{"title":"Deformation of the spectrum for Darboux–Treibich–Verdier potential along $operatorname{Re}tau=frac{1}{2}$","authors":"Erjuan Fu","doi":"10.4171/jst/453","DOIUrl":"https://doi.org/10.4171/jst/453","url":null,"abstract":"","PeriodicalId":48789,"journal":{"name":"Journal of Spectral Theory","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-07-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46327074","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":"Scaling inequalities for spherical and hyperbolic eigenvalues","authors":"J. Langford, R. Laugesen","doi":"10.4171/jst/447","DOIUrl":"https://doi.org/10.4171/jst/447","url":null,"abstract":"","PeriodicalId":48789,"journal":{"name":"Journal of Spectral Theory","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-07-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45337471","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":"Spectral estimates for the Dirichlet Laplacian on spiral-shaped regions","authors":"Diana Barseghyan, Pavel Exner","doi":"10.4171/jst/454","DOIUrl":"https://doi.org/10.4171/jst/454","url":null,"abstract":"","PeriodicalId":48789,"journal":{"name":"Journal of Spectral Theory","volume":"20 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-07-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135658512","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}
In this paper we prove Lieb--Thirring inequalities for magnetic Schr"odinger operators on the torus, where the constants in the inequalities depend on the magnetic flux.
{"title":"Magnetic Lieb–Thirring inequalities on the torus","authors":"A. Ilyin, A. Laptev","doi":"10.4171/jst/470","DOIUrl":"https://doi.org/10.4171/jst/470","url":null,"abstract":"In this paper we prove Lieb--Thirring inequalities for magnetic Schr\"odinger operators on the torus, where the constants in the inequalities depend on the magnetic flux.","PeriodicalId":48789,"journal":{"name":"Journal of Spectral Theory","volume":"125 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139372293","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 periodic and antiperiodic spectra of non-self-adjoint Dirac operators","authors":"A. Makin","doi":"10.4171/jst/435","DOIUrl":"https://doi.org/10.4171/jst/435","url":null,"abstract":"","PeriodicalId":48789,"journal":{"name":"Journal of Spectral Theory","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-05-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45356789","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":"Edge states for second order elliptic operators in a channel","authors":"D. Gontier","doi":"10.4171/jst/430","DOIUrl":"https://doi.org/10.4171/jst/430","url":null,"abstract":"","PeriodicalId":48789,"journal":{"name":"Journal of Spectral Theory","volume":"1 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-04-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43281003","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 well-prepared states along Martinet singular geodesics","authors":"Yves Colin de Verdière, Cyril Letrouit","doi":"10.4171/jst/421","DOIUrl":"https://doi.org/10.4171/jst/421","url":null,"abstract":"","PeriodicalId":48789,"journal":{"name":"Journal of Spectral Theory","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-03-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45606964","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}
Let {pn} denote the orthonormal polynomials associated with a measure μ with compact support on the real line. Let μ be regular in the sense of Stahl, Totik, and Ullmann, and I be a subinterval of the support in which μ is absolutely continuous, while μ′ is positive and continuous there. We show that boundedness of the {pn} in that subinterval is closely related to the spacing of zeros of pn and pn−1 in that interval. One ingredient is proving that "local limits" imply universality limits. Abstract. Research supported by NSF grant DMS1800251 Research supported by NSF grant DMS1800251 1. Results Let μ be a finite positive Borel measure with compact support, which we denote by supp[μ]. Then we may define orthonormal polynomials pn (x) = γnx n + ..., γn > 0, n = 0, 1, 2, ... satisfying the orthonormality conditions ∫ pnpmdμ = δmn. The zeros of pn are real and simple. We list them in decreasing order: x1n > x2n > ... > xn−1,n > xnn. They interlace the zeros yjn of pn : pn (yjn) = 0 and yjn ∈ (xj+1,n, xjn) , 1 ≤ j ≤ n− 1. It is a classic result that the zeros of pn and pn−1 also interlace. The three term recurrence relation has the form (x− bn) pn (x) = an+1pn+1 (x) + anpn−1 (x) , where for n ≥ 1, an = γn−1 γn = ∫ xpn−1 (x) pn (x) dμ (x) ; bn = ∫ xpn (x) dμ (x) . 1 2 ELI LEVIN AND D. S. LUBINSKY Uniform boundedness of orthonormal polynomials is a long studied topic. For example, given an interval I, one asks whether sup n≥1 ‖pn‖L∞(I) <∞. There is an extensive literature on this fundamental question see for example [1], [2], [3], [4], [12]. In this paper, we establish a connection to the distance between zeros of pn and pn−1. The results require more terminology: we let dist (a,Z) denote the distance from a real number a to the integers. We say that μ is regular (in the sense of Stahl, Totik, and Ullmann) if for every sequence of non-zero polynomials {Pn} with degree Pn at most n, lim sup n→∞ ( |Pn (x)| (∫ |Pn| dμ )1/2 )1/n ≤ 1 for quasi-every x ∈supp[μ] (that is except in a set of logarithmic capacity 0). If the support consists of finitely many intervals, and μ′ > 0 a.e. in each subinterval, then μ is regular, though much less is required [15]. An equivalent formulation involves the leading coeffi cients {γn} of the orthonormal polynomials for μ : lim n→∞ γ n = 1 cap (supp [μ]) , where cap denotes logarithmic capacity. Recall that the equilibrium measure for the compact set supp[μ] is the probability measure that minimizes the energy integral ∫ ∫ log 1 |x− y| (x) dν (y) amongst all probability measures ν supported on supp[μ]. If I is an interval contained in supp[μ], then the equilibrium measure is absolutely continuous in I, and moreover its density, which we denote throughout by ω, is positive and continuous in the interior I of I [13, p.216, Thm. IV.2.5]. Given sequences {xn} , {yn} of non-0 real numbers, we write xn ∼ yn if there exists C > 1 such that for n ≥ 1, C−1 ≤ xn/yn < C. Similar notation is used for functions and sequences of functions.
{"title":"Bounds on orthogonal polynomials and separation of their zeros","authors":"E. Levin, D. Lubinsky","doi":"10.4171/jst/408","DOIUrl":"https://doi.org/10.4171/jst/408","url":null,"abstract":"Let {pn} denote the orthonormal polynomials associated with a measure μ with compact support on the real line. Let μ be regular in the sense of Stahl, Totik, and Ullmann, and I be a subinterval of the support in which μ is absolutely continuous, while μ′ is positive and continuous there. We show that boundedness of the {pn} in that subinterval is closely related to the spacing of zeros of pn and pn−1 in that interval. One ingredient is proving that \"local limits\" imply universality limits. Abstract. Research supported by NSF grant DMS1800251 Research supported by NSF grant DMS1800251 1. Results Let μ be a finite positive Borel measure with compact support, which we denote by supp[μ]. Then we may define orthonormal polynomials pn (x) = γnx n + ..., γn > 0, n = 0, 1, 2, ... satisfying the orthonormality conditions ∫ pnpmdμ = δmn. The zeros of pn are real and simple. We list them in decreasing order: x1n > x2n > ... > xn−1,n > xnn. They interlace the zeros yjn of pn : pn (yjn) = 0 and yjn ∈ (xj+1,n, xjn) , 1 ≤ j ≤ n− 1. It is a classic result that the zeros of pn and pn−1 also interlace. The three term recurrence relation has the form (x− bn) pn (x) = an+1pn+1 (x) + anpn−1 (x) , where for n ≥ 1, an = γn−1 γn = ∫ xpn−1 (x) pn (x) dμ (x) ; bn = ∫ xpn (x) dμ (x) . 1 2 ELI LEVIN AND D. S. LUBINSKY Uniform boundedness of orthonormal polynomials is a long studied topic. For example, given an interval I, one asks whether sup n≥1 ‖pn‖L∞(I) <∞. There is an extensive literature on this fundamental question see for example [1], [2], [3], [4], [12]. In this paper, we establish a connection to the distance between zeros of pn and pn−1. The results require more terminology: we let dist (a,Z) denote the distance from a real number a to the integers. We say that μ is regular (in the sense of Stahl, Totik, and Ullmann) if for every sequence of non-zero polynomials {Pn} with degree Pn at most n, lim sup n→∞ ( |Pn (x)| (∫ |Pn| dμ )1/2 )1/n ≤ 1 for quasi-every x ∈supp[μ] (that is except in a set of logarithmic capacity 0). If the support consists of finitely many intervals, and μ′ > 0 a.e. in each subinterval, then μ is regular, though much less is required [15]. An equivalent formulation involves the leading coeffi cients {γn} of the orthonormal polynomials for μ : lim n→∞ γ n = 1 cap (supp [μ]) , where cap denotes logarithmic capacity. Recall that the equilibrium measure for the compact set supp[μ] is the probability measure that minimizes the energy integral ∫ ∫ log 1 |x− y| (x) dν (y) amongst all probability measures ν supported on supp[μ]. If I is an interval contained in supp[μ], then the equilibrium measure is absolutely continuous in I, and moreover its density, which we denote throughout by ω, is positive and continuous in the interior I of I [13, p.216, Thm. IV.2.5]. Given sequences {xn} , {yn} of non-0 real numbers, we write xn ∼ yn if there exists C > 1 such that for n ≥ 1, C−1 ≤ xn/yn < C. Similar notation is used for functions and sequences of functions. ","PeriodicalId":48789,"journal":{"name":"Journal of Spectral Theory","volume":"1 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2022-09-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41705052","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}
In this paper, we show that the 2-sphere does not exhibit symmetry of $L^p$ norms of eigenfunctions of the Laplacian for $pgeq 6$. In other words, there exists a sequence of spherical eigenfunctions $psi_n$, with eigenvalues $lambda_ntoinfty$ as $ntoinfty$, such that the ratio of the $L^p$ norms of the positive and negative parts of the eigenfunctions does not tend to $1$ as $ntoinfty$ when $pgeq 6$. Our proof relies on fundamental properties of the Legendre polynomials and Bessel functions of the first kind.
{"title":"Failure of $L^p$ symmetry of zonal spherical harmonics","authors":"G. Beiner, William Verreault","doi":"10.4171/jst/446","DOIUrl":"https://doi.org/10.4171/jst/446","url":null,"abstract":"In this paper, we show that the 2-sphere does not exhibit symmetry of $L^p$ norms of eigenfunctions of the Laplacian for $pgeq 6$. In other words, there exists a sequence of spherical eigenfunctions $psi_n$, with eigenvalues $lambda_ntoinfty$ as $ntoinfty$, such that the ratio of the $L^p$ norms of the positive and negative parts of the eigenfunctions does not tend to $1$ as $ntoinfty$ when $pgeq 6$. Our proof relies on fundamental properties of the Legendre polynomials and Bessel functions of the first kind.","PeriodicalId":48789,"journal":{"name":"Journal of Spectral Theory","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2022-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43156117","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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：