{"title":"On the Bloch eigenvalues, band functions and bands of the differential operator of odd order with the periodic matrix coefficients","authors":"O. A. Veliev","doi":"10.1007/s11005-024-01810-2","DOIUrl":"10.1007/s11005-024-01810-2","url":null,"abstract":"<div><p>In this paper, we consider the Bloch eigenvalues, band functions and bands of the self-adjoint differential operator <i>L</i> generated by the differential expression of odd order <i>n</i> with the <span>(mtimes m)</span> periodic matrix coefficients, where <span>(n>1.)</span> We study the localizations of the Bloch eigenvalues and continuity of the band functions and prove that each point of the set <span>(left[ (2pi N)^{n},infty right) cup (-infty ,(-2pi N)^{n}])</span> belongs to at least <i>m</i> bands, where <i>N</i> is the smallest integer satisfying <span>(Nge pi ^{-2}M+1)</span> and <i>M</i> is the sum of the norms of the coefficients. Moreover, we prove that if <span>(Mle pi ^{2}2^{-n+1/2})</span>, then each point of the real line belong to at least <i>m</i> bands.</p></div>","PeriodicalId":685,"journal":{"name":"Letters in Mathematical Physics","volume":"114 3","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140939283","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Detecting causality with symplectic quandles","authors":"Ayush Jain","doi":"10.1007/s11005-024-01808-w","DOIUrl":"10.1007/s11005-024-01808-w","url":null,"abstract":"<div><p>We investigate the capability of symplectic quandles to detect causality for (2+1)-dimensional globally hyperbolic spacetimes (X). Allen and Swenberg showed that the Alexander–Conway polynomial is insufficient to distinguish connected sum of two Hopf links from the links in the family of Allen–Swenberg 2-sky like links, suggesting that it cannot always detect causality in X. We find that symplectic quandles, combined with Alexander–Conway polynomial, can distinguish these two types of links, thereby suggesting their ability to detect causality in X. The fact that symplectic quandles can capture causality in the Allen–Swenberg example is intriguing since the theorem of Chernov and Nemirovski, which states that Legendrian linking equals causality, is proved using Contact Geometry methods. \u0000</p></div>","PeriodicalId":685,"journal":{"name":"Letters in Mathematical Physics","volume":"114 3","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140885255","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Topological twists of massive SQCD, Part I","authors":"Johannes Aspman, Elias Furrer, Jan Manschot","doi":"10.1007/s11005-024-01803-1","DOIUrl":"10.1007/s11005-024-01803-1","url":null,"abstract":"<div><p>We consider topological twists of four-dimensional <span>(mathcal {N}=2)</span> supersymmetric QCD with gauge group SU(2) and <span>(N_fle 3)</span> fundamental hypermultiplets. The twists are labelled by a choice of background fluxes for the flavour group, which provides an infinite family of topological partition functions. In this Part I, we demonstrate that in the presence of such fluxes the theories can be formulated for arbitrary gauge bundles on a compact four-manifold. Moreover, we consider arbitrary masses for the hypermultiplets, which introduce new intricacies for the evaluation of the low-energy path integral on the Coulomb branch. We develop techniques for the evaluation of these path integrals. In the forthcoming Part II, we will deal with the explicit evaluation.\u0000</p></div>","PeriodicalId":685,"journal":{"name":"Letters in Mathematical Physics","volume":"114 3","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s11005-024-01803-1.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140885257","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Local scattering matrix for a degenerate avoided-crossing in the non-coupled regime","authors":"Kenta Higuchi","doi":"10.1007/s11005-024-01807-x","DOIUrl":"10.1007/s11005-024-01807-x","url":null,"abstract":"<div><p>A Landau–Zener-type formula for a degenerate avoided-crossing is studied in the non-coupled regime. More precisely, a <span>(2times 2)</span> system of first-order <i>h</i>-differential operator with <span>(mathcal {O}(varepsilon ))</span> off-diagonal part is considered in 1D. Asymptotic behavior as <span>(varepsilon h^{m/(m+1)}rightarrow 0^+)</span> of the local scattering matrix near an avoided-crossing is given, where <i>m</i> stands for the contact order of two curves of the characteristic set. A generalization including the cases with vanishing off-diagonals and non-Hermitian symbols is also given.</p></div>","PeriodicalId":685,"journal":{"name":"Letters in Mathematical Physics","volume":"114 3","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-04-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140826825","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Correction to: Local index theorem for orbifold Riemann surfaces","authors":"Leon A. Takhtajan, Peter Zograf","doi":"10.1007/s11005-024-01809-9","DOIUrl":"10.1007/s11005-024-01809-9","url":null,"abstract":"","PeriodicalId":685,"journal":{"name":"Letters in Mathematical Physics","volume":"114 3","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-04-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142415004","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Fermionic construction of the (frac{{{mathbb {Z}}}}{2})-graded meromorphic open-string vertex algebra and its ({{mathbb {Z}}}_2)-twisted module, I","authors":"Francesco Fiordalisi, Fei Qi","doi":"10.1007/s11005-024-01794-z","DOIUrl":"10.1007/s11005-024-01794-z","url":null,"abstract":"<div><p>We define the <span>(frac{{{mathbb {Z}}}}{2})</span>-graded meromorphic open-string vertex algebra that is an appropriate noncommutative generalization of the vertex operator superalgebra. We also illustrate an example that can be viewed as a noncommutative generalization of the free fermion vertex operator superalgebra. The example is built upon a universal half-integer-graded non-anti-commutative Fock space where a creation operator and an annihilation operator satisfy the fermionic anti-commutativity relation, while no relations exist among the creation operators. The former feature allows us to define the normal ordering, while the latter feature allows us to describe interactions among the fermions. With respect to the normal ordering, Wick’s theorem holds and leads to a proof of weak associativity and a closed formula of correlation functions.</p></div>","PeriodicalId":685,"journal":{"name":"Letters in Mathematical Physics","volume":"114 2","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-04-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140810949","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Clemens Sämann, Benedict Schinnerl, Roland Steinbauer, Robert Švarc
{"title":"Cut-and-paste for impulsive gravitational waves with (Lambda ): the mathematical analysis","authors":"Clemens Sämann, Benedict Schinnerl, Roland Steinbauer, Robert Švarc","doi":"10.1007/s11005-024-01804-0","DOIUrl":"10.1007/s11005-024-01804-0","url":null,"abstract":"<div><p>Impulsive gravitational waves are theoretical models of short but violent bursts of gravitational radiation. They are commonly described by two distinct spacetime metrics, one of local Lipschitz regularity and the other one even distributional. These two metrics are thought to be ‘physically equivalent’ since they can be formally related by a ‘discontinuous coordinate transformation’. In this paper we provide a mathematical analysis of this issue for the entire class of nonexpanding impulsive gravitational waves propagating in a background spacetime of constant curvature. We devise a natural geometric regularisation procedure to show that the notorious change of variables arises as the distributional limit of a family of smooth coordinate transformations. In other words, we establish that both spacetimes arise as distributional limits of a smooth sandwich wave taken in different coordinate systems which are diffeomorphically related.\u0000</p></div>","PeriodicalId":685,"journal":{"name":"Letters in Mathematical Physics","volume":"114 2","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s11005-024-01804-0.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140799163","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Petz–Rényi relative entropy of thermal states and their displacements","authors":"George Androulakis, Tiju Cherian John","doi":"10.1007/s11005-024-01805-z","DOIUrl":"10.1007/s11005-024-01805-z","url":null,"abstract":"<div><p>In this letter, we obtain the precise range of the values of the parameter <span>(alpha )</span> such that Petz–Rényi <span>(alpha )</span>-relative entropy <span>(D_{alpha }(rho ||sigma ))</span> of two faithful displaced thermal states is finite. More precisely, we prove that, given two displaced thermal states <span>(rho )</span> and <span>(sigma )</span> with inverse temperature parameters <span>(r_1, r_2,ldots , r_n)</span> and <span>(s_1,s_2, ldots , s_n)</span>, respectively, <span>(0<r_j,s_j<infty )</span>, for all <i>j</i>, we have </p><div><div><span>$$begin{aligned} D_{alpha }(rho ||sigma )<infty Leftrightarrow alpha< min left{ frac{s_j}{s_j-r_j}: j in { 1, ldots , n } text { such that } r_j<s_j right} , end{aligned}$$</span></div></div><p>where we adopt the convention that the minimum of an empty set is equal to infinity. This result is particularly useful in the light of operational interpretations of the Petz–Rényi <span>(alpha )</span>-relative entropy in the regime <span>(alpha >1 )</span>. Along the way, we also prove a special case of a conjecture of Seshadreesan et al. (J Math Phys 59(7):072204, 2018. https://doi.org/10.1063/1.5007167).</p></div>","PeriodicalId":685,"journal":{"name":"Letters in Mathematical Physics","volume":"114 2","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-04-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140612242","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Marek Mozrzymas, Michał Horodecki, Michał Studziński
{"title":"From port-based teleportation to Frobenius reciprocity theorem: partially reduced irreducible representations and their applications","authors":"Marek Mozrzymas, Michał Horodecki, Michał Studziński","doi":"10.1007/s11005-024-01800-4","DOIUrl":"10.1007/s11005-024-01800-4","url":null,"abstract":"<div><p>In this paper, we present the connection of two concepts as induced representation and partially reduced irreducible representations (PRIR) appear in the context of port-based teleportation protocols. Namely, for a given finite group <i>G</i> with arbitrary subgroup <i>H</i>, we consider a particular case of matrix irreducible representations, whose restriction to the subgroup <i>H</i>, as a matrix representation of <i>H</i>, is completely reduced to diagonal block form with an irreducible representation of <i>H</i> in the blocks. The basic properties of such representations are given. Then as an application of this concept, we show that the spectrum of the port-based teleportation operator acting on <i>n</i> systems is connected in a very simple way with the spectrum of the corresponding Jucys–Murphy operator for the symmetric group <span>(S(n-1)subset S(n))</span>. This shows on the technical level relation between teleporation and one of the basic objects from the point of view of the representation theory of the symmetric group. This shows a deep connection between the central object describing properties of deterministic PBT schemes and objects appearing naturally in the abstract representation theory of the symmetric group. In particular, we present a new expression for the eigenvalues of the Jucys–Murphy operators based on the irreducible characters of the symmetric group. As an additional but not trivial result, we give also purely matrix proof of the Frobenius reciprocity theorem for characters with explicit construction of the unitary matrix that realizes the reduction in the natural basis of induced representation to the reduced one.</p></div>","PeriodicalId":685,"journal":{"name":"Letters in Mathematical Physics","volume":"114 2","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-04-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s11005-024-01800-4.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140589305","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Nonlocal Kundu–Eckhaus equation: integrability, Riemann–Hilbert approach and Cauchy problem with step-like initial data","authors":"Bei-Bei Hu, Zu-Yi Shen, Ling Zhang","doi":"10.1007/s11005-024-01802-2","DOIUrl":"10.1007/s11005-024-01802-2","url":null,"abstract":"<div><p>The main purpose of this paper is to discuss the Cauchy problem of integrable nonlocal (reverse-space-time) Kundu–Eckhaus (KE) equation through the Riemann–Hilbert (RH) method. Firstly, based on the zero-curvature equation, we present an integrable nonlocal KE equation and its Lax pair. Then, we discuss the properties of eigenfunctions and scattering matrix, such as analyticity, asymptotic behavior, and symmetry. Finally, for the prescribed step-like initial value: <span>(u(z,t)=o(1))</span>, <span>(zrightarrow -infty )</span> and <span>(u(z,t)=R+o(1))</span>, <span>(zrightarrow +infty )</span>, where <span>(R>0)</span> is an arbitrary constant, we consider the initial value problem of the nonlocal KE equation. The paramount techniques is the asymptotic analysis of the associated RH problem.</p></div>","PeriodicalId":685,"journal":{"name":"Letters in Mathematical Physics","volume":"114 2","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-04-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140589278","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}