We discuss some perturbation results concerning certain pairs of sequences of vectors in a Hilbert space [Formula: see text] and producing new sequences, which share, with the original ones, reconstruction formulas on a dense subspace of [Formula: see text] or on the whole space. We also propose some preliminary results on the same issue, but in a distributional settings.
{"title":"Some perturbation results for quasi-bases and other sequences of vectors","authors":"F. Bagarello, R. Corso","doi":"10.1063/5.0131314","DOIUrl":"https://doi.org/10.1063/5.0131314","url":null,"abstract":"We discuss some perturbation results concerning certain pairs of sequences of vectors in a Hilbert space [Formula: see text] and producing new sequences, which share, with the original ones, reconstruction formulas on a dense subspace of [Formula: see text] or on the whole space. We also propose some preliminary results on the same issue, but in a distributional settings.","PeriodicalId":50141,"journal":{"name":"Journal of Mathematical Physics Analysis Geometry","volume":"71 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-03-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73910170","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Short-ranged and line-gapped non-Hermitian Hamiltonians have strong topological invariants given by an index of an associated Fredholm operator. It is shown how these invariants can be accessed via the signature of a suitable spectral localizer. This numerical technique is implemented in an example with relevance to the design of topological photonic systems, such as topological lasers.
{"title":"Spectral localizer for line-gapped non-Hermitian systems","authors":"A. Cerjan, L. Koekenbier, H. Schulz-Baldes","doi":"10.1063/5.0150995","DOIUrl":"https://doi.org/10.1063/5.0150995","url":null,"abstract":"Short-ranged and line-gapped non-Hermitian Hamiltonians have strong topological invariants given by an index of an associated Fredholm operator. It is shown how these invariants can be accessed via the signature of a suitable spectral localizer. This numerical technique is implemented in an example with relevance to the design of topological photonic systems, such as topological lasers.","PeriodicalId":50141,"journal":{"name":"Journal of Mathematical Physics Analysis Geometry","volume":"39 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-03-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72769067","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We provide an elementary proof and refinement of a well-known idea from physics: a chiral-symmetric local Hamiltonian on a half-space has the same signed number of edge-localized states with energies in the bulk bandgap as its bulk winding number. The requirement of non-elementary methods to relate generic and non-generic cases is emphasized. Our hands-on approach complements a quick abstract proof based on the classical index theory of Toeplitz operators.
{"title":"Topological edge states of 1D chains and index theory","authors":"Guo Chuan Thiang","doi":"10.1063/5.0150870","DOIUrl":"https://doi.org/10.1063/5.0150870","url":null,"abstract":"We provide an elementary proof and refinement of a well-known idea from physics: a chiral-symmetric local Hamiltonian on a half-space has the same signed number of edge-localized states with energies in the bulk bandgap as its bulk winding number. The requirement of non-elementary methods to relate generic and non-generic cases is emphasized. Our hands-on approach complements a quick abstract proof based on the classical index theory of Toeplitz operators.","PeriodicalId":50141,"journal":{"name":"Journal of Mathematical Physics Analysis Geometry","volume":"88 6 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-03-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84056057","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A discrete quantum process is represented by a sequence of quantum operations, which are completely positive maps that are not necessarily trace preserving. We consider quantum processes that are obtained by repeated iterations of a quantum operation with noise. Such ergodic quantum processes generalize independent quantum processes. An ergodic theorem describing convergence to equilibrium for a general class of such processes has been recently obtained by Movassagh and Schenker. Under irreducibility and mixing conditions, we obtain a central limit type theorem describing fluctuations around the ergodic limit.
{"title":"Law of large numbers and central limit theorem for ergodic quantum processes","authors":"Lubashan Pathirana, J. Schenker","doi":"10.1063/5.0153483","DOIUrl":"https://doi.org/10.1063/5.0153483","url":null,"abstract":"A discrete quantum process is represented by a sequence of quantum operations, which are completely positive maps that are not necessarily trace preserving. We consider quantum processes that are obtained by repeated iterations of a quantum operation with noise. Such ergodic quantum processes generalize independent quantum processes. An ergodic theorem describing convergence to equilibrium for a general class of such processes has been recently obtained by Movassagh and Schenker. Under irreducibility and mixing conditions, we obtain a central limit type theorem describing fluctuations around the ergodic limit.","PeriodicalId":50141,"journal":{"name":"Journal of Mathematical Physics Analysis Geometry","volume":"20 3 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77825084","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We define a Z2-valued topological and gauge invariant associated with any one-dimensional, translation-invariant topological insulator that satisfies either particle–hole symmetry or chiral symmetry. The invariant can be computed from the Berry phase associated with a suitable basis of Bloch functions that is compatible with the symmetries. We compute the invariant in the Su–Schrieffer–Heeger model for chiral symmetric insulators and in the Kitaev model for particle–hole symmetric insulators. We show that in both cases, the Z2 invariant predicts the existence of zero-energy boundary states for the corresponding truncated models.
{"title":"A Z2 invariant for chiral and particle–hole symmetric topological chains","authors":"Domenico Monaco, Gabriele Peluso","doi":"10.1063/5.0138647","DOIUrl":"https://doi.org/10.1063/5.0138647","url":null,"abstract":"We define a Z2-valued topological and gauge invariant associated with any one-dimensional, translation-invariant topological insulator that satisfies either particle–hole symmetry or chiral symmetry. The invariant can be computed from the Berry phase associated with a suitable basis of Bloch functions that is compatible with the symmetries. We compute the invariant in the Su–Schrieffer–Heeger model for chiral symmetric insulators and in the Kitaev model for particle–hole symmetric insulators. We show that in both cases, the Z2 invariant predicts the existence of zero-energy boundary states for the corresponding truncated models.","PeriodicalId":50141,"journal":{"name":"Journal of Mathematical Physics Analysis Geometry","volume":"33 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73970812","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We introduce a family of quantum spin Hamiltonians on Z2 that can be regarded as perturbations of Kitaev’s Abelian quantum double models that preserve the gauge and duality symmetries of these models. We analyze in detail the sector with one electric charge and one magnetic flux and show that the spectrum in this sector consists of both bound states and scattering states of Abelian anyons. Concretely, we have defined a family of lattice models in which Abelian anyons arise naturally as finite-size quasi-particles with non-trivial dynamics that consist of a charge-flux pair. In particular, the anyons exhibit a non-trivial holonomy with a quantized phase, consistent with the gauge and duality symmetries of the Hamiltonian.
{"title":"Dynamical Abelian anyons with bound states and scattering states","authors":"S. Bachmann, B. Nachtergaele, Siddharth Vadnerkar","doi":"10.1063/5.0151232","DOIUrl":"https://doi.org/10.1063/5.0151232","url":null,"abstract":"We introduce a family of quantum spin Hamiltonians on Z2 that can be regarded as perturbations of Kitaev’s Abelian quantum double models that preserve the gauge and duality symmetries of these models. We analyze in detail the sector with one electric charge and one magnetic flux and show that the spectrum in this sector consists of both bound states and scattering states of Abelian anyons. Concretely, we have defined a family of lattice models in which Abelian anyons arise naturally as finite-size quasi-particles with non-trivial dynamics that consist of a charge-flux pair. In particular, the anyons exhibit a non-trivial holonomy with a quantized phase, consistent with the gauge and duality symmetries of the Hamiltonian.","PeriodicalId":50141,"journal":{"name":"Journal of Mathematical Physics Analysis Geometry","volume":"143 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85352107","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Emmanuel Alejandro Avila-Vargas, Claudia Moreno, Rafael Hern'andez-Jim'enez
In this work, we describe the partial collapse of a compact object and the emission of spacetime waves as a result of back-reaction effects. As a source mass term, we propose a non-smooth continuous function that describes a mass-loss, and we then obtain the solution of such setting. We present three distinct examples of the evolution of the norm |Rnl(t, r*)| in terms of t, and four different results are shown for the parameter l = 1, 2, 5, 10; here, r* is the fixed radius of an observer outside the compact object. In all cases, the decay behavior is actually present at t ≫ 1 and becomes more evident for larger l. In addition, for the results that have smaller l’s, their amplitudes are larger when the asymptotic character of |Rnl(t, r*)| clearly appears. Finally, the farther away an observer is set, the fewer oscillations are perceived; however, from our particular fixed set of parameters, the best spot to observe the wiggles of the emitted spacetime waves is close to r* ≃ α.
{"title":"Emission of spacetime waves from the partial collapse of a compact object","authors":"Emmanuel Alejandro Avila-Vargas, Claudia Moreno, Rafael Hern'andez-Jim'enez","doi":"10.1063/5.0155046","DOIUrl":"https://doi.org/10.1063/5.0155046","url":null,"abstract":"In this work, we describe the partial collapse of a compact object and the emission of spacetime waves as a result of back-reaction effects. As a source mass term, we propose a non-smooth continuous function that describes a mass-loss, and we then obtain the solution of such setting. We present three distinct examples of the evolution of the norm |Rnl(t, r*)| in terms of t, and four different results are shown for the parameter l = 1, 2, 5, 10; here, r* is the fixed radius of an observer outside the compact object. In all cases, the decay behavior is actually present at t ≫ 1 and becomes more evident for larger l. In addition, for the results that have smaller l’s, their amplitudes are larger when the asymptotic character of |Rnl(t, r*)| clearly appears. Finally, the farther away an observer is set, the fewer oscillations are perceived; however, from our particular fixed set of parameters, the best spot to observe the wiggles of the emitted spacetime waves is close to r* ≃ α.","PeriodicalId":50141,"journal":{"name":"Journal of Mathematical Physics Analysis Geometry","volume":"40 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83645497","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The relationship between symmetry fields and first integrals of divergence-free vector fields is explored in three dimensions in light of its relevance to plasma physics and magnetic confinement fusion. A Noether-type theorem is known: for each such symmetry, there corresponds a first integral. The extent to which the converse is true is investigated. In doing so, a reformulation of this Noether-type theorem is found for which the converse holds on what is called the toroidal region. Some consequences of the methods presented are quick proofs of the existence of flux coordinates for magnetic fields in high generality, without needing to assume a symmetry such as in the cases of magneto-hydrostatics or quasi-symmetry.
{"title":"Existence of global symmetries of divergence-free fields with first integrals","authors":"D. Perrella, Nathan Duignan, David Pfefferl'e","doi":"10.1063/5.0152213","DOIUrl":"https://doi.org/10.1063/5.0152213","url":null,"abstract":"The relationship between symmetry fields and first integrals of divergence-free vector fields is explored in three dimensions in light of its relevance to plasma physics and magnetic confinement fusion. A Noether-type theorem is known: for each such symmetry, there corresponds a first integral. The extent to which the converse is true is investigated. In doing so, a reformulation of this Noether-type theorem is found for which the converse holds on what is called the toroidal region. Some consequences of the methods presented are quick proofs of the existence of flux coordinates for magnetic fields in high generality, without needing to assume a symmetry such as in the cases of magneto-hydrostatics or quasi-symmetry.","PeriodicalId":50141,"journal":{"name":"Journal of Mathematical Physics Analysis Geometry","volume":"27 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79912646","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Proofs of localization for random Schrödinger operators with sufficiently regular distribution of the potential can take advantage of the fractional moment method introduced by Aizenman–Molchanov [Commun. Math. Phys. 157(2), 245–278 (1993)] or use the classical Wegner estimate as part of another method, e.g., the multi-scale analysis introduced by Fröhlich–Spencer [Commun. Math. Phys. 88, 151–184 (1983)] and significantly developed by Klein and his collaborators. When the potential distribution is singular, most proofs rely crucially on exponential estimates of events corresponding to finite truncations of the operator in question; these estimates in some sense substitute for the classical Wegner estimate. We introduce a method to “lift” such estimates, which have been obtained for many stationary models, to certain closely related non-stationary models. As an application, we use this method to derive Anderson localization on the 1D lattice for certain non-stationary potentials along the lines of the non-perturbative approach developed by Jitomirskaya–Zhu [Commun. Math. Physics 370, 311–324 (2019)] in 2019.
{"title":"A “lifting” method for exponential large deviation estimates and an application to certain non-stationary 1D lattice Anderson models","authors":"O. Hurtado","doi":"10.1063/5.0150430","DOIUrl":"https://doi.org/10.1063/5.0150430","url":null,"abstract":"Proofs of localization for random Schrödinger operators with sufficiently regular distribution of the potential can take advantage of the fractional moment method introduced by Aizenman–Molchanov [Commun. Math. Phys. 157(2), 245–278 (1993)] or use the classical Wegner estimate as part of another method, e.g., the multi-scale analysis introduced by Fröhlich–Spencer [Commun. Math. Phys. 88, 151–184 (1983)] and significantly developed by Klein and his collaborators. When the potential distribution is singular, most proofs rely crucially on exponential estimates of events corresponding to finite truncations of the operator in question; these estimates in some sense substitute for the classical Wegner estimate. We introduce a method to “lift” such estimates, which have been obtained for many stationary models, to certain closely related non-stationary models. As an application, we use this method to derive Anderson localization on the 1D lattice for certain non-stationary potentials along the lines of the non-perturbative approach developed by Jitomirskaya–Zhu [Commun. Math. Physics 370, 311–324 (2019)] in 2019.","PeriodicalId":50141,"journal":{"name":"Journal of Mathematical Physics Analysis Geometry","volume":"7 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75721258","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper rigorously justifies the incompressible limit of strong solutions to isentropic compressible magnetohydrodynamic equations with ill-prepared initial data in a three-dimensional bounded domain as the Mach number goes to zero. In both cases of viscous and inviscid magnetic fields, we establish a new energy functional with weight to obtain uniform estimates for strong solutions with respect to the Mach number. Then, we prove the weak convergence of a velocity and the strong convergence of a magnetic field and the divergence-free component of a velocity field, which yields the corresponding incompressible limit.
{"title":"Incompressible limit of isentropic magnetohydrodynamic equations with ill-prepared data in bounded domains","authors":"Xiaoyu Gu, Yaobin Ou, Lu Yang","doi":"10.1063/5.0140349","DOIUrl":"https://doi.org/10.1063/5.0140349","url":null,"abstract":"This paper rigorously justifies the incompressible limit of strong solutions to isentropic compressible magnetohydrodynamic equations with ill-prepared initial data in a three-dimensional bounded domain as the Mach number goes to zero. In both cases of viscous and inviscid magnetic fields, we establish a new energy functional with weight to obtain uniform estimates for strong solutions with respect to the Mach number. Then, we prove the weak convergence of a velocity and the strong convergence of a magnetic field and the divergence-free component of a velocity field, which yields the corresponding incompressible limit.","PeriodicalId":50141,"journal":{"name":"Journal of Mathematical Physics Analysis Geometry","volume":"21 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74184175","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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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