Reports on Mathematical Physics最新文献

{"title":"On Localization of Eigenfunctions of The Magnetic Laplacian","authors":"Jeffrey S. Ovall,&nbsp;Hadrian Quan,&nbsp;Robyn Reid,&nbsp;Stefan Steinerberger","doi":"10.1016/S0034-4877(24)00078-8","DOIUrl":"10.1016/S0034-4877(24)00078-8","url":null,"abstract":"<div><div>Let Ω ⊂ ℝ<em>^d</em> and consider the magnetic Laplace operator given by <em>H</em>(<em>A</em>) = (–<em>i</em>∇ – <em>A</em>(<em>x</em>))², where <em>A</em> : Ω <em>→</em> ℝ<em>^d</em>, subject to Dirichlet boundary conditions. For certain vector fields <em>A</em>, this operator can have eigenfunctions, <em>H</em>(<em>A</em>)ψ = λψ, that are highly localized in a small region of Ω. The main goal of this paper is to show that if |ψ| assumes its maximum at <em>x</em>₀ ∈ Ω, then <em>A</em> behaves 'almost' like a conservative vector field in a \u0000\t\t\t\t<span><math><mrow><mn>1</mn><mo>/</mo><msqrt><mi>λ</mi></msqrt></mrow></math></span>-neighborhood of <em>x</em>₀ in a precise sense. In particular, we expect localization in regions where |curl <em>A</em>| is small. The result is illustrated with numerical examples.</div></div>","PeriodicalId":49630,"journal":{"name":"Reports on Mathematical Physics","volume":"94 2","pages":"Pages 235-257"},"PeriodicalIF":1.0,"publicationDate":"2024-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142698848","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}

{"title":"Exact Solution to Bratu Second Order Differential Equation","authors":"Adam R. Szewczyk","doi":"10.1016/S0034-4877(24)00075-2","DOIUrl":"10.1016/S0034-4877(24)00075-2","url":null,"abstract":"<div><div>This paper deals with the temperature profile of a simple combustion and presents the alternative exact formulas for the temperature profile of the planar vessel. The differential equation that describes this system is referred as a Bratu equation or Poisson's equation in one-dimensional steady state case. In this present study, new solutions with general boundary conditions are developed. The results are compared with numerical solutions using Maxima, a computer algebra system program capable of numerical and symbolic computation. The new solutions yield formula that may provide a valuable information about relationship between terms, variables and coefficients which can be useful for theoretical physics.</div></div>","PeriodicalId":49630,"journal":{"name":"Reports on Mathematical Physics","volume":"94 2","pages":"Pages 175-188"},"PeriodicalIF":1.0,"publicationDate":"2024-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142698846","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}

{"title":"Applications of Buschman–fox H-Function in Nuclear Physics","authors":"Ashik A. Kabeer,&nbsp;Dilip Kumar","doi":"10.1016/S0034-4877(24)00079-X","DOIUrl":"10.1016/S0034-4877(24)00079-X","url":null,"abstract":"<div><div>The paper is devoted to presenting a novel closed-form representation of the resonant thermonuclear functions and the nonrelativistic Voigt function, which are essential tools in nuclear physics. Understanding thermonuclear fusion reaction rates within solar analogs is crucial for understanding stellar evolution and energy production mechanisms. Initially, this paper focuses on evaluating fusion reaction rates, particularly emphasizing resonant reactions, which play pivotal roles in stellar evolution phases. A key challenge lies in solving reaction rate integrals in closed form. The Buschman–Fox <em>H</em>-function of two variables is employed to address this issue. Conventionally, it is assumed that the plasma particles' velocity follows the Maxwell–Boltzmann distribution. However, it is acknowledged that particles may deviate from this assumed equilibrium state in actual scenarios, leading to nonequilibrium situations. The study also aims to address these nonequilibrium situations by utilizing appropriate velocity models from the existing literature. Utilizing the Mellin transform technique, we achieve the closed-form representation of the resonant reaction rate integral. Furthermore, we address the nonrelativistic Voigt profile and, in particular, Voigt function. The Voigt profile, resulting from the convolution of Gaussian and Lorentzian distributions, effectively captures the intricate shapes of spectral lines encountered in spectroscopy. Apart from its significance in spectroscopy, the Voigt function finds application in various areas such as plasma nuclear studies, acoustics, and radiation transfer. Many approximations of the Voigt function can be found in the literature, yet currently, there is no existing closed-form expression. This paper also sets out to fill this gap by deriving the exact closed-form expressions for the Voigt function and its conjugate in terms of Buschman–Fox <em>H</em>-function, employing the Mellin convolution theorem. This paper marks the first instance in the literature where the applications of Buschman Fox's <em>H</em>-function has been documented.</div></div>","PeriodicalId":49630,"journal":{"name":"Reports on Mathematical Physics","volume":"94 2","pages":"Pages 259-278"},"PeriodicalIF":1.0,"publicationDate":"2024-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142698849","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}

{"title":"Nonisospectral equations from the Cauchy matrix approach","authors":"Alemu Yilma Tefera,&nbsp;Shangshuai Li,&nbsp;Da-jun Zhang","doi":"10.1016/S0034-4877(24)00055-7","DOIUrl":"10.1016/S0034-4877(24)00055-7","url":null,"abstract":"<div><div>The Cauchy matrix approach is developed to construct explicit solutions for some nonisospectral equations, including the nonisospectral Korteweg–de Vries (KdV) equation, the nonisospectral modified KdV equation, and the nonisospectral sine-Gordon equation. By means of a Sylvester equation, a set of scalar master functions {<em>S</em>^{(<em>i,j</em>)}} is defined. We show how nonisospectral dispersion relations are introduced such that the evolutions of {<em>S</em>^{(<em>i,j</em>)}} can be derived. Some identities of {<em>S</em>^{(<em>i,j</em>)}} are employed in verifying solutions. Some explicit one-soliton and two-soliton solutions are illustrated together with analysis of their dynamics.</div></div>","PeriodicalId":49630,"journal":{"name":"Reports on Mathematical Physics","volume":"94 1","pages":"Pages 47-72"},"PeriodicalIF":1.0,"publicationDate":"2024-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142311096","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}

{"title":"Certain advancements in multidimensional q-hermite polynomials","authors":"Shahid Ahmad Wani,&nbsp;Mumtaz Riyasat,&nbsp;Subuhi Khan,&nbsp;William Ramírez","doi":"10.1016/S0034-4877(24)00059-4","DOIUrl":"10.1016/S0034-4877(24)00059-4","url":null,"abstract":"<div><div>In the realm of specialized functions, the allure of <em>q</em>-calculus beckons to many scholars, captivating them with its prowess in shaping models of quantum computing, noncommutative probability, combinatorics, functional analysis, mathematical physics, approximation theory, and beyond. This study explores a new idea called the multidimensional <em>q</em>-Hermite polynomials, using different <em>q</em>-calculus techniques. Numerous properties and novel findings regarding these polynomials are derived, encompassing their generating function, series representations, recurrence relations, <em>q</em>-differential formula, and operational principles. Further, we proved that these polynomials are quasi-monomial in <em>q</em>-aspect. As the applications, these findings are subsequently employed to address connection between the multidimensional <em>q</em>-Hermite polynomials and the two-variable <em>q</em>-Legendre polynomials for the first time. Various characterizations are examined, as well the graphical representations of the two-variable <em>q</em>-Legendre polynomials are provided by the surface plots and graphs of distribution of zeros for the <em>q</em>-Legendre polynomials with some specific set of parameters are shown using Mathematica. Our investigations shed light on the intricate nature of these polynomials, elucidating their behaviour and facilitating deeper understanding within the realm of <em>q</em>-calculus.</div></div>","PeriodicalId":49630,"journal":{"name":"Reports on Mathematical Physics","volume":"94 1","pages":"Pages 117-141"},"PeriodicalIF":1.0,"publicationDate":"2024-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142311093","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}

{"title":"New solutions of the lattice Kadomtsev–Petviashvili system associated with an elliptic curve","authors":"Ying-ying Sun ,&nbsp;Xinyi Wang,&nbsp;Da-jun Zhang","doi":"10.1016/S0034-4877(24)00053-3","DOIUrl":"10.1016/S0034-4877(24)00053-3","url":null,"abstract":"<div><div>A generalization of the lattice Kadomtsev–Petviashvili system associated with an elliptic curve, that is referred to as an elliptic integrable system, has been revisited by means of the Cauchy matrix scheme. Various types of explicit solutions are obtained, some of which offer new insights of both mathematical and physical significance. The construction of exact solutions to the elliptic lattice Kadomtsev–Petviashvili system is closely connected to that of a special Sylvester-type matrix equation.</div></div>","PeriodicalId":49630,"journal":{"name":"Reports on Mathematical Physics","volume":"94 1","pages":"Pages 11-33"},"PeriodicalIF":1.0,"publicationDate":"2024-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142311094","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}

{"title":"Weakly periodic gibbs measures for the HC model with a countable set of spin values","authors":"Muhtorjon Makhammadaliev","doi":"10.1016/S0034-4877(24)00057-0","DOIUrl":"10.1016/S0034-4877(24)00057-0","url":null,"abstract":"<div><div>In this paper, we study the weakly periodic (nonperiodic) Gibbs measures for the Hard Core (HC) model with a countable set ℤ of spin values and with a countable set of parameters <em>λ_i &gt;</em> 0, <em>i</em> ∈ ℤ, on a Cayley tree of order <em>k</em> ≥ 2. For the considered model in the case ∑<em>_i</em> λ<em>_i</em> &lt; +∞, a complete description of weakly periodic Gibbs measures is obtained for any normal divisor of index two and in the case ∑<em>_i</em>; λ<em>_i</em> = +∞, it is shown that there is no weakly periodic Gibbs measure. Moreover, in the case of a normal divisor of index four the uniqueness conditions for weakly periodic Gibbs measures are found. Further, under certain conditions an exact critical value is found that ensures the existence of weakly periodic Gibbs measures.</div></div>","PeriodicalId":49630,"journal":{"name":"Reports on Mathematical Physics","volume":"94 1","pages":"Pages 83-103"},"PeriodicalIF":1.0,"publicationDate":"2024-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142311092","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}

{"title":"Construction of quadratic invariants for time-dependent systems in complex phase space using Struckmeier and Riedel approach","authors":"Vipin Kumar,&nbsp;S.B. Bhardwaj,&nbsp;Ram Mehar Singh,&nbsp;Shalini Gupta,&nbsp;Fakir Chand","doi":"10.1016/S0034-4877(24)00052-1","DOIUrl":"10.1016/S0034-4877(24)00052-1","url":null,"abstract":"<div><div>In this paper, we deal with the construction of quadratic invariants in a complex phase space under the transformation <em>z</em> = (<em>x</em> + <em>iy</em>) and\u0000<span><math><mrow><mover><mi>z</mi><mo>¯</mo></mover><mo>=</mo><mrow><mo>(</mo><mrow><mi>x</mi><mo>-</mo><mi>i</mi><mi>y</mi></mrow><mo>)</mo></mrow></mrow></math></span> for various time-dependent systems. For this purpose, Struckmeier and Riedel (SR) approach [<span><span>1</span></span>, <span><span>2</span></span>] is used. The constructed invariants include an unknown function <em>f</em>₂(<em>t</em>) that is a solution of a third-order differential equation and its coefficients can be determined by the trajectories of the particle. The invariants play an important role in the study of a dynamical system, to access the accuracy in numerical simulations and to investigate the classical and quantum integrability of a system.</div></div>","PeriodicalId":49630,"journal":{"name":"Reports on Mathematical Physics","volume":"94 1","pages":"Pages 1-10"},"PeriodicalIF":1.0,"publicationDate":"2024-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142311090","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}

{"title":"Higher-order squeezing of both quadrature components in superposition of orthogonal even coherent state and vacuum state","authors":"Pankaj Kumar,&nbsp;Rakesh Kumar","doi":"10.1016/S0034-4877(24)00056-9","DOIUrl":"10.1016/S0034-4877(24)00056-9","url":null,"abstract":"<div><div>We study the Hong–Mandel higher-order squeezing of both quadrature components for an arbitrary 2<em>n</em>^th-order (<em>n</em> ≠1) considering the most general Hermitian operator, <em>X_θ = X</em>₁ cos <em>θ + iX</em>₂ sin <em>θ</em>, in the superposed state, |<em>Ψ</em>〉 = <em><strong>K</strong></em> [|Ψ₀) + <em>re</em>^<em>i</em>ϕ |0〉] of the orthogonal even coherent state and vacuum state. Here | Ψ₀〉 = <strong><em>K</em></strong>[|α, +〉 + |iα, +〉] is the orthogonal coherent state, |α, +〉 = <strong><em>K</em></strong>′[|α) + | – α〉] and |<em>i</em>α, +〉 = <strong><em>K</em></strong><em>″</em> [|<em>i</em>α, +〉 + | – <em>i</em>α, +〉] are even coherent states, operators <em>X</em>_1,2 are defined by <em>X</em>₁ + <em>iX</em>₂ = <em>a, a</em> is the annihilation operator, α, <em>θ</em>, <em>r</em> and ϕ are arbitrary parameters and the only restriction on these is the normalization condition of the superposed state |<em>Ψ</em>〉. We find that maximum simultaneous 2<em>n</em>^th-order Hong–Mandel squeezing of both quadrature components <em>X_θ</em> and <em>X_θ+π</em>/2 exhibited by the orthogonal even coherent state enhances in its superposition with vacuum state. We conclude that the values of higher-order momenta in the superposed state become much closer to the best minimum values of the corresponding values of higher-order momenta explored numerically so far than that obtained in orthogonal even coherent state. Variations of 2<em>n</em>^th-order squeezing for <em>n</em> = 2,3 and 4, i.e. fourth, sixth and eighth-order squeezing with different parameters have also been discussed.</div></div>","PeriodicalId":49630,"journal":{"name":"Reports on Mathematical Physics","volume":"94 1","pages":"Pages 73-82"},"PeriodicalIF":1.0,"publicationDate":"2024-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142311091","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}

{"title":"New operator realization of angular momentum for description of electron's motion in uniform magnetic field","authors":"Cheng Da ,&nbsp;Hony-Yi Fan","doi":"10.1016/S0034-4877(24)00058-2","DOIUrl":"10.1016/S0034-4877(24)00058-2","url":null,"abstract":"<div><div>By introducing appropriate electron's coordinate eigenstate and momentum eigenstate we propose new realization of angular momentum operators for describing electron's motion in uniform magnetic field. The coordinate eigenstates make up a representation and embody quantum entanglement between magnetic field and electron. The eigenstate of angular momentum's lowering-ascending operator <strong><em>L</em>_±</strong> are derived, the way we tackle this problem is to make an analogue between the transform\u0000<span><math><mrow><msup><mi>e</mi><mrow><mn>2</mn><mi>f</mi><msub><mi>L</mi><mi>z</mi></msub></mrow></msup><msub><mi>L</mi><mo>±</mo></msub><msup><mi>e</mi><mrow><mo>-</mo><mn>2</mn><mi>f</mi><msub><mi>L</mi><mi>z</mi></msub></mrow></msup></mrow></math></span> to the squeezing mechanism. All these discussions reveal that the quantum theory for charged particles' motion in magnetic field needs to be developed.</div></div>","PeriodicalId":49630,"journal":{"name":"Reports on Mathematical Physics","volume":"94 1","pages":"Pages 105-115"},"PeriodicalIF":1.0,"publicationDate":"2024-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142311087","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}