We develop a theory of Wilson's adelic Grassmannian ${mathrm{Gr}}^{mathrm{ad}}(R)$ and Segal-Wilson's rational Grasssmannian ${mathrm{Gr}}^ {mathrm{rat}}(R)$ associated to an arbitrary finite dimensional complex algebra $R$. We provide several equivalent descriptions of the former in terms of the indecomposable projective modules of $R$ and its primitive idempotents, and prove that it classifies the bispectral Darboux transformations of the $R$-valued exponential function. The rational Grasssmannian $ {mathrm{Gr}}^{mathrm{rat}}(R)$ is defined by using certain free submodules of $R(z)$ and it is proved that it can be alternatively defined via Wilson type conditions imposed in a representation theoretic settings. A canonical embedding ${mathrm{Gr}}^{mathrm{ad}}(R) hookrightarrow {mathrm{Gr}}^{mathrm{rat}}(R)$ is constructed based on a perfect pairing between the $R$-bimodule of quasiexponentials with values in $R$ and the $R$-bimodule $R[z]$.
{"title":"Adelic and Rational Grassmannians for finite dimensional algebras","authors":"Emil Horozov, Milen Yakimov","doi":"arxiv-2408.04355","DOIUrl":"https://doi.org/arxiv-2408.04355","url":null,"abstract":"We develop a theory of Wilson's adelic Grassmannian\u0000${mathrm{Gr}}^{mathrm{ad}}(R)$ and Segal-Wilson's rational Grasssmannian\u0000${mathrm{Gr}}^ {mathrm{rat}}(R)$ associated to an arbitrary finite\u0000dimensional complex algebra $R$. We provide several equivalent descriptions of\u0000the former in terms of the indecomposable projective modules of $R$ and its\u0000primitive idempotents, and prove that it classifies the bispectral Darboux\u0000transformations of the $R$-valued exponential function. The rational\u0000Grasssmannian $ {mathrm{Gr}}^{mathrm{rat}}(R)$ is defined by using certain\u0000free submodules of $R(z)$ and it is proved that it can be alternatively defined\u0000via Wilson type conditions imposed in a representation theoretic settings. A\u0000canonical embedding ${mathrm{Gr}}^{mathrm{ad}}(R) hookrightarrow\u0000{mathrm{Gr}}^{mathrm{rat}}(R)$ is constructed based on a perfect pairing\u0000between the $R$-bimodule of quasiexponentials with values in $R$ and the\u0000$R$-bimodule $R[z]$.","PeriodicalId":501373,"journal":{"name":"arXiv - MATH - Spectral Theory","volume":"168 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141943885","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 obtain geometric lower bounds for the low Steklov eigenvalues of finite-volume hyperbolic surfaces with geodesic boundary. The bounds we obtain depend on the length of a shortest multi-geodesic disconnecting the surfaces into connected components each containing a boundary component and the rate of dependency on it is sharp. Our result also identifies situations when the bound is independent of the length of this multi-geodesic. The bounds also hold when the Gaussian curvature is bounded between two negative constants and can be viewed as a counterpart of the well-known Schoen-Wolpert-Yau inequality for Laplace eigenvalues. The proof is based on analysing the behaviour of the {corresponding Steklov} eigenfunction on an adapted version of thick-thin decomposition for hyperbolic surfaces with geodesic boundary. Our results extend and improve the previously known result in the compact case obtained by a different method.
{"title":"Geometric bounds for low Steklov eigenvalues of finite volume hyperbolic surfaces","authors":"Asma Hassannezhad, Antoine Métras, Hélène Perrin","doi":"arxiv-2408.04534","DOIUrl":"https://doi.org/arxiv-2408.04534","url":null,"abstract":"We obtain geometric lower bounds for the low Steklov eigenvalues of\u0000finite-volume hyperbolic surfaces with geodesic boundary. The bounds we obtain\u0000depend on the length of a shortest multi-geodesic disconnecting the surfaces\u0000into connected components each containing a boundary component and the rate of\u0000dependency on it is sharp. Our result also identifies situations when the bound\u0000is independent of the length of this multi-geodesic. The bounds also hold when\u0000the Gaussian curvature is bounded between two negative constants and can be\u0000viewed as a counterpart of the well-known Schoen-Wolpert-Yau inequality for\u0000Laplace eigenvalues. The proof is based on analysing the behaviour of the\u0000{corresponding Steklov} eigenfunction on an adapted version of thick-thin\u0000decomposition for hyperbolic surfaces with geodesic boundary. Our results\u0000extend and improve the previously known result in the compact case obtained by\u0000a different method.","PeriodicalId":501373,"journal":{"name":"arXiv - MATH - Spectral Theory","volume":"58 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141943887","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 develop the wave packet decomposition to study the Schrodinger evolution with rough potential. As an application, we obtain the improved bound on the wave propagation for the generic value of a parameter.
我们利用波包分解来研究具有粗糙势的薛定谔演化。作为应用，我们获得了参数一般值的改进波传播约束。
{"title":"Wave packet decomposition for Schrodinger evolution with rough potential and generic value of parameter","authors":"Sergey A. Denisov","doi":"arxiv-2408.03470","DOIUrl":"https://doi.org/arxiv-2408.03470","url":null,"abstract":"We develop the wave packet decomposition to study the Schrodinger evolution\u0000with rough potential. As an application, we obtain the improved bound on the\u0000wave propagation for the generic value of a parameter.","PeriodicalId":501373,"journal":{"name":"arXiv - MATH - Spectral Theory","volume":"18 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141943897","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 prove a general abstract theorem deducing wave expansions as time goes to infinity from resolvent expansions as energy goes to zero, under an assumption of polynomial boundedness of the resolvent at high energy. We give applications to obstacle scattering, to Aharonov--Bohm Hamiltonians, to scattering in a sector, and to scattering by a compactly supported potential.
{"title":"From resolvent expansions at zero to long time wave expansions","authors":"T. J. Christiansen, K. Datchev, M. Yang","doi":"arxiv-2408.03234","DOIUrl":"https://doi.org/arxiv-2408.03234","url":null,"abstract":"We prove a general abstract theorem deducing wave expansions as time goes to\u0000infinity from resolvent expansions as energy goes to zero, under an assumption\u0000of polynomial boundedness of the resolvent at high energy. We give applications\u0000to obstacle scattering, to Aharonov--Bohm Hamiltonians, to scattering in a\u0000sector, and to scattering by a compactly supported potential.","PeriodicalId":501373,"journal":{"name":"arXiv - MATH - Spectral Theory","volume":"23 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141943888","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 compute low energy asymptotics for the resolvent of the Aharonov--Bohm Hamiltonian with multiple poles for both integer and non-integer total fluxes. For integral total flux we reduce to prior results in black-box scattering while for non-integral total flux we build on the corresponding techniques using an appropriately chosen model resolvent. The resolvent expansion can be used to obtain long-time wave asymptotics for the Aharonov--Bohm Hamiltonian with multiple poles. An interesting phenomenon is that if the total flux is an integer then the scattering resembles even-dimensional Euclidean scattering, while if it is half an odd integer then it resembles odd-dimensional Euclidean scattering. The behavior for other values of total flux thus provides an `interpolation' between these.
{"title":"Low energy resolvent asymptotics of the multipole Aharonov--Bohm Hamiltonian","authors":"T. J. Christiansen, K. Datchev, M. Yang","doi":"arxiv-2408.03233","DOIUrl":"https://doi.org/arxiv-2408.03233","url":null,"abstract":"We compute low energy asymptotics for the resolvent of the Aharonov--Bohm\u0000Hamiltonian with multiple poles for both integer and non-integer total fluxes.\u0000For integral total flux we reduce to prior results in black-box scattering\u0000while for non-integral total flux we build on the corresponding techniques\u0000using an appropriately chosen model resolvent. The resolvent expansion can be\u0000used to obtain long-time wave asymptotics for the Aharonov--Bohm Hamiltonian\u0000with multiple poles. An interesting phenomenon is that if the total flux is an\u0000integer then the scattering resembles even-dimensional Euclidean scattering,\u0000while if it is half an odd integer then it resembles odd-dimensional Euclidean\u0000scattering. The behavior for other values of total flux thus provides an\u0000`interpolation' between these.","PeriodicalId":501373,"journal":{"name":"arXiv - MATH - Spectral Theory","volume":"10 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141943735","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 develop a complete stationary scattering theory for Schr"odinger operators on $mathbb R^d$, $dge 2$, with $C^2$ long-range potentials. This extends former results in the literature, in particular [Is1, Is2, II, GY], which all require a higher degree of smoothness. In this sense the spirit of our paper is similar to [H"o2, Chapter XXX], which also develops a scattering theory under the $C^2$ condition, however being very different from ours. While the Agmon-H"ormander theory is based on the Fourier transform, our theory is not and may be seen as more related to our previous approach to scattering theory on manifolds [IS1,IS2,IS3]. The $C^2$ regularity is natural in the Agmon-H"ormander theory as well as in our theory, in fact probably being `optimal' in the Euclidean setting. We prove equivalence of the stationary and time-dependent theories by giving stationary representations of associated time-dependent wave operators. Furthermore we develop a related stationary scattering theory at fixed energy in terms of asymptotics of generalized eigenfunctions of minimal growth. A basic ingredient of our approach is a solution to the eikonal equation constructed from the geometric variational scheme of [CS]. Another key ingredient is strong radiation condition bounds for the limiting resolvents originating in [HS]. They improve formerly known ones [Is1, Sa] and considerably simplify the stationary approach. We obtain the bounds by a new commutator scheme whose elementary form allows a small degree of smoothness.
{"title":"Scattering theory for $C^2$ long-range potentials","authors":"K. Ito, E. Skibsted","doi":"arxiv-2408.02979","DOIUrl":"https://doi.org/arxiv-2408.02979","url":null,"abstract":"We develop a complete stationary scattering theory for Schr\"odinger\u0000operators on $mathbb R^d$, $dge 2$, with $C^2$ long-range potentials. This\u0000extends former results in the literature, in particular [Is1, Is2, II, GY],\u0000which all require a higher degree of smoothness. In this sense the spirit of\u0000our paper is similar to [H\"o2, Chapter XXX], which also develops a scattering\u0000theory under the $C^2$ condition, however being very different from ours. While\u0000the Agmon-H\"ormander theory is based on the Fourier transform, our theory is\u0000not and may be seen as more related to our previous approach to scattering\u0000theory on manifolds [IS1,IS2,IS3]. The $C^2$ regularity is natural in the\u0000Agmon-H\"ormander theory as well as in our theory, in fact probably being\u0000`optimal' in the Euclidean setting. We prove equivalence of the stationary and\u0000time-dependent theories by giving stationary representations of associated\u0000time-dependent wave operators. Furthermore we develop a related stationary\u0000scattering theory at fixed energy in terms of asymptotics of generalized\u0000eigenfunctions of minimal growth. A basic ingredient of our approach is a\u0000solution to the eikonal equation constructed from the geometric variational\u0000scheme of [CS]. Another key ingredient is strong radiation condition bounds for\u0000the limiting resolvents originating in [HS]. They improve formerly known ones\u0000[Is1, Sa] and considerably simplify the stationary approach. We obtain the\u0000bounds by a new commutator scheme whose elementary form allows a small degree\u0000of smoothness.","PeriodicalId":501373,"journal":{"name":"arXiv - MATH - Spectral Theory","volume":"30 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141943890","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}
Let $(X,g)$ be a closed, connected surface, with variable negative curvature. We consider the distribution of eigenvalues of the Laplacian on random covers $X_nto X$ of degree $n$. We focus on the ensemble variance of the smoothed number of eigenvalues of the square root of the positive Laplacian $sqrt{Delta}$ in windows $[lambda-frac 1L,lambda+frac 1L]$, over the set of $n$-sheeted covers of $X$. We first take the limit of large degree $nto +infty$, then we let the energy $lambda$ go to $+infty$ while the window size $frac 1L$ goes to $0$. In this ad hoc limit, local energy averages of the variance converge to an expression corresponding to the variance of the same statistic when considering instead spectra of large random matrices of the Gaussian Orthogonal Ensemble (GOE). By twisting the Laplacian with unitary representations, we are able to observe different statistics, corresponding to the Gaussian Unitary Ensemble (GUE) when time reversal symmetry is broken. These results were shown by F. Naud for the model of random covers of a hyperbolic surface. For an individual cover $X_nto X$, we consider spectral fluctuations of the counting function on $X_n$ around the ensemble average. In the large energy regime, for a typical cover $X_nto X$ of large degree, these fluctuations are shown to approach the GOE result, a phenomenon called ergodicity in Random Matrix Theory. An analogous result for random covers of hyperbolic surfaces was obtained by Y. Maoz.
{"title":"Spectral statistics of the Laplacian on random covers of a closed negatively curved surface","authors":"Julien Moy","doi":"arxiv-2408.02808","DOIUrl":"https://doi.org/arxiv-2408.02808","url":null,"abstract":"Let $(X,g)$ be a closed, connected surface, with variable negative curvature.\u0000We consider the distribution of eigenvalues of the Laplacian on random covers\u0000$X_nto X$ of degree $n$. We focus on the ensemble variance of the smoothed\u0000number of eigenvalues of the square root of the positive Laplacian\u0000$sqrt{Delta}$ in windows $[lambda-frac 1L,lambda+frac 1L]$, over the set\u0000of $n$-sheeted covers of $X$. We first take the limit of large degree $nto\u0000+infty$, then we let the energy $lambda$ go to $+infty$ while the window\u0000size $frac 1L$ goes to $0$. In this ad hoc limit, local energy averages of the\u0000variance converge to an expression corresponding to the variance of the same\u0000statistic when considering instead spectra of large random matrices of the\u0000Gaussian Orthogonal Ensemble (GOE). By twisting the Laplacian with unitary\u0000representations, we are able to observe different statistics, corresponding to\u0000the Gaussian Unitary Ensemble (GUE) when time reversal symmetry is broken.\u0000These results were shown by F. Naud for the model of random covers of a\u0000hyperbolic surface. For an individual cover $X_nto X$, we consider spectral fluctuations of the\u0000counting function on $X_n$ around the ensemble average. In the large energy\u0000regime, for a typical cover $X_nto X$ of large degree, these fluctuations are\u0000shown to approach the GOE result, a phenomenon called ergodicity in Random\u0000Matrix Theory. An analogous result for random covers of hyperbolic surfaces was\u0000obtained by Y. Maoz.","PeriodicalId":501373,"journal":{"name":"arXiv - MATH - Spectral Theory","volume":"34 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141943886","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 establish quantitative Green's function estimates for a class of quasi-periodic (QP) operators on $mathbb{Z}^d$ with power-law long-range hopping and analytic cosine type potentials. As applications, we prove the arithmetic version of localization, the finite volume version of $(frac12-)$-H"older continuity of the IDS, and the absence of eigenvalues (for Aubry dual operators).
{"title":"Green's function estimates for quasi-periodic operators on $mathbb{Z}^d$ with power-law long-range hopping","authors":"Yunfeng Shi, Li Wen","doi":"arxiv-2408.01913","DOIUrl":"https://doi.org/arxiv-2408.01913","url":null,"abstract":"We establish quantitative Green's function estimates for a class of\u0000quasi-periodic (QP) operators on $mathbb{Z}^d$ with power-law long-range\u0000hopping and analytic cosine type potentials. As applications, we prove the\u0000arithmetic version of localization, the finite volume version of\u0000$(frac12-)$-H\"older continuity of the IDS, and the absence of eigenvalues\u0000(for Aubry dual operators).","PeriodicalId":501373,"journal":{"name":"arXiv - MATH - Spectral Theory","volume":"43 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141943894","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 note we study Landis conjecture for positive Schr"odinger operators on graphs. More precisely, we give a decay criterion that ensures when $ mathcal{H} $-harmonic functions for a positive Schr"odinger operator $ mathcal{H} $ with potentials bounded from above by $ 1 $ are trivial. We then specifically look at the special cases of $ mathbb{Z}^{d} $ and regular trees for which we get explicit decay criterion. Moreover, we consider the fractional analogue of Landis conjecture on $ mathbb{Z}^{d} $. Our approach relies on the discrete version of Liouville comparison principle which is also proved in this article.
{"title":"On Landis conjecture for positive Schrödinger operators on graphs","authors":"Ujjal Das, Matthias Keller, Yehuda Pinchover","doi":"arxiv-2408.02149","DOIUrl":"https://doi.org/arxiv-2408.02149","url":null,"abstract":"In this note we study Landis conjecture for positive Schr\"odinger operators\u0000on graphs. More precisely, we give a decay criterion that ensures when $\u0000mathcal{H} $-harmonic functions for a positive Schr\"odinger operator $\u0000mathcal{H} $ with potentials bounded from above by $ 1 $ are trivial. We then\u0000specifically look at the special cases of $ mathbb{Z}^{d} $ and regular trees\u0000for which we get explicit decay criterion. Moreover, we consider the fractional\u0000analogue of Landis conjecture on $ mathbb{Z}^{d} $. Our approach relies on the\u0000discrete version of Liouville comparison principle which is also proved in this\u0000article.","PeriodicalId":501373,"journal":{"name":"arXiv - MATH - Spectral Theory","volume":"100 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141943902","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}
Burq-G'erard-Tzvetkov and Hu established $L^p$ estimates for the restriction of Laplace-Beltrami eigenfunctions to submanifolds. We investigate the eigenfunctions of the Schr"odinger operators with critically singular potentials, and estimate the $L^p$ norms and period integrals for their restriction to submanifolds. Recently, Blair-Sire-Sogge obtained global $L^p$ bounds for Schr"odinger eigenfunctions by the resolvent method. Due to the Sobolev trace inequalities, the resolvent method cannot work for submanifolds of all dimensions. We get around this difficulty and establish spectral projection bounds by the wave kernel techniques and the bootstrap argument involving an induction on the dimensions of the submanifolds.
{"title":"Restriction of Schrödinger eigenfunctions to submanifolds","authors":"Xiaoqi Huang, Xing Wang, Cheng Zhang","doi":"arxiv-2408.01947","DOIUrl":"https://doi.org/arxiv-2408.01947","url":null,"abstract":"Burq-G'erard-Tzvetkov and Hu established $L^p$ estimates for the restriction\u0000of Laplace-Beltrami eigenfunctions to submanifolds. We investigate the\u0000eigenfunctions of the Schr\"odinger operators with critically singular\u0000potentials, and estimate the $L^p$ norms and period integrals for their\u0000restriction to submanifolds. Recently, Blair-Sire-Sogge obtained global $L^p$\u0000bounds for Schr\"odinger eigenfunctions by the resolvent method. Due to the\u0000Sobolev trace inequalities, the resolvent method cannot work for submanifolds\u0000of all dimensions. We get around this difficulty and establish spectral\u0000projection bounds by the wave kernel techniques and the bootstrap argument\u0000involving an induction on the dimensions of the submanifolds.","PeriodicalId":501373,"journal":{"name":"arXiv - MATH - Spectral Theory","volume":"17 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141943893","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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