Lax pairs are one of the most important features of integrable system. In this work, we propose the Lax pairs informed neural networks (LPNNs) tailored for the integrable systems with Lax pairs by designing novel network architectures and loss functions, comprising LPNN-v1 and LPNN-v2. The most noteworthy advantage of LPNN-v1 is that it can transform the solving of nonlinear integrable systems into the solving of a linear Lax pairs spectral problems, and it not only efficiently solves data-driven localized wave solutions, but also obtains spectral parameter and corresponding spectral function in Lax pairs spectral problems of the integrable systems. On the basis of LPNN-v1, we additionally incorporate the compatibility condition/zero curvature equation of Lax pairs in LPNN-v2, its major advantage is the ability to solve and explore high-accuracy data-driven localized wave solutions and associated spectral problems for integrable systems with Lax pairs. The numerical experiments focus on studying abundant localized wave solutions for very important and representative integrable systems with Lax pairs spectral problems, including the soliton solution of the Korteweg-de Vries (KdV) euqation and modified KdV equation, rogue wave solution of the nonlinear Schr"odinger equation, kink solution of the sine-Gordon equation, non-smooth peakon solution of the Camassa-Holm equation and pulse solution of the short pulse equation, as well as the line-soliton solution of Kadomtsev-Petviashvili equation and lump solution of high-dimensional KdV equation. The innovation of this work lies in the pioneering integration of Lax pairs informed of integrable systems into deep neural networks, thereby presenting a fresh methodology and pathway for investigating data-driven localized wave solutions and Lax pairs spectral problems.
{"title":"Lax pairs informed neural networks solving integrable systems","authors":"Juncai Pu, Yong Chen","doi":"arxiv-2401.04982","DOIUrl":"https://doi.org/arxiv-2401.04982","url":null,"abstract":"Lax pairs are one of the most important features of integrable system. In\u0000this work, we propose the Lax pairs informed neural networks (LPNNs) tailored\u0000for the integrable systems with Lax pairs by designing novel network\u0000architectures and loss functions, comprising LPNN-v1 and LPNN-v2. The most\u0000noteworthy advantage of LPNN-v1 is that it can transform the solving of\u0000nonlinear integrable systems into the solving of a linear Lax pairs spectral\u0000problems, and it not only efficiently solves data-driven localized wave\u0000solutions, but also obtains spectral parameter and corresponding spectral\u0000function in Lax pairs spectral problems of the integrable systems. On the basis\u0000of LPNN-v1, we additionally incorporate the compatibility condition/zero\u0000curvature equation of Lax pairs in LPNN-v2, its major advantage is the ability\u0000to solve and explore high-accuracy data-driven localized wave solutions and\u0000associated spectral problems for integrable systems with Lax pairs. The\u0000numerical experiments focus on studying abundant localized wave solutions for\u0000very important and representative integrable systems with Lax pairs spectral\u0000problems, including the soliton solution of the Korteweg-de Vries (KdV)\u0000euqation and modified KdV equation, rogue wave solution of the nonlinear\u0000Schr\"odinger equation, kink solution of the sine-Gordon equation, non-smooth\u0000peakon solution of the Camassa-Holm equation and pulse solution of the short\u0000pulse equation, as well as the line-soliton solution of Kadomtsev-Petviashvili\u0000equation and lump solution of high-dimensional KdV equation. The innovation of\u0000this work lies in the pioneering integration of Lax pairs informed of\u0000integrable systems into deep neural networks, thereby presenting a fresh\u0000methodology and pathway for investigating data-driven localized wave solutions\u0000and Lax pairs spectral problems.","PeriodicalId":501592,"journal":{"name":"arXiv - PHYS - Exactly Solvable and Integrable Systems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-01-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139422841","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}
Chenan Wei, Vagharsh V. Mkhitaryan, Tigran A. Sedrakyan
We study the low-energy properties of the chiral Heisenberg chain, namely, a one-dimensional spin-1/2 isotropic Heisenberg chain with time-reversal symmetry-breaking pseudo-scalar chiral interaction. We employ the thermodynamic Bethe ansatz to find "chiralization", the response of the ground state versus the strength of the chiral interaction of a chiral Heisenberg chain. Unlike the magnetization case, the chirality of the ground state remains zero until the transition point corresponding to critical coupling $alpha_c=2J/pi$ with $J$ being the antiferromagnetic spin-exchange interaction. The central-charge $c=1$ conformal field theories (CFTs) describe the two phases with zero and finite chirality. We suggest that the difference lies in the symmetry of their ground state (lightest weight) primary fields, i.e., the two phases are symmetry-enriched CFTs. At finite but small temperatures, the non-chiral Heisenberg phase acquires a finite chirality that scales with the temperature quadratically. We show that the finite-size effect around the transition point probes the transition.
{"title":"Unveiling chiral phases: Finite-size scaling as a probe of quantum phase transition in symmetry-enriched $c=1$ conformal field theories","authors":"Chenan Wei, Vagharsh V. Mkhitaryan, Tigran A. Sedrakyan","doi":"arxiv-2312.16660","DOIUrl":"https://doi.org/arxiv-2312.16660","url":null,"abstract":"We study the low-energy properties of the chiral Heisenberg chain, namely, a\u0000one-dimensional spin-1/2 isotropic Heisenberg chain with time-reversal\u0000symmetry-breaking pseudo-scalar chiral interaction. We employ the thermodynamic\u0000Bethe ansatz to find \"chiralization\", the response of the ground state versus\u0000the strength of the chiral interaction of a chiral Heisenberg chain. Unlike the\u0000magnetization case, the chirality of the ground state remains zero until the\u0000transition point corresponding to critical coupling $alpha_c=2J/pi$ with $J$\u0000being the antiferromagnetic spin-exchange interaction. The central-charge $c=1$\u0000conformal field theories (CFTs) describe the two phases with zero and finite\u0000chirality. We suggest that the difference lies in the symmetry of their ground\u0000state (lightest weight) primary fields, i.e., the two phases are\u0000symmetry-enriched CFTs. At finite but small temperatures, the non-chiral\u0000Heisenberg phase acquires a finite chirality that scales with the temperature\u0000quadratically. We show that the finite-size effect around the transition point\u0000probes the transition.","PeriodicalId":501592,"journal":{"name":"arXiv - PHYS - Exactly Solvable and Integrable Systems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-12-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139064000","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 paper, we show that all the bilinear Adler-Bobenko-Suris (ABS) equations (except Q2 and Q4) can be obtained from symmetric discrete AKP system by taking proper reductions and continuum limits. Among the bilinear ABS equations, a simpler bilinear form of the ABS H2 equation is given. In addition, an 8-point 3-dimensional lattice equation and an 8-point 4-dimensional lattice equation are obtained as by-products. Both of them can be considered as extensions of the symmetric discrete AKP equation.
{"title":"Connection between the symmetric discrete AKP system and bilinear ABS lattice equations","authors":"Jing Wang, Da-jun Zhang, Ken-ichi Maruno","doi":"arxiv-2312.15669","DOIUrl":"https://doi.org/arxiv-2312.15669","url":null,"abstract":"In this paper, we show that all the bilinear Adler-Bobenko-Suris (ABS)\u0000equations (except Q2 and Q4) can be obtained from symmetric discrete AKP system\u0000by taking proper reductions and continuum limits. Among the bilinear ABS\u0000equations, a simpler bilinear form of the ABS H2 equation is given. In\u0000addition, an 8-point 3-dimensional lattice equation and an 8-point\u00004-dimensional lattice equation are obtained as by-products. Both of them can be\u0000considered as extensions of the symmetric discrete AKP equation.","PeriodicalId":501592,"journal":{"name":"arXiv - PHYS - Exactly Solvable and Integrable Systems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-12-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139053971","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}
Frederik Møller, Federica Cataldini, Jörg Schmiedmayer
In the hydrodynamics of integrable models, diffusion is a subleading correction to ballistic propagation. Here we quantify the diffusive contribution for one-dimensional Bose gases and find it most influential in the crossover between the main thermodynamic regimes of the gas. Analysing the experimentally measured dynamics of a single density mode, we find diffusion to be relevant only for high wavelength excitations. Instead, the observed relaxation is solely caused by a ballistically driven dephasing process, whose time scale is related to the phonon lifetime of the system and is thus useful to evaluate the applicability of the phonon bases typically used in quantum field simulators.
{"title":"Identifying diffusive length scales in one-dimensional Bose gases","authors":"Frederik Møller, Federica Cataldini, Jörg Schmiedmayer","doi":"arxiv-2312.14007","DOIUrl":"https://doi.org/arxiv-2312.14007","url":null,"abstract":"In the hydrodynamics of integrable models, diffusion is a subleading\u0000correction to ballistic propagation. Here we quantify the diffusive\u0000contribution for one-dimensional Bose gases and find it most influential in the\u0000crossover between the main thermodynamic regimes of the gas. Analysing the\u0000experimentally measured dynamics of a single density mode, we find diffusion to\u0000be relevant only for high wavelength excitations. Instead, the observed\u0000relaxation is solely caused by a ballistically driven dephasing process, whose\u0000time scale is related to the phonon lifetime of the system and is thus useful\u0000to evaluate the applicability of the phonon bases typically used in quantum\u0000field simulators.","PeriodicalId":501592,"journal":{"name":"arXiv - PHYS - Exactly Solvable and Integrable Systems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-12-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139031178","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}
Integrable quantum field theories can be regularized on the lattice while preserving integrability. The resulting theory on the lattice are integrable lattice models. A prototype of such a regularization is the correspondence between sine-Gordon model and 6-vertex model on a light-cone lattice. We propose an integrable deformation of the light-cone lattice model such that in the continuum limit we obtain the $Tbar{T}$-deformed sine-Gordon model. Under this deformation, the cut-off momentum becomes energy dependent while the underlying Yang-Baxter integrability is preserved. Therefore this deformation is integrable but non-local, similar to the $Tbar{T}$-deformation of quantum field theory.
{"title":"$Tbar{T}$-deformation: a lattice approach","authors":"Yunfeng Jiang","doi":"arxiv-2312.12078","DOIUrl":"https://doi.org/arxiv-2312.12078","url":null,"abstract":"Integrable quantum field theories can be regularized on the lattice while\u0000preserving integrability. The resulting theory on the lattice are integrable\u0000lattice models. A prototype of such a regularization is the correspondence\u0000between sine-Gordon model and 6-vertex model on a light-cone lattice. We\u0000propose an integrable deformation of the light-cone lattice model such that in\u0000the continuum limit we obtain the $Tbar{T}$-deformed sine-Gordon model. Under\u0000this deformation, the cut-off momentum becomes energy dependent while the\u0000underlying Yang-Baxter integrability is preserved. Therefore this deformation\u0000is integrable but non-local, similar to the $Tbar{T}$-deformation of quantum\u0000field theory.","PeriodicalId":501592,"journal":{"name":"arXiv - PHYS - Exactly Solvable and Integrable Systems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138816569","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}
Mark J. Ablowitz, Ziad H. Musslimani, Nicholas J. Ossi
Nonlocal integrable partial differential equations possessing a spatial or temporal reflection have constituted an active research area for the past decade. Recently, more general classes of these nonlocal equations have been proposed, wherein the nonlocality appears as a combination of a shift (by a real or a complex parameter) and a reflection. This new shifting parameter manifests itself in the inverse scattering transform (IST) as an additional phase factor in an analogous way to the classical Fourier transform. In this paper, the IST is analyzed in detail for several examples of such systems. Particularly, time, space, and space-time shifted nonlinear Schr"odinger (NLS) and space-time shifted modified Korteweg-de Vries (mKdV) equations are studied. Additionally, the semi-discrete IST is developed for the time, space and space-time shifted variants of the Ablowitz-Ladik integrable discretization of the NLS. One soliton solutions are constructed for all continuous and discrete cases.
{"title":"Inverse scattering transform for continuous and discrete space-time shifted integrable equations","authors":"Mark J. Ablowitz, Ziad H. Musslimani, Nicholas J. Ossi","doi":"arxiv-2312.11780","DOIUrl":"https://doi.org/arxiv-2312.11780","url":null,"abstract":"Nonlocal integrable partial differential equations possessing a spatial or\u0000temporal reflection have constituted an active research area for the past\u0000decade. Recently, more general classes of these nonlocal equations have been\u0000proposed, wherein the nonlocality appears as a combination of a shift (by a\u0000real or a complex parameter) and a reflection. This new shifting parameter\u0000manifests itself in the inverse scattering transform (IST) as an additional\u0000phase factor in an analogous way to the classical Fourier transform. In this\u0000paper, the IST is analyzed in detail for several examples of such systems.\u0000Particularly, time, space, and space-time shifted nonlinear Schr\"odinger (NLS)\u0000and space-time shifted modified Korteweg-de Vries (mKdV) equations are studied.\u0000Additionally, the semi-discrete IST is developed for the time, space and\u0000space-time shifted variants of the Ablowitz-Ladik integrable discretization of\u0000the NLS. One soliton solutions are constructed for all continuous and discrete\u0000cases.","PeriodicalId":501592,"journal":{"name":"arXiv - PHYS - Exactly Solvable and Integrable Systems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138821808","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 consider the Painleve asymptotics for a solution of integrable coupled Hirota equationwith a 3*3 Lax pair whose initial data decay rapidly at infinity. Using Riemann-Hilbert techniques and Deift-Zhou nonlinear steepest descent arguments, in a transition zone defined by /x/t-1/(12a)/t^2/3<=C, where C>0 is a constant, it turns out that the leading-order term to the solution can be expressed in terms of the solution of a coupled Painleve II equation associated with a 3*3 matrix Riemann-Hilbert problem.
{"title":"The coupled hirota equation with a 3*3 lax pair: painleve-type asymptotics in transition zone","authors":"Xao-Dan Zhao, Lei Wang","doi":"arxiv-2312.07185","DOIUrl":"https://doi.org/arxiv-2312.07185","url":null,"abstract":"We consider the Painleve asymptotics for a solution of integrable coupled\u0000Hirota equationwith a 3*3 Lax pair whose initial data decay rapidly at\u0000infinity. Using Riemann-Hilbert techniques and Deift-Zhou nonlinear steepest\u0000descent arguments, in a transition zone defined by /x/t-1/(12a)/t^2/3<=C, where\u0000C>0 is a constant, it turns out that the leading-order term to the solution can\u0000be expressed in terms of the solution of a coupled Painleve II equation\u0000associated with a 3*3 matrix Riemann-Hilbert problem.","PeriodicalId":501592,"journal":{"name":"arXiv - PHYS - Exactly Solvable and Integrable Systems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-12-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138628798","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}
The study of the set-theoretic solutions of the reflection equation, also known as reflection maps, is closely related to that of the Yang-Baxter maps. In this work, we construct reflection maps on various geometrical objects, associated with factorization problems on rational loop groups and involutions. We show that such reflection maps are smoothly conjugate to the composite of permutation maps, with corresponding reduced Yang-Baxter maps. In the case when the reduced Yang-Baxter maps are independent of parameters, the latter are just braiding operators. We also study the symplectic and Poisson geometry of such reflection maps. In a special case, the factorization problems are associated with the collision of N-solitons of the n-Manakov system with a boundary, and in this context the N-body polarization reflection map is a symplectomorphism.
反射方程的集合论解（又称反射映射）的研究与杨-巴克斯特映射的研究密切相关。在这项工作中，我们构建了各种几何对象上的反射映射，这些对象与有理环群和渐开线上的因式分解问题相关。在还原的杨-巴克斯特映射与参数无关的情况下，后者只是制导算子。我们还研究了这种反射图的交映几何和泊松几何。在一种特殊情况下，因式分解问题与 n-Manakov 系统的 N 粒子与边界的碰撞有关，在这种情况下，N 体极化反射图是一种交映射。
{"title":"Reflection Maps Associated with Involutions and Factorization Problems, and Their Poisson Geometry","authors":"Luen-Chau Li, Vincent Caudrelier","doi":"arxiv-2312.05164","DOIUrl":"https://doi.org/arxiv-2312.05164","url":null,"abstract":"The study of the set-theoretic solutions of the reflection equation, also\u0000known as reflection maps, is closely related to that of the Yang-Baxter maps.\u0000In this work, we construct reflection maps on various geometrical objects,\u0000associated with factorization problems on rational loop groups and involutions.\u0000We show that such reflection maps are smoothly conjugate to the composite of\u0000permutation maps, with corresponding reduced Yang-Baxter maps. In the case when\u0000the reduced Yang-Baxter maps are independent of parameters, the latter are just\u0000braiding operators. We also study the symplectic and Poisson geometry of such\u0000reflection maps. In a special case, the factorization problems are associated\u0000with the collision of N-solitons of the n-Manakov system with a boundary, and\u0000in this context the N-body polarization reflection map is a symplectomorphism.","PeriodicalId":501592,"journal":{"name":"arXiv - PHYS - Exactly Solvable and Integrable Systems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-12-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138569319","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 propose commuting set of matrix-valued difference operators in terms of trigonometric ${rm GL}(N|M)$-valued $R$-matrices providing quantum supersymmetric (and possibly anisotropic) spin Ruijsenaars-Macdonald operators. Two types of trigonometric supersymmetric $R$-matrices are used. The first is the one related to the affine quantized algebra ${hat{mathcal U}}_q({rm gl}(N|M))$. The second is a graded version of the standard $mathbb Z_n$-invariant $A_{n-1}$ type $R$-matrix. We show that being properly normalized the latter graded $R$-matrix satisfies the associative Yang-Baxter equation. Next, we proceed to construction of long-range spin chains using the Polychronakos freezing trick. As a result we obtain a new family of spin chains, which extend the ${rm gl}(N|M)$-invariant Haldane-Shastry spin chain to q-deformed case with possible presence of anisotropy.
{"title":"Supersymmetric generalization of q-deformed long-range spin chains of Haldane-Shastry type and trigonometric GL(N|M) solution of associative Yang-Baxter equation","authors":"M. Matushko, A. Zotov","doi":"arxiv-2312.04525","DOIUrl":"https://doi.org/arxiv-2312.04525","url":null,"abstract":"We propose commuting set of matrix-valued difference operators in terms of\u0000trigonometric ${rm GL}(N|M)$-valued $R$-matrices providing quantum\u0000supersymmetric (and possibly anisotropic) spin Ruijsenaars-Macdonald operators.\u0000Two types of trigonometric supersymmetric $R$-matrices are used. The first is\u0000the one related to the affine quantized algebra ${hat{mathcal U}}_q({rm\u0000gl}(N|M))$. The second is a graded version of the standard $mathbb\u0000Z_n$-invariant $A_{n-1}$ type $R$-matrix. We show that being properly\u0000normalized the latter graded $R$-matrix satisfies the associative Yang-Baxter\u0000equation. Next, we proceed to construction of long-range spin chains using the\u0000Polychronakos freezing trick. As a result we obtain a new family of spin\u0000chains, which extend the ${rm gl}(N|M)$-invariant Haldane-Shastry spin chain\u0000to q-deformed case with possible presence of anisotropy.","PeriodicalId":501592,"journal":{"name":"arXiv - PHYS - Exactly Solvable and Integrable Systems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-12-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138555149","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}
This article deals with two classes of quasi-exactly solvable (QES) trigonometric potentials for which the one-dimensional Schroedinger equation reduces to a confluent Heun equation (CHE) where the independent variable takes only finite values. Power series for the CHE are used to get finite- and infinite-series eigenfunctions. Finite series occur only for special sets of parameters and characterize the quasi-exact solvability. Infinite series occur for all admissible values of the parameters (even values involving finite series), and are bounded and convergent in the entire range of the independent variable. Moreover, throughout the article we examine other QES trigonometric and hyperbolic potentials. In all cases, for a finite series there is a convergent infinite series.
{"title":"Schroedinger equation as a confluent Heun equation","authors":"Bartolomeu Donatila Bonorino Figueiredo","doi":"arxiv-2312.03569","DOIUrl":"https://doi.org/arxiv-2312.03569","url":null,"abstract":"This article deals with two classes of quasi-exactly solvable (QES)\u0000trigonometric potentials for which the one-dimensional Schroedinger equation\u0000reduces to a confluent Heun equation (CHE) where the independent variable takes\u0000only finite values. Power series for the CHE are used to get finite- and\u0000infinite-series eigenfunctions. Finite series occur only for special sets of\u0000parameters and characterize the quasi-exact solvability. Infinite series occur\u0000for all admissible values of the parameters (even values involving finite\u0000series), and are bounded and convergent in the entire range of the independent\u0000variable. Moreover, throughout the article we examine other QES trigonometric\u0000and hyperbolic potentials. In all cases, for a finite series there is a\u0000convergent infinite series.","PeriodicalId":501592,"journal":{"name":"arXiv - PHYS - Exactly Solvable and Integrable Systems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138547580","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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