In this article, we investigate spectral properties of the sublaplacian $-Delta_{G}$ on the Engel group, which is the main example of a Carnot group of step 3. We develop a new approach to the Fourier analysis on the Engel group in terms of a frequency set. This enables us to give fine estimates on the convolution kernel satisfying $F(-Delta_{G})u=ustar k_{F}$, for suitable scalar functions $F$, and in turn to obtain proofs of classical functional embeddings, via Fourier techniques. This analysis requires a summability property on the spectrum of the quartic oscillator, which we obtain by means of semiclassical techniques and which is of independent interest.
{"title":"Spectral summability for the quartic oscillator with applications to the Engel group","authors":"Hajer Bahouri, Davide Barilari, Isabelle Gallagher, Matthieu Léautaud","doi":"10.4171/jst/464","DOIUrl":"https://doi.org/10.4171/jst/464","url":null,"abstract":"In this article, we investigate spectral properties of the sublaplacian $-Delta_{G}$ on the Engel group, which is the main example of a Carnot group of step 3. We develop a new approach to the Fourier analysis on the Engel group in terms of a frequency set. This enables us to give fine estimates on the convolution kernel satisfying $F(-Delta_{G})u=ustar k_{F}$, for suitable scalar functions $F$, and in turn to obtain proofs of classical functional embeddings, via Fourier techniques. This analysis requires a summability property on the spectrum of the quartic oscillator, which we obtain by means of semiclassical techniques and which is of independent interest.","PeriodicalId":48789,"journal":{"name":"Journal of Spectral Theory","volume":"68 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135251905","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 investigate two-dimensional canonical systems $y'=zJHy$ on an interval, with positive semi-definite Hamiltonian $H$. Let $q_H$ be the Weyl coefficient of the system. We prove a formula that determines the imaginary part of $q_H$ along the imaginary axis up to multiplicative constants, which are independent of $H$. We also provide versions of this result for Sturm–Liouville operators and Krein strings. Using classical Abelian–Tauberian theorems, we deduce characterizations of spectral properties such as integrability of a given comparison function with respect to the spectral measure $mu_H$, and boundedness of the distribution function of $mu_H$ relative to a given comparison function. We study in depth Hamiltonians for which $arg q_H(ir)$ approaches $0$ or $pi$ (at least on a subsequence). It turns out that this behavior of $q_H(ir)$ imposes a substantial restriction on the growth of $|q_H(ir)|$. Our results in this context are interesting also from a function theoretic point of view.
{"title":"A quantitative formula for the imaginary part of a Weyl coefficient","authors":"Jakob Reiffenstein","doi":"10.4171/jst/457","DOIUrl":"https://doi.org/10.4171/jst/457","url":null,"abstract":"We investigate two-dimensional canonical systems $y'=zJHy$ on an interval, with positive semi-definite Hamiltonian $H$. Let $q_H$ be the Weyl coefficient of the system. We prove a formula that determines the imaginary part of $q_H$ along the imaginary axis up to multiplicative constants, which are independent of $H$. We also provide versions of this result for Sturm–Liouville operators and Krein strings. Using classical Abelian–Tauberian theorems, we deduce characterizations of spectral properties such as integrability of a given comparison function with respect to the spectral measure $mu_H$, and boundedness of the distribution function of $mu_H$ relative to a given comparison function. We study in depth Hamiltonians for which $arg q_H(ir)$ approaches $0$ or $pi$ (at least on a subsequence). It turns out that this behavior of $q_H(ir)$ imposes a substantial restriction on the growth of $|q_H(ir)|$. Our results in this context are interesting also from a function theoretic point of view.","PeriodicalId":48789,"journal":{"name":"Journal of Spectral Theory","volume":"40 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135252933","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 present conditions for a family $(A(x))_{xinmathbb{R}^{d}}$ of self-adjoint operators in $H^{r}=mathbb{C}^{r}otimes H$ for a separable complex Hilbert space $H$, such that the Callias operator $D=icnabla+A(X)$ satisfies that $(D^{ast}D+1)^{-N}-(DD^{ast}+1)^{-N}$ is trace class in $L^2(mathbb{R}^{d},H^{r})$. Here, $cnabla$ is the Dirac operator associated to a Clifford multiplication $c$ of rank $r$ on $mathbb{R}^{d}$, and $A(X)$ is fibre-wise multiplication with $A(x)$ in $L^2(mathbb{R}^{d},H^{r})$.
{"title":"Trace class properties of resolvents of Callias operators","authors":"Oliver Fürst","doi":"10.4171/jst/451","DOIUrl":"https://doi.org/10.4171/jst/451","url":null,"abstract":"We present conditions for a family $(A(x))_{xinmathbb{R}^{d}}$ of self-adjoint operators in $H^{r}=mathbb{C}^{r}otimes H$ for a separable complex Hilbert space $H$, such that the Callias operator $D=icnabla+A(X)$ satisfies that $(D^{ast}D+1)^{-N}-(DD^{ast}+1)^{-N}$ is trace class in $L^2(mathbb{R}^{d},H^{r})$. Here, $cnabla$ is the Dirac operator associated to a Clifford multiplication $c$ of rank $r$ on $mathbb{R}^{d}$, and $A(X)$ is fibre-wise multiplication with $A(x)$ in $L^2(mathbb{R}^{d},H^{r})$.","PeriodicalId":48789,"journal":{"name":"Journal of Spectral Theory","volume":"25 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135252673","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 consider the transport properties of the class of limit-periodic continuum Schrödinger operators whose potentials are approximated exponentially quickly by a sequence of periodic functions. For such an operator $H$, and $X_H(t)$ the Heisenberg evolution of the position operator, we show the limit of $frac{1}{t}X_H(t)psi$ as $ttoinfty$ exists and is nonzero for $psine 0$ belonging to a dense subspace of initial states which are sufficiently regular and of suitably rapid decay. This is viewed as a particularly strong form of ballistic transport, and this is the first time it has been proven in a continuum almost periodic non-periodic setting. In particular, this statement implies that for the initial states considered, the second moment grows quadratically in time.
{"title":"Ballistic transport for limit-periodic Schrödinger operators in one dimension","authors":"Giorgio Young","doi":"10.4171/jst/463","DOIUrl":"https://doi.org/10.4171/jst/463","url":null,"abstract":"In this paper, we consider the transport properties of the class of limit-periodic continuum Schrödinger operators whose potentials are approximated exponentially quickly by a sequence of periodic functions. For such an operator $H$, and $X_H(t)$ the Heisenberg evolution of the position operator, we show the limit of $frac{1}{t}X_H(t)psi$ as $ttoinfty$ exists and is nonzero for $psine 0$ belonging to a dense subspace of initial states which are sufficiently regular and of suitably rapid decay. This is viewed as a particularly strong form of ballistic transport, and this is the first time it has been proven in a continuum almost periodic non-periodic setting. In particular, this statement implies that for the initial states considered, the second moment grows quadratically in time.","PeriodicalId":48789,"journal":{"name":"Journal of Spectral Theory","volume":"57 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136264000","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 linear symmetric operator in a Hilbert space that is neither bounded from above nor from below, admits a block decomposition corresponding to an orthogonal splitting of the Hilbert space and has a variational gap property associated with the block decomposition. A typical example is the Dirac–Coulomb operator defined on $C^infty_c(mathbb{R}^3setminus{0}, mathbb{C}^4)$. In this paper we define a distinguished self-adjoint extension with a spectral gap and characterize its eigenvalues in that gap by a min-max principle. This has been done in the past under technical conditions. Here we use a different, geometric strategy, to achieve that goal by making only minimal assumptions. Our result applied to the Dirac–Coulomb-like Hamitonians covers sign-changing potentials as well as molecules with an arbitrary number of nuclei having atomic numbers less than or equal to 137
{"title":"Distinguished self-adjoint extension and eigenvalues of operators with gaps. Application to Dirac–Coulomb operators","authors":"Jean Dolbeault, Maria J Esteban, Eric Séré","doi":"10.4171/jst/461","DOIUrl":"https://doi.org/10.4171/jst/461","url":null,"abstract":"We consider a linear symmetric operator in a Hilbert space that is neither bounded from above nor from below, admits a block decomposition corresponding to an orthogonal splitting of the Hilbert space and has a variational gap property associated with the block decomposition. A typical example is the Dirac–Coulomb operator defined on $C^infty_c(mathbb{R}^3setminus{0}, mathbb{C}^4)$. In this paper we define a distinguished self-adjoint extension with a spectral gap and characterize its eigenvalues in that gap by a min-max principle. This has been done in the past under technical conditions. Here we use a different, geometric strategy, to achieve that goal by making only minimal assumptions. Our result applied to the Dirac–Coulomb-like Hamitonians covers sign-changing potentials as well as molecules with an arbitrary number of nuclei having atomic numbers less than or equal to 137","PeriodicalId":48789,"journal":{"name":"Journal of Spectral Theory","volume":"8 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136263697","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 nonlinear Helmholtz equation $(Delta - lambda^2)u = pm |u|^{p-1}u$ on $mathbb{R}^n$, $lambda > 0$, $p in mathbb{N}$ odd, and more generally $(Delta_g + V - lambda^2)u = N[u]$, where $Delta_g$ is the (positive) Laplace–Beltrami operator on an asymptotically Euclidean or conic manifold, $V$ is a short range potential, and $N[u]$ is a more general polynomial nonlinearity. Under the conditions $(p-1)(n-1)/2 > 2$ and $k > (n-1)/2$, for every $f in H^k(mathbb{S}^{n-1}_omega)$ of sufficiently small norm, we show there is a nonlinear Helmholtz eigenfunction taking the form $$ u(r, omega) = r^{-(n-1)/2} ( e^{-ilambda r} f(omega) + e^{+ilambda r} b(omega) + O(r^{-epsilon}) ), quad text{as } r to infty, $$ for some $b in H^k(mathbb{S}_omega^{n-1})$ and $epsilon > 0$. That is, the nonlinear scattering matrix $f mapsto b$ preserves Sobolev regularity, which is an improvement over the authors' previous work (2020) with Zhang, that proved a similar result with a loss of four derivatives.
我们在$mathbb{R}^n$, $lambda > 0$, $p in mathbb{N}$奇数和更一般的$(Delta_g + V - lambda^2)u = N[u]$上研究非线性亥姆霍兹方程$(Delta - lambda^2)u = pm |u|^{p-1}u$，其中$Delta_g$是渐近欧几里得流形或二次流形上的(正)拉普拉斯-贝尔特拉米算子，$V$是短程势，$N[u]$是更一般的多项式非线性。在$(p-1)(n-1)/2 > 2$和$k > (n-1)/2$条件下，对于每一个足够小范数的$f in H^k(mathbb{S}^{n-1}_omega)$，我们证明了对于某些$b in H^k(mathbb{S}_omega^{n-1})$和$epsilon > 0$存在一个非线性亥姆霍兹特征函数，其形式为$$ u(r, omega) = r^{-(n-1)/2} ( e^{-ilambda r} f(omega) + e^{+ilambda r} b(omega) + O(r^{-epsilon}) ), quad text{as } r to infty, $$。也就是说，非线性散射矩阵$f mapsto b$保留了Sobolev正则性，这是作者与Zhang之前的工作(2020)的改进，该工作证明了类似的结果，但损失了四个导数。
{"title":"Regularity of the scattering matrix for nonlinear Helmholtz eigenfunctions","authors":"Jesse Gell-Redman, Andrew Hassell, Jacob Shapiro","doi":"10.4171/jst/460","DOIUrl":"https://doi.org/10.4171/jst/460","url":null,"abstract":"We study the nonlinear Helmholtz equation $(Delta - lambda^2)u = pm |u|^{p-1}u$ on $mathbb{R}^n$, $lambda > 0$, $p in mathbb{N}$ odd, and more generally $(Delta_g + V - lambda^2)u = N[u]$, where $Delta_g$ is the (positive) Laplace–Beltrami operator on an asymptotically Euclidean or conic manifold, $V$ is a short range potential, and $N[u]$ is a more general polynomial nonlinearity. Under the conditions $(p-1)(n-1)/2 > 2$ and $k > (n-1)/2$, for every $f in H^k(mathbb{S}^{n-1}_omega)$ of sufficiently small norm, we show there is a nonlinear Helmholtz eigenfunction taking the form $$ u(r, omega) = r^{-(n-1)/2} ( e^{-ilambda r} f(omega) + e^{+ilambda r} b(omega) + O(r^{-epsilon}) ), quad text{as } r to infty, $$ for some $b in H^k(mathbb{S}_omega^{n-1})$ and $epsilon > 0$. That is, the nonlinear scattering matrix $f mapsto b$ preserves Sobolev regularity, which is an improvement over the authors' previous work (2020) with Zhang, that proved a similar result with a loss of four derivatives.","PeriodicalId":48789,"journal":{"name":"Journal of Spectral Theory","volume":"72 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136263992","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}
This paper proves a probabilistic Weyl-law for the spectrum of randomly perturbed Berezin–Toeplitz operators, generalizing a result proven by Martin Vogel (2020). This is done following Vogel’s strategy using the exotic symbol calculus developed by the author (2022).
{"title":"A probabilistic Weyl-law for perturbed Berezin–Toeplitz operators","authors":"Izak Oltman","doi":"10.4171/jst/459","DOIUrl":"https://doi.org/10.4171/jst/459","url":null,"abstract":"This paper proves a probabilistic Weyl-law for the spectrum of randomly perturbed Berezin–Toeplitz operators, generalizing a result proven by Martin Vogel (2020). This is done following Vogel’s strategy using the exotic symbol calculus developed by the author (2022).","PeriodicalId":48789,"journal":{"name":"Journal of Spectral Theory","volume":"344 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136306644","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 investigate the existence or non-existence of spectral minimal partitions of unbounded metric graphs, where the operator applied to each of the partition elements is a Schrödinger operator of the form $-Delta + V$ with suitable (electric) potential $V$, which is taken as a fixed, underlying function on the whole graph. We show that there is a strong link between spectral minimal partitions and infimal partition energies on the one hand, and the infimum $Sigma$ of the essential spectrum of the corresponding Schrödinger operator on the whole graph on the other. Namely, we show that for any $kinmathbb{N}$, the infimal energy among all admissible $k$-partitions is bounded from above by $Sigma$, and if it is strictly below $Sigma$, then a spectral minimal $k$-partition exists. We illustrate our results with several examples of existence and non-existence of minimal partitions of unbounded and infinite graphs, with and without potentials. The nature of the proofs, a key ingredient of which is a version of the characterization of the infimum of the essential spectrum known as Persson’s theorem for quantum graphs, strongly suggests that corresponding results should hold for Schrödinger operator-based partitions of unbounded domains in Euclidean space.
{"title":"Spectral minimal partitions of unbounded metric graphs","authors":"Matthias Hofmann, James Kennedy, Andrea Serio","doi":"10.4171/jst/462","DOIUrl":"https://doi.org/10.4171/jst/462","url":null,"abstract":"We investigate the existence or non-existence of spectral minimal partitions of unbounded metric graphs, where the operator applied to each of the partition elements is a Schrödinger operator of the form $-Delta + V$ with suitable (electric) potential $V$, which is taken as a fixed, underlying function on the whole graph. We show that there is a strong link between spectral minimal partitions and infimal partition energies on the one hand, and the infimum $Sigma$ of the essential spectrum of the corresponding Schrödinger operator on the whole graph on the other. Namely, we show that for any $kinmathbb{N}$, the infimal energy among all admissible $k$-partitions is bounded from above by $Sigma$, and if it is strictly below $Sigma$, then a spectral minimal $k$-partition exists. We illustrate our results with several examples of existence and non-existence of minimal partitions of unbounded and infinite graphs, with and without potentials. The nature of the proofs, a key ingredient of which is a version of the characterization of the infimum of the essential spectrum known as Persson’s theorem for quantum graphs, strongly suggests that corresponding results should hold for Schrödinger operator-based partitions of unbounded domains in Euclidean space.","PeriodicalId":48789,"journal":{"name":"Journal of Spectral Theory","volume":"34 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136306455","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 describe the spectrum structure for the restricted Dirichlet fractional Laplacian in multi-tubes, i.e., domains with cylindrical outlets to infinity. Some new effects in comparison with the local case are discovered.
{"title":"Dirichlet fractional Laplacian in multi-tubes","authors":"Fedor Bakharev, Alexander Nazarov","doi":"10.4171/jst/458","DOIUrl":"https://doi.org/10.4171/jst/458","url":null,"abstract":"We describe the spectrum structure for the restricted Dirichlet fractional Laplacian in multi-tubes, i.e., domains with cylindrical outlets to infinity. Some new effects in comparison with the local case are discovered.","PeriodicalId":48789,"journal":{"name":"Journal of Spectral Theory","volume":"24 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135690424","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 Jacobi matrices on star-like graphs, which are graphs that are given by the pasting of a finite number of half-lines to a compact graph. Specifically, we extend subordinacy theory to this type of graphs, that is, we find a connection between asymptotic properties of solutions to the eigenvalue equations and continuity properties of the spectral measure with respect to the Lebesgue measure. We also use this theory in order to derive results regarding the multiplicity of the singular spectrum.
{"title":"Subordinacy theory on star-like graphs","authors":"Netanel Levi","doi":"10.4171/jst/450","DOIUrl":"https://doi.org/10.4171/jst/450","url":null,"abstract":"We study Jacobi matrices on star-like graphs, which are graphs that are given by the pasting of a finite number of half-lines to a compact graph. Specifically, we extend subordinacy theory to this type of graphs, that is, we find a connection between asymptotic properties of solutions to the eigenvalue equations and continuity properties of the spectral measure with respect to the Lebesgue measure. We also use this theory in order to derive results regarding the multiplicity of the singular spectrum.","PeriodicalId":48789,"journal":{"name":"Journal of Spectral Theory","volume":"37 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135734872","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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