr K$ has an absolutely continuous component $[0,1]$ if and only if $J$ and $E$ have common endpoints, and its multiplicity equals to their number. If there are no common endpoints, the spectrum of $mathscr K$ consists only of eigenvalues and $0$. If there are common endpoints, then $mathscr K$ may have eigenvalues imbedded in the continuous spectrum, each of them has a finite multiplicity, and the eigenvalues may accumulate only at $0$. In all cases, $mathscr K$ does not have a singular continuous spectrum. The spectral properties of $A^dagger A$ and $A A^dagger$, which are very similar to those of $mathscr K$, are obtained as well.","PeriodicalId":48789,"journal":{"name":"Journal of Spectral Theory","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2020-08-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48253080","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Ivica Nakić, Matthias Täufer, Martin Tautenhahn, I. Veselić
{"title":"Unique continuation and lifting of spectral band edges of Schrödinger operators on unbounded domains","authors":"Ivica Nakić, Matthias Täufer, Martin Tautenhahn, I. Veselić","doi":"10.4171/jst/314","DOIUrl":"https://doi.org/10.4171/jst/314","url":null,"abstract":"We prove and apply two theorems: First, a quantitative, scale-free unique continuation estimate for functions in a spectral subspace of a Schrödinger operator on a bounded or unbounded domain; second, a perturbation and lifting estimate for edges of the essential spectrum of a self-adjoint operator under a semi-definite perturbation. These two results are combined to obtain lower and upper Lipschitz bounds on the function parametrizing locally a chosen edge of the essential spectrum of a Schrödinger operator in dependence of a coupling constant. Analogous estimates for eigenvalues, possibly in gaps of the essential spectrum, are exhibited as well.","PeriodicalId":48789,"journal":{"name":"Journal of Spectral Theory","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2020-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/jst/314","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46228788","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The Hartree functional in a double well","authors":"A. Olgiati, N. Rougerie","doi":"10.4171/jst/385","DOIUrl":"https://doi.org/10.4171/jst/385","url":null,"abstract":"We consider a non-linear Hartree energy for bosonic particles in a symmetric double-well potential. In the limit where the wells are fare apart and the potential barrier is high, we prove that the ground state and first excited state are given to leading order by an even, respectively odd, superposition of ground states in single wells. We evaluate the resulting tunneling term splitting the corresponding energies precisely.","PeriodicalId":48789,"journal":{"name":"Journal of Spectral Theory","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2020-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46300660","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
S. Jitomirskaya, Lyuben Konstantinov, I. Krasovsky
{"title":"On the spectrum of critical almost Mathieu operators in the rational case","authors":"S. Jitomirskaya, Lyuben Konstantinov, I. Krasovsky","doi":"10.4171/jst/392","DOIUrl":"https://doi.org/10.4171/jst/392","url":null,"abstract":"We derive a new Chambers-type formula and prove sharper upper bounds on the measure of the spectrum of critical almost Mathieu operators with rational frequencies.","PeriodicalId":48789,"journal":{"name":"Journal of Spectral Theory","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2020-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47988315","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Eigenfunction asymptotics and spectral rigidity of the ellipse","authors":"Hamid Hezari, S. Zelditch","doi":"10.4171/jst/393","DOIUrl":"https://doi.org/10.4171/jst/393","url":null,"abstract":"This paper is part of a series concerning the isospectral problem for an ellipse. In this paper, we study Cauchy data of eigenfunctions of the ellipse with Dirichlet or Neumann boundary conditions. Using many classical results on ellipse eigenfunctions, we determine the microlocal defect measures of the Cauchy data of the eigenfunctions. The ellipse has integrable billiards, i.e. the boundary phase space is foliated by invariant curves of the billiard map. We prove that, for any invariant curve $C$, there exists a sequence of eigenfunctions whose Cauchy data concentrates on $C$. We use this result to give a new proof that ellipses are infinitesimally spectrally rigid among $C^{infty}$ domains with the symmetries of the ellipse.","PeriodicalId":48789,"journal":{"name":"Journal of Spectral Theory","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2020-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42780289","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"A Szegő type theorem and distribution of symplectic eigenvalues","authors":"R. Bhatia, Tanvi Jain, R. Sengupta","doi":"10.4171/JST/377","DOIUrl":"https://doi.org/10.4171/JST/377","url":null,"abstract":"We study the properties of stationary G-chains in terms of their generating functions. In particular, we prove an analogue of the Szegő limit theorem for symplectic eigenvalues, derive an expression for the entropy rate of stationary quantum Gaussian processes, and study the distribution of symplectic eigenvalues of truncated block Toeplitz matrices. We also introduce a concept of symplectic numerical range, analogous to that of numerical range, and study some of its basic properties, mainly in the context of block Toeplitz operators.","PeriodicalId":48789,"journal":{"name":"Journal of Spectral Theory","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2020-06-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47567426","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Uniform resolvent estimates for the discrete Schrödinger operator in dimension three","authors":"Kouichi Taira","doi":"10.4171/jst/387","DOIUrl":"https://doi.org/10.4171/jst/387","url":null,"abstract":"In this note, we prove the uniform resolvent estimate of the discrete Schrodinger operator with dimension three. To do this, we show a Fourier decay of the surface measure on the Fermi surface.","PeriodicalId":48789,"journal":{"name":"Journal of Spectral Theory","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2020-05-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44142202","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On Lieb–Thirring inequalities for one-dimensional non-self-adjoint Jacobi and Schrödinger operators","authors":"Sabine Bogli, F. vStampach","doi":"10.4171/jst/378","DOIUrl":"https://doi.org/10.4171/jst/378","url":null,"abstract":"We study to what extent Lieb--Thirring inequalities are extendable from self-adjoint to general (possibly non-self-adjoint) Jacobi and Schrodinger operators. Namely, we prove the conjecture of Hansmann and Katriel from [Complex Anal. Oper. Theory 5, No. 1 (2011), 197-218] and answer another open question raised therein. The results are obtained by means of asymptotic analysis of eigenvalues of discrete Schrodinger operators with rectangular barrier potential and complex coupling. Applying the ideas in the continuous setting, we also solve a similar open problem for one-dimensional Schrodinger operators with complex-valued potentials published by Demuth, Hansmann, and Katriel in [Integral Equations Operator Theory 75, No. 1 (2013), 1-5].","PeriodicalId":48789,"journal":{"name":"Journal of Spectral Theory","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2020-04-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46870378","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Propagation of polyhomogeneity, diffraction, and scattering on product cones","authors":"Mengxuan Yang","doi":"10.4171/jst/404","DOIUrl":"https://doi.org/10.4171/jst/404","url":null,"abstract":"We consider diffraction of waves on a product cone. We first show that diffractive waves enjoy a one-step polyhomogeneous asymptotic expansion, which is an improvement of Cheeger-Taylor's classical result of half-step polyhomogeneity of diffractive waves in [CT82a], [CT82b]. We also conclude that on product cones, the scattering matrix is the diffraction coefficient, which is the principal symbol of the diffractive half wave kernel, for strictly diffractively related points on the cross section. This generalize the result of Ford, Hassell and Hillairet in 2-dimensional flat cone settings [FHH18]. In the last section, we also give a radiation field interpretation of the relationship between the scattering matrix and the diffraction coefficient.","PeriodicalId":48789,"journal":{"name":"Journal of Spectral Theory","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2020-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46687792","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Jacobi matrices on trees generated by Angelesco systems: asymptotics of coefficients and essential spectrum","authors":"A. Aptekarev, S. Denisov, M. Yattselev","doi":"10.4171/jst/380","DOIUrl":"https://doi.org/10.4171/jst/380","url":null,"abstract":"We continue studying the connection between Jacobi matrices defined on a tree and multiple orthogonal polynomials (MOPs) that was discovered previously by the authors. In this paper, we consider Angelesco systems formed by two analytic weights and obtain asymptotics of the recurrence coefficients and strong asymptotics of MOPs along all directions (including the marginal ones). These results are then applied to show that the essential spectrum of the related Jacobi matrix is the union of intervals of orthogonality.","PeriodicalId":48789,"journal":{"name":"Journal of Spectral Theory","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2020-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42417815","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}