In the recent paper [32] the authors have considered the Birman-Schwinger (Cwikel) type operators in a domain Ω ⊆ R, having the form TP = A∗PA. Here A is a pseudodifferential operator in Ω of order −l = −N/2 and P = V μ is a finite signed measure containing a singular part. We found out there that for such operators, properly defined using quadratic forms, for a special class of measures, an estimate for eigenvalues λk = λ ± k (TP ) holds with order λ ± k = O(k −1) with coefficient involving an Orlicz norm of the weight function V . For a subclass of such measures, namely, for the ones whose singular part is a finite sum of measures absolutely continuous with respect to the surface measures on compact Lipschitz surfaces of arbitrary dimension, an asymptotic formula for eigenvalues was proved, with all surfaces, independently of their dimension, making the same order contributions. In the present paper we discuss some generalizations of these results and their consequences for introducing Connes’ integration with respect to singular measures. Our considerations are based upon the variational (via quadratic forms) approach to the spectral analysis of differential operators in a singular setting, in the form developed in 60-s and 70-s by M.Sh. Birman and M.Z. Solomyak. This approach enables one to obtain, for rather general spectral problems, eigenvalue estimates, sharp both in order and in the class of coefficients involved, this sharpness confirmed by exact asymptotic eigenvalue formulas. In the initial setting, this approach was applied to measures P absolutely continuous with respect to Lebesgue measure. Passing to singular measures, it was found that, for the equation −λ∆(X) = Pu(X), X ∈ Ω ⊆ R, if the singular part of P is concentrated on a smooth compact surface inside Ω (or on the boundary of Ω, provided the latter is smooth enough), it makes contribution of the order, different from the one produced by the absolutely continuous part, see, e.g., [1] or [18]. It happens always, with the only exception of the case N = 2, where the above orders are the same. For a class of singular self-similar measures P , K.Naimark and M.Solomyak established in [28] two-sided estimates for eigenvalues. And it turned out there that the order of two-sided eigenvalue estimates depends generally on the parameters used in the construction of the measure, in particular, on the Hausdorff dimension of its support. However, in the single case, again of the dimension
{"title":"Eigenvalues of singular measures and Connes’ noncommutative integration","authors":"G. Rozenblum","doi":"10.4171/jst/401","DOIUrl":"https://doi.org/10.4171/jst/401","url":null,"abstract":"In the recent paper [32] the authors have considered the Birman-Schwinger (Cwikel) type operators in a domain Ω ⊆ R, having the form TP = A∗PA. Here A is a pseudodifferential operator in Ω of order −l = −N/2 and P = V μ is a finite signed measure containing a singular part. We found out there that for such operators, properly defined using quadratic forms, for a special class of measures, an estimate for eigenvalues λk = λ ± k (TP ) holds with order λ ± k = O(k −1) with coefficient involving an Orlicz norm of the weight function V . For a subclass of such measures, namely, for the ones whose singular part is a finite sum of measures absolutely continuous with respect to the surface measures on compact Lipschitz surfaces of arbitrary dimension, an asymptotic formula for eigenvalues was proved, with all surfaces, independently of their dimension, making the same order contributions. In the present paper we discuss some generalizations of these results and their consequences for introducing Connes’ integration with respect to singular measures. Our considerations are based upon the variational (via quadratic forms) approach to the spectral analysis of differential operators in a singular setting, in the form developed in 60-s and 70-s by M.Sh. Birman and M.Z. Solomyak. This approach enables one to obtain, for rather general spectral problems, eigenvalue estimates, sharp both in order and in the class of coefficients involved, this sharpness confirmed by exact asymptotic eigenvalue formulas. In the initial setting, this approach was applied to measures P absolutely continuous with respect to Lebesgue measure. Passing to singular measures, it was found that, for the equation −λ∆(X) = Pu(X), X ∈ Ω ⊆ R, if the singular part of P is concentrated on a smooth compact surface inside Ω (or on the boundary of Ω, provided the latter is smooth enough), it makes contribution of the order, different from the one produced by the absolutely continuous part, see, e.g., [1] or [18]. It happens always, with the only exception of the case N = 2, where the above orders are the same. For a class of singular self-similar measures P , K.Naimark and M.Solomyak established in [28] two-sided estimates for eigenvalues. And it turned out there that the order of two-sided eigenvalue estimates depends generally on the parameters used in the construction of the measure, in particular, on the Hausdorff dimension of its support. However, in the single case, again of the dimension","PeriodicalId":48789,"journal":{"name":"Journal of Spectral Theory","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2021-03-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45683274","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}
A. Girouard, M. Karpukhin, M. Levitin, I. Polterovich
How close is the Dirichlet-to-Neumann (DtN) map to the square root of the corresponding boundary Laplacian? This question has been actively investigated in recent years. Somewhat surprisingly, a lot of techniques involved can be traced back to a newly rediscovered manuscript of H"ormander from the 1950s. We present H"ormander's approach and its applications, with an emphasis on eigenvalue estimates and spectral asymptotics. In particular, we obtain results for the DtN maps on non-smooth boundaries in the Riemannian setting, the DtN operators for the Helmholtz equation and the DtN operators on differential forms.
{"title":"The Dirichlet-to-Neumann map, the boundary Laplacian, and Hörmander’s rediscovered manuscript","authors":"A. Girouard, M. Karpukhin, M. Levitin, I. Polterovich","doi":"10.4171/jst/399","DOIUrl":"https://doi.org/10.4171/jst/399","url":null,"abstract":"How close is the Dirichlet-to-Neumann (DtN) map to the square root of the corresponding boundary Laplacian? This question has been actively investigated in recent years. Somewhat surprisingly, a lot of techniques involved can be traced back to a newly rediscovered manuscript of H\"ormander from the 1950s. We present H\"ormander's approach and its applications, with an emphasis on eigenvalue estimates and spectral asymptotics. In particular, we obtain results for the DtN maps on non-smooth boundaries in the Riemannian setting, the DtN operators for the Helmholtz equation and the DtN operators on differential forms.","PeriodicalId":48789,"journal":{"name":"Journal of Spectral Theory","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2021-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47482365","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}
We affirmatively settle the question on existence of a real-valued higher order spectral shift function for a pair of self-adjoint operators $H$ and $V$ such that $V$ is bounded and $V(H-iI)^{-1}$ belongs to a Schatten-von Neumann ideal $mathcal{S}^n$ of compact operators in a separable Hilbert space. We also show that the function satisfies the same trace formula as in the known case of $Vinmathcal{S}^n$ and that it is unique up to a polynomial summand of order $n-1$. Our result significantly advances earlier partial results where counterparts of the spectral shift function for noncompact perturbations lacked real-valuedness and aforementioned uniqueness as well as appeared in more complicated trace formulas for much more restrictive sets of functions. Our result applies to models arising in noncommutative geometry and mathematical physics.
{"title":"Spectral shift for relative Schatten class perturbations","authors":"T. V. Nuland, A. Skripka","doi":"10.4171/jst/425","DOIUrl":"https://doi.org/10.4171/jst/425","url":null,"abstract":"We affirmatively settle the question on existence of a real-valued higher order spectral shift function for a pair of self-adjoint operators $H$ and $V$ such that $V$ is bounded and $V(H-iI)^{-1}$ belongs to a Schatten-von Neumann ideal $mathcal{S}^n$ of compact operators in a separable Hilbert space. We also show that the function satisfies the same trace formula as in the known case of $Vinmathcal{S}^n$ and that it is unique up to a polynomial summand of order $n-1$. Our result significantly advances earlier partial results where counterparts of the spectral shift function for noncompact perturbations lacked real-valuedness and aforementioned uniqueness as well as appeared in more complicated trace formulas for much more restrictive sets of functions. Our result applies to models arising in noncommutative geometry and mathematical physics.","PeriodicalId":48789,"journal":{"name":"Journal of Spectral Theory","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2021-01-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46207659","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}
We study the discrete Gel'fand's inverse boundary spectral problem of determining a finite weighted graph. Suppose that the set of vertices of the graph is a union of two disjoint sets: $X=Bcup G$, where $B$ is called the set of the boundary vertices and $G$ is called the set of the interior vertices. We consider the case where the vertices in the set $G$ and the edges connecting them are unknown. Assume that we are given the set $B$ and the pairs $(lambda_j,phi_j|_B)$, where $lambda_j$ are the eigenvalues of the graph Laplacian and $phi_j|_B$ are the values of the corresponding eigenfunctions at the vertices in $B$. We show that the graph structure, namely the unknown vertices in $G$ and the edges connecting them, along with the weights, can be uniquely determined from the given data, if every boundary vertex is connected to only one interior vertex and the graph satisfies the following property: any subset $Ssubseteq G$ of cardinality $|S|geqslant 2$ contains two extreme points. A point $xin S$ is called an extreme point of $S$ if there exists a point $zin B$ such that $x$ is the unique nearest point in $S$ from $z$ with respect to the graph distance. This property is valid for several standard types of lattices and their perturbations.
{"title":"Gelfand’s inverse problem for the graph Laplacian","authors":"Emilia Blaasten, H. Isozaki, M. Lassas, Jin Lu","doi":"10.4171/jst/455","DOIUrl":"https://doi.org/10.4171/jst/455","url":null,"abstract":"We study the discrete Gel'fand's inverse boundary spectral problem of determining a finite weighted graph. Suppose that the set of vertices of the graph is a union of two disjoint sets: $X=Bcup G$, where $B$ is called the set of the boundary vertices and $G$ is called the set of the interior vertices. We consider the case where the vertices in the set $G$ and the edges connecting them are unknown. Assume that we are given the set $B$ and the pairs $(lambda_j,phi_j|_B)$, where $lambda_j$ are the eigenvalues of the graph Laplacian and $phi_j|_B$ are the values of the corresponding eigenfunctions at the vertices in $B$. We show that the graph structure, namely the unknown vertices in $G$ and the edges connecting them, along with the weights, can be uniquely determined from the given data, if every boundary vertex is connected to only one interior vertex and the graph satisfies the following property: any subset $Ssubseteq G$ of cardinality $|S|geqslant 2$ contains two extreme points. A point $xin S$ is called an extreme point of $S$ if there exists a point $zin B$ such that $x$ is the unique nearest point in $S$ from $z$ with respect to the graph distance. This property is valid for several standard types of lattices and their perturbations.","PeriodicalId":48789,"journal":{"name":"Journal of Spectral Theory","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2021-01-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47462875","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}
We study the trace class perturbations of the half-line, discrete Laplacian and obtain a new bound for the perturbation determinant of the corresponding non-self-adjoint Jacobi operator. Based on this bound, we obtain the Lieb–Thirring inequalities for such operators. The spectral enclosure for the discrete spectrum and embedded eigenvalues are also discussed. In memory of Sergey Naboko (1950–2020) Introduction In the last two decades there was a splash of activity around the spectral theory of non-self-adjoint perturbations of some classical operators of mathematical physics, such as the Laplace and Dirac operators on the whole space, their fractional powers, and others. Recently, there has been some interest in studying certain discrete models of the above problem. In particular, the structure of the spectrum for compact, non-self-adjoint perturbations of the free Jacobi and the discrete Dirac operators has attracted much attention lately. Actually the problem concerns the discrete component of the spectrum and the rate of its accumulation to the essential spectrum. Such type of results under the various assumptions on the perturbations are united under a common name Lieb–Thirring inequalities. For a nice account of the existing results on the problem for non-self-adjoint, two-sided Jacobi operators, the reader may consult two recent surveys [7] and [10, Section 5.13] and references therein. The spectral theory of semi-infinite, self-adjoint Jacobi matrices is quite popular owing to their tight relation to the theory of orthogonal polynomials on the real line [19]. In contrast, there are only a few papers where semiinfinite, non-self-adjoint Jacobi matrices are examined [18, 1, 2, 8, 13, 14, 4]. The main object under consideration is a semi-infinite Jacobi matrix (0.1) J({aj}, {bj}, {cj})j∈N = b1 c1 a1 b2 c2 a2 b3 c3 . . . . . . . . . , Date: August 11, 2021. 2010 Mathematics Subject Classification. 47B36, 47A10, 47A75.
{"title":"Perturbation determinants and discrete spectra of semi-infinite non-self-adjoint Jacobi operators","authors":"L. Golinskii","doi":"10.4171/jst/420","DOIUrl":"https://doi.org/10.4171/jst/420","url":null,"abstract":"We study the trace class perturbations of the half-line, discrete Laplacian and obtain a new bound for the perturbation determinant of the corresponding non-self-adjoint Jacobi operator. Based on this bound, we obtain the Lieb–Thirring inequalities for such operators. The spectral enclosure for the discrete spectrum and embedded eigenvalues are also discussed. In memory of Sergey Naboko (1950–2020) Introduction In the last two decades there was a splash of activity around the spectral theory of non-self-adjoint perturbations of some classical operators of mathematical physics, such as the Laplace and Dirac operators on the whole space, their fractional powers, and others. Recently, there has been some interest in studying certain discrete models of the above problem. In particular, the structure of the spectrum for compact, non-self-adjoint perturbations of the free Jacobi and the discrete Dirac operators has attracted much attention lately. Actually the problem concerns the discrete component of the spectrum and the rate of its accumulation to the essential spectrum. Such type of results under the various assumptions on the perturbations are united under a common name Lieb–Thirring inequalities. For a nice account of the existing results on the problem for non-self-adjoint, two-sided Jacobi operators, the reader may consult two recent surveys [7] and [10, Section 5.13] and references therein. The spectral theory of semi-infinite, self-adjoint Jacobi matrices is quite popular owing to their tight relation to the theory of orthogonal polynomials on the real line [19]. In contrast, there are only a few papers where semiinfinite, non-self-adjoint Jacobi matrices are examined [18, 1, 2, 8, 13, 14, 4]. The main object under consideration is a semi-infinite Jacobi matrix (0.1) J({aj}, {bj}, {cj})j∈N = b1 c1 a1 b2 c2 a2 b3 c3 . . . . . . . . . , Date: August 11, 2021. 2010 Mathematics Subject Classification. 47B36, 47A10, 47A75.","PeriodicalId":48789,"journal":{"name":"Journal of Spectral Theory","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2021-01-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45994933","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}
Consider a random Schr"odinger-type operator of the form $H:=-H_X+V+xi$ acting on a general graph $mathscr G=(mathscr V,mathscr E)$, where $H_X$ is the generator of a Markov process $X$ on $mathscr G$, $V$ is a deterministic potential with sufficient growth (so that $H$ has a purely discrete spectrum), and $xi$ is a random noise with at-most-exponential tails. We prove that $H$'s eigenvalue point process is number rigid in the sense of Ghosh and Peres (Duke Math. J. 166 (2017), no. 10, 1789--1858); that is, the number of eigenvalues in any bounded domain $Bsubsetmathbb C$ is determined by the configuration of eigenvalues outside of $B$. Our general setting allows to treat cases where $X$ could be non-symmetric (hence $H$ is non-self-adjoint) and $xi$ has long-range dependence. Our strategy of proof consists of controlling the variance of the trace of the semigroup $mathrm e^{-t H}$ using the Feynman-Kac formula.
{"title":"On spatial conditioning of the spectrum of discrete Random Schrödinger operators","authors":"P. Lamarre, Promit Ghosal, Yuchen Liao","doi":"10.4171/jst/415","DOIUrl":"https://doi.org/10.4171/jst/415","url":null,"abstract":"Consider a random Schr\"odinger-type operator of the form $H:=-H_X+V+xi$ acting on a general graph $mathscr G=(mathscr V,mathscr E)$, where $H_X$ is the generator of a Markov process $X$ on $mathscr G$, $V$ is a deterministic potential with sufficient growth (so that $H$ has a purely discrete spectrum), and $xi$ is a random noise with at-most-exponential tails. We prove that $H$'s eigenvalue point process is number rigid in the sense of Ghosh and Peres (Duke Math. J. 166 (2017), no. 10, 1789--1858); that is, the number of eigenvalues in any bounded domain $Bsubsetmathbb C$ is determined by the configuration of eigenvalues outside of $B$. Our general setting allows to treat cases where $X$ could be non-symmetric (hence $H$ is non-self-adjoint) and $xi$ has long-range dependence. Our strategy of proof consists of controlling the variance of the trace of the semigroup $mathrm e^{-t H}$ using the Feynman-Kac formula.","PeriodicalId":48789,"journal":{"name":"Journal of Spectral Theory","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44432032","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 construct a family of Berezin-Toeplitz type quantizations of a compact symplectic manifold. For this, we choose a Riemannian metric on the manifold such that the associated Bochner Laplacian has the same local model at each point (this is slightly more general than in almost-Kahler quantization). Then the spectrum of the Bochner Laplacian on high tensor powers $L^p$ of the prequantum line bundle $L$ asymptotically splits into clusters of size ${mathcal O}(p^{3/4})$ around the points $pLambda$, where $Lambda$ is an eigenvalue of the model operator (which can be naturally called a Landau level). We develop the Toeplitz operator calculus with the quantum space, which is the eigenspace of the Bochner Laplacian corresponding to the eigebvalues frrom the cluster. We show that it provides a Berezin-Toeplitz quantization. If the cluster corresponds to a Landau level of multiplicity one, we obtain an algebra of Toeplitz operators and a formal star-product. For the lowest Landau level, it recovers the almost Kahler quantization.
{"title":"Berezin–Toeplitz quantization associated with higher Landau levels of the Bochner Laplacian","authors":"Y. Kordyukov","doi":"10.4171/jst/397","DOIUrl":"https://doi.org/10.4171/jst/397","url":null,"abstract":"In this paper, we construct a family of Berezin-Toeplitz type quantizations of a compact symplectic manifold. For this, we choose a Riemannian metric on the manifold such that the associated Bochner Laplacian has the same local model at each point (this is slightly more general than in almost-Kahler quantization). Then the spectrum of the Bochner Laplacian on high tensor powers $L^p$ of the prequantum line bundle $L$ asymptotically splits into clusters of size ${mathcal O}(p^{3/4})$ around the points $pLambda$, where $Lambda$ is an eigenvalue of the model operator (which can be naturally called a Landau level). We develop the Toeplitz operator calculus with the quantum space, which is the eigenspace of the Bochner Laplacian corresponding to the eigebvalues frrom the cluster. We show that it provides a Berezin-Toeplitz quantization. If the cluster corresponds to a Landau level of multiplicity one, we obtain an algebra of Toeplitz operators and a formal star-product. For the lowest Landau level, it recovers the almost Kahler quantization.","PeriodicalId":48789,"journal":{"name":"Journal of Spectral Theory","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2020-12-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45715486","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}
We study the Schr"odinger operator in the plane with a step magnetic field function. The bottom of its spectrum is described by the infimum of the lowest eigenvalue band function, for which we establish the existence and uniqueness of the non-degenerate minimum. We discuss the curvature effects on the localization properties of magnetic ground states, among other applications.
{"title":"Lowest energy band function for magnetic steps","authors":"W. Assaad, Ayman Kachmar","doi":"10.4171/jst/419","DOIUrl":"https://doi.org/10.4171/jst/419","url":null,"abstract":"We study the Schr\"odinger operator in the plane with a step magnetic field function. The bottom of its spectrum is described by the infimum of the lowest eigenvalue band function, for which we establish the existence and uniqueness of the non-degenerate minimum. We discuss the curvature effects on the localization properties of magnetic ground states, among other applications.","PeriodicalId":48789,"journal":{"name":"Journal of Spectral Theory","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2020-12-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49236046","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}
We show that a weighted manifold which admits a relative Faber Krahn inequality admits the Fefferman Phong inequality V $psi$, $psi$ $le$ CV $psi$ 2 , with the constant depending on a Morrey norm of V , and we deduce from it a condition for a L 2 Hardy inequality to holds, as well as conditions for Schr{"o}dinger operators to be positive. We also obtain an estimate on the bottom of the spectrum for Schr{"o}dinger operators.
{"title":"Lower bound of Schrödinger operators on Riemannian manifolds","authors":"Mael Lansade","doi":"10.4171/jst/448","DOIUrl":"https://doi.org/10.4171/jst/448","url":null,"abstract":"We show that a weighted manifold which admits a relative Faber Krahn inequality admits the Fefferman Phong inequality V $psi$, $psi$ $le$ CV $psi$ 2 , with the constant depending on a Morrey norm of V , and we deduce from it a condition for a L 2 Hardy inequality to holds, as well as conditions for Schr{\"o}dinger operators to be positive. We also obtain an estimate on the bottom of the spectrum for Schr{\"o}dinger operators.","PeriodicalId":48789,"journal":{"name":"Journal of Spectral Theory","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2020-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45249007","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}
We consider a compact perturbation $H_0 = S + K_0^* K_0$ of a self-adjoint operator $S$ with an eigenvalue $lambda^circ$ below its essential spectrum and the corresponding eigenfunction $f$. The perturbation is assumed to be "along" the eigenfunction $f$, namely $K_0f=0$. The eigenvalue $lambda^circ$ belongs to the spectra of both $H_0$ and $S$. Let $S$ have $sigma$ more eigenvalues below $lambda^circ$ than $H_0$; $sigma$ is known as the spectral shift at $lambda^circ$. We now allow the perturbation to vary in a suitable operator space and study the continuation of the eigenvalue $lambda^circ$ in the spectrum of $H(K)=S + K^* K$. We show that the eigenvalue as a function of $K$ has a critical point at $K=K_0$ and the Morse index of this critical point is the spectral shift $sigma$. A version of this theorem also holds for some non-positive perturbations.
{"title":"Spectral shift via “lateral” perturbation","authors":"G. Berkolaiko, P. Kuchment","doi":"10.4171/jst/395","DOIUrl":"https://doi.org/10.4171/jst/395","url":null,"abstract":"We consider a compact perturbation $H_0 = S + K_0^* K_0$ of a self-adjoint operator $S$ with an eigenvalue $lambda^circ$ below its essential spectrum and the corresponding eigenfunction $f$. The perturbation is assumed to be \"along\" the eigenfunction $f$, namely $K_0f=0$. The eigenvalue $lambda^circ$ belongs to the spectra of both $H_0$ and $S$. Let $S$ have $sigma$ more eigenvalues below $lambda^circ$ than $H_0$; $sigma$ is known as the spectral shift at $lambda^circ$. \u0000We now allow the perturbation to vary in a suitable operator space and study the continuation of the eigenvalue $lambda^circ$ in the spectrum of $H(K)=S + K^* K$. We show that the eigenvalue as a function of $K$ has a critical point at $K=K_0$ and the Morse index of this critical point is the spectral shift $sigma$. A version of this theorem also holds for some non-positive perturbations.","PeriodicalId":48789,"journal":{"name":"Journal of Spectral Theory","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2020-11-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41540216","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}
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