Pub Date : 2025-08-01DOI: 10.1016/S0034-4877(25)00054-0
Sedef Karakiliç, Sedef Özcan, Setenay Akduman
We explore the asymptotic behaviour of the so-called unstable Bloch eigenvalues of the polyharmonic matrix operator , in the single resonance domain which is a subset of resonance domain — the set of eigenvalues situated close to the diffraction hyperplanes. The single resonance domain approaches full measure asymptotically across the entire resonance domain. In our analysis, we discover a significant trend: as energy levels increase, the eigenvalues are related to those of a Sturm–Liouville operator.
{"title":"On The Asymptotic Behaviour of The Unstable Bloch Eigenvalues of A Polyharmonic Matrix Operator","authors":"Sedef Karakiliç, Sedef Özcan, Setenay Akduman","doi":"10.1016/S0034-4877(25)00054-0","DOIUrl":"10.1016/S0034-4877(25)00054-0","url":null,"abstract":"<div><div>We explore the asymptotic behaviour of the so-called unstable Bloch eigenvalues of the polyharmonic matrix operator \u0000\t\t\t\t<span><math><mrow><msup><mrow><mo>(</mo><mo>−</mo><mi>Δ</mi><mo>)</mo></mrow><mi>l</mi></msup><mo>+</mo><mi>V</mi><mo>(</mo><mi>x</mi><mo>)</mo><mtext> with </mtext><mfrac><mn>1</mn><mn>2</mn></mfrac><mo><</mo><mi>l</mi><mo><</mo><mn>1</mn></mrow></math></span>, in the single resonance domain which is a subset of resonance domain — the set of eigenvalues situated close to the diffraction hyperplanes. The single resonance domain approaches full measure asymptotically across the entire resonance domain. In our analysis, we discover a significant trend: as energy levels increase, the eigenvalues are related to those of a Sturm–Liouville operator.</div></div>","PeriodicalId":49630,"journal":{"name":"Reports on Mathematical Physics","volume":"96 1","pages":"Pages 55-74"},"PeriodicalIF":1.2,"publicationDate":"2025-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145371392","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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}
Pub Date : 2025-08-01DOI: 10.1016/S0034-4877(25)00052-7
Yuki Ueda, Satoshi Okumura , Daiju Funakawa
We investigate the existence of discrete positive or negative energy ground states of the Dirac operator H which describe the fermion scattering on topological solitons in the nonlinear O (3) σ-model. Additionally, we provide a sufficient condition to ensure that the lowest positive and the largest negative energies of the Dirac operator H are nonzero.
{"title":"Spectral Analysis of Dirac Operators for Fermion Scattering on Topological Solitons in The Nonlinear O(3) σ-Model","authors":"Yuki Ueda, Satoshi Okumura , Daiju Funakawa","doi":"10.1016/S0034-4877(25)00052-7","DOIUrl":"10.1016/S0034-4877(25)00052-7","url":null,"abstract":"<div><div>We investigate the existence of discrete positive or negative energy ground states of the Dirac operator <em>H</em> which describe the fermion scattering on topological solitons in the nonlinear O (3) σ-model. Additionally, we provide a sufficient condition to ensure that the lowest positive and the largest negative energies of the Dirac operator <em>H</em> are nonzero.</div></div>","PeriodicalId":49630,"journal":{"name":"Reports on Mathematical Physics","volume":"96 1","pages":"Pages 1-17"},"PeriodicalIF":1.2,"publicationDate":"2025-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145370998","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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}
Pub Date : 2025-08-01DOI: 10.1016/S0034-4877(25)00056-4
Shigeru Furuichi, Frank Hansen
A lower bound of the reduced relative entropy is given by the use of a variational expression. The reduced Tsallis relative entropy is defined and some results are given. In particular, the convexity of the reduced Tsallis relative entropy is obtained. Finally, an upper bound of the reduced Tsallis relative entropy is given.
{"title":"Bounds for The Reduced Relative Entropies","authors":"Shigeru Furuichi, Frank Hansen","doi":"10.1016/S0034-4877(25)00056-4","DOIUrl":"10.1016/S0034-4877(25)00056-4","url":null,"abstract":"<div><div>A lower bound of the reduced relative entropy is given by the use of a variational expression. The reduced Tsallis relative entropy is defined and some results are given. In particular, the convexity of the reduced Tsallis relative entropy is obtained. Finally, an upper bound of the reduced Tsallis relative entropy is given.</div></div>","PeriodicalId":49630,"journal":{"name":"Reports on Mathematical Physics","volume":"96 1","pages":"Pages 85-99"},"PeriodicalIF":1.2,"publicationDate":"2025-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145371394","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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}
Pub Date : 2025-08-01DOI: 10.1016/S0034-4877(25)00058-8
Jan Naudts, Jun Zhang
We revisit the work of Rieffel and van Daele on pairs of subspaces of a real Hilbert space, while relaxing as much as possible the assumption that all the relevant subspaces are in general position with respect to each other. We work out, in detail, how two real projection operators lead to the construction of a complex Hilbert space where the theory of the modular operator is applicable, with emphasis on the relevance of a central extension of the group of split quaternions. Two examples are given for which the subspaces have unequal dimension and therefore are not in generic position.
{"title":"Pairs of Subspaces, Split Quaternions and The Modular Operator","authors":"Jan Naudts, Jun Zhang","doi":"10.1016/S0034-4877(25)00058-8","DOIUrl":"10.1016/S0034-4877(25)00058-8","url":null,"abstract":"<div><div>We revisit the work of Rieffel and van Daele on pairs of subspaces of a real Hilbert space, while relaxing as much as possible the assumption that all the relevant subspaces are in general position with respect to each other. We work out, in detail, how two real projection operators lead to the construction of a complex Hilbert space where the theory of the modular operator is applicable, with emphasis on the relevance of a central extension of the group of split quaternions. Two examples are given for which the subspaces have unequal dimension and therefore are not in generic position.</div></div>","PeriodicalId":49630,"journal":{"name":"Reports on Mathematical Physics","volume":"96 1","pages":"Pages 115-145"},"PeriodicalIF":1.2,"publicationDate":"2025-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145371395","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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}
Pub Date : 2025-06-01DOI: 10.1016/S0034-4877(25)00036-9
Tuna Bayrakdar, Z. Ok Bayrakdar
In this work we show that the Hunter–Saxton equation appears as the identity for the curvature of a two-dimensional Riemannian manifold. As being motivated by this result we show that the time evolution of a curve along a geodesic curve on the Riemannian manifold is governed by the Hunter–Saxton equation.
{"title":"A Moving Curve Description of The Hunter–Saxton Equation","authors":"Tuna Bayrakdar, Z. Ok Bayrakdar","doi":"10.1016/S0034-4877(25)00036-9","DOIUrl":"10.1016/S0034-4877(25)00036-9","url":null,"abstract":"<div><div>In this work we show that the Hunter–Saxton equation appears as the identity for the curvature of a two-dimensional Riemannian manifold. As being motivated by this result we show that the time evolution of a curve along a geodesic curve on the Riemannian manifold is governed by the Hunter–Saxton equation.</div></div>","PeriodicalId":49630,"journal":{"name":"Reports on Mathematical Physics","volume":"95 3","pages":"Pages 381-391"},"PeriodicalIF":1.0,"publicationDate":"2025-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144523226","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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}
Pub Date : 2025-06-01DOI: 10.1016/S0034-4877(25)00037-0
Mohammad Bagher Kazemi Balgeshir, Sara Miri
In the present paper, we study invariant and screen real lightlike statistical submersions h from a mixed 3-Sasakian statistical manifold. We prove that the fibers of an invariant lightlike statistical submersion are totally geodesic. We obtain some properties of screen real lightlike statistical submersions from a mixed 3-Sasakian statistical manifold. Some examples related to these notions are also constructed. Finally, we investigate warped product manifolds of the type M = Δ ×ϑs (Ker h*).
本文研究了混合3-Sasakian统计流形的不变量和屏蔽真实类光统计淹没。我们证明了不变类光统计浸没的纤维是完全测地线的。从混合3-Sasakian统计流形中得到了屏实类光统计淹没的一些性质。本文还构造了一些与这些概念有关的例子。最后，我们研究了M = Δ ×ϑ s （Ker h*）型的翘曲积流形。
{"title":"Lightlike Statistical Submersions from A Mixed 3-Sasakian Statistical Manifold","authors":"Mohammad Bagher Kazemi Balgeshir, Sara Miri","doi":"10.1016/S0034-4877(25)00037-0","DOIUrl":"10.1016/S0034-4877(25)00037-0","url":null,"abstract":"<div><div>In the present paper, we study invariant and screen real lightlike statistical submersions <em>h</em> from a mixed 3-Sasakian statistical manifold. We prove that the fibers of an invariant lightlike statistical submersion are totally geodesic. We obtain some properties of screen real lightlike statistical submersions from a mixed 3-Sasakian statistical manifold. Some examples related to these notions are also constructed. Finally, we investigate warped product manifolds of the type <em>M</em> = Δ ×<sub>ϑ</sub> <em>s</em> (Ker <em>h</em><sub>*</sub>).</div></div>","PeriodicalId":49630,"journal":{"name":"Reports on Mathematical Physics","volume":"95 3","pages":"Pages 393-409"},"PeriodicalIF":1.0,"publicationDate":"2025-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144523259","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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}
Pub Date : 2025-06-01DOI: 10.1016/S0034-4877(25)00035-7
Paulo A. Faria da Veiga, Michael O'Carroll
We consider the local gauge-invariant Yang-Mills quantum field theory on the finite hyper-cubic lattice Λ ⊂ aℤd ⊂ ℝd, d = 2, 3, 4, a ∊ (0, 1], with L (even) sites on a side and with the gauge Lie groups = U(N), SU(N). To each Λ bond b, there is a unitary matrix gauge variable Ub from an irrep of . The vector gauge potentials (gluon fields) are parameters in the Lie algebra of . The Wilson finite lattice partition function ZΛ (a) is used. The action AΛ (a) is a sum of gauge-invariant plaquette actions times , , . Each plaquette action has the product of four bond variables; the partition function is the integral over the Boltzmann factor with a product over bonds of Haar measures. Formally, in the continuum, ultraviolet (UV) limit a &drarr; 0, the action gives the YM classical continuum action. For free and periodic boundary conditions (b.c.), and using scaled fields, defined with an a-dependent noncanonical scaling, we show thermodynamic and UV stable (TUV) stability bounds for a scaled partition function, with constants independent of L, a and g. Passing to scaled fields does not alter the model energy-momentum spectrum and can be interpreted as an a priori field strength renormalization, making the action more regular. With scaled fields, we can isolate the UV singularity of the finite lattice physical, unscaled free energy fΛ(a) = [ln ZΛ ]/Λs, where Λs = Ld is the total number of lattice sites. With this, we show the existence of, at least, the subsequential thermodynamic (Λ ↗ dℤd) and UV limits of a scaled free energy. To obtain the TUV bounds, the Weyl integration formula is used in the gauge integral and the random matrix probability distributions of the CUE and GUE appear naturally. Using periodic b.c. and the multireflection method, the generating function of r scaled plaquette field correlations is bounded uniformly in L, a, g and the location/orientation of the r plaquette fields. Consequently, r-scaled plaquette field correlati
{"title":"On Yang-Mills Stability Bounds and Plaquette Field Generating Function","authors":"Paulo A. Faria da Veiga, Michael O'Carroll","doi":"10.1016/S0034-4877(25)00035-7","DOIUrl":"10.1016/S0034-4877(25)00035-7","url":null,"abstract":"<div><div>We consider the local gauge-invariant Yang-Mills quantum field theory on the finite hyper-cubic lattice Λ ⊂ <em>aℤ<sup>d</sup></em> ⊂ <em>ℝ<sup>d</sup></em>, <em>d</em> = 2, 3, 4, <em>a</em> ∊ (0, 1], with <em>L</em> (even) sites on a side and with the gauge Lie groups \u0000\t\t\t\t<span><math><mi>G</mi></math></span> = U(<em>N</em>), <em>SU</em>(<em>N</em>). To each Λ bond <em>b</em>, there is a unitary matrix gauge variable <em>U<sub>b</sub></em> from an irrep of \u0000\t\t\t\t<span><math><mi>G</mi></math></span>. The vector gauge potentials (gluon fields) are parameters in the Lie algebra of \u0000\t\t\t\t<span><math><mi>G</mi></math></span>. The Wilson finite lattice partition function Z<sub>Λ</sub> (<em>a</em>) is used. The action A<sub>Λ</sub> (<em>a</em>) is a sum of gauge-invariant plaquette actions times \u0000\t\t\t\t<span><math><mrow><mrow><mo>[</mo><mrow><msup><mi>a</mi><mrow><mi>d</mi><mo>-</mo><mn>4</mn></mrow></msup><mo>/</mo><msup><mi>g</mi><mn>2</mn></msup></mrow><mo>]</mo></mrow></mrow></math></span>, \u0000\t\t\t\t<span><math><mrow><msup><mi>g</mi><mn>2</mn></msup><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><msubsup><mi>g</mi><mn>0</mn><mn>2</mn></msubsup><mo>]</mo></mrow></math></span>, \u0000\t\t\t\t<span><math><mrow><mn>0</mn><mo><</mo><msubsup><mi>g</mi><mn>0</mn><mn>2</mn></msubsup><mo><</mo><mo>∞</mo></mrow></math></span>. Each plaquette action has the product of four bond variables; the partition function is the integral over the Boltzmann factor with a product over bonds of \u0000\t\t\t\t<span><math><mi>G</mi></math></span> Haar measures. Formally, in the continuum, ultraviolet (UV) limit <em>a &</em>drarr; 0, the action gives the YM classical continuum action. For free and periodic boundary conditions (b.c.), and using scaled fields, defined with an <em>a-</em>dependent noncanonical scaling, we show thermodynamic and UV stable (TUV) stability bounds for a scaled partition function, with constants independent of <em>L, a</em> and <em>g.</em> Passing to scaled fields does not alter the model energy-momentum spectrum and can be interpreted as an a priori field strength renormalization, making the action more regular. With scaled fields, we can isolate the UV singularity of the finite lattice physical, unscaled free energy <em>f</em>Λ(<em>a</em>) = [ln ZΛ ]/Λ<em><sub>s</sub></em>, where Λ<sub>s</sub> = <em>L<sup>d</sup></em> is the total number of lattice sites. With this, we show the existence of, at least, the subsequential thermodynamic (Λ &nearr; <em>dℤ<sup>d</sup></em>) and UV limits of a scaled free energy. To obtain the TUV bounds, the Weyl integration formula is used in the gauge integral and the random matrix probability distributions of the CUE and GUE appear naturally. Using periodic b.c. and the multireflection method, the generating function of <em>r</em> scaled plaquette field correlations is bounded uniformly in <em>L, a, g</em> and the location/orientation of the <em>r</em> plaquette fields. Consequently, <em>r</em>-scaled plaquette field correlati","PeriodicalId":49630,"journal":{"name":"Reports on Mathematical Physics","volume":"95 3","pages":"Pages 303-380"},"PeriodicalIF":1.0,"publicationDate":"2025-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144523169","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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}
Pub Date : 2025-06-01DOI: 10.1016/S0034-4877(25)00029-1
Tomoya Tagawa
For the Schrödinger operator generalized from the harmonic oscillator, we prove a Rellich type theorem, which characterizes the order of growth of eigenfunctions at infinity. The proofs are given by an extensive use of commutator arguments invented recently by Ito and Skibsted. These arguments are simple and elementary and do not employ energy cut-offs or microlocal analysis.
{"title":"A Rellich Type Theorem for The Generalized Oscillator","authors":"Tomoya Tagawa","doi":"10.1016/S0034-4877(25)00029-1","DOIUrl":"10.1016/S0034-4877(25)00029-1","url":null,"abstract":"<div><div>For the Schrödinger operator generalized from the harmonic oscillator, we prove a Rellich type theorem, which characterizes the order of growth of eigenfunctions at infinity. The proofs are given by an extensive use of commutator arguments invented recently by Ito and Skibsted. These arguments are simple and elementary and do not employ energy cut-offs or microlocal analysis.</div></div>","PeriodicalId":49630,"journal":{"name":"Reports on Mathematical Physics","volume":"95 3","pages":"Pages 281-302"},"PeriodicalIF":1.0,"publicationDate":"2025-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144523168","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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}
Pub Date : 2025-04-01DOI: 10.1016/S0034-4877(25)00027-8
Jen-Hsu Chang
The B-type Kadomtsev–Petviashvili equation (BKP) is obtained from the reduction of Kadomtsev–Petviashvili (KP) hierarchy under the orthogonal type transformation group. The skew Schur's Q functions can be used to construct the t-functions of solitons in the BKP equation. Then the totally nonnegative Pfaffian can be defined via the skew Schur's Q functions to obtain nonsingular line-solitons solution in the BKP equation. The totally nonnegative Pfaffians are investigated. The line solitons interact to form web-like structure in the near field region and their resonances appearing in soliton graph could be investigated by the totally nonnegative Pfaffians.
{"title":"TOTALLY NONNEGATIVE PFAFFIAN FOR SOLITONS IN 5-TYPE KADOMTSEV–PETVIASHVILI EQUATION","authors":"Jen-Hsu Chang","doi":"10.1016/S0034-4877(25)00027-8","DOIUrl":"10.1016/S0034-4877(25)00027-8","url":null,"abstract":"<div><div>The <em>B</em>-type Kadomtsev–Petviashvili equation (BKP) is obtained from the reduction of Kadomtsev–Petviashvili (KP) hierarchy under the orthogonal type transformation group. The skew Schur's <em>Q</em> functions can be used to construct the t-functions of solitons in the BKP equation. Then the totally nonnegative Pfaffian can be defined via the skew Schur's <em>Q</em> functions to obtain nonsingular line-solitons solution in the BKP equation. The totally nonnegative Pfaffians are investigated. The line solitons interact to form web-like structure in the near field region and their resonances appearing in soliton graph could be investigated by the totally nonnegative Pfaffians.</div></div>","PeriodicalId":49630,"journal":{"name":"Reports on Mathematical Physics","volume":"95 2","pages":"Pages 259-279"},"PeriodicalIF":1.0,"publicationDate":"2025-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144107403","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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}
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