Pub Date : 2021-01-01DOI: 10.7494/OPMATH.2021.41.3.413
A. Główczyk, S. Kużel
Schrödinger operators with nonlocal (delta)-interaction are studied with the use of the Lax-Phillips scattering theory methods. The condition of applicability of the Lax-Phillips approach in terms of non-cyclic functions is established. Two formulas for the (S)-matrix are obtained. The first one deals with the Krein-Naimark resolvent formula and the Weyl-Titchmarsh function, whereas the second one is based on modified reflection and transmission coefficients. The (S)-matrix (S(z)) is analytical in the lower half-plane (mathbb{C}_{−}) when the Schrödinger operator with nonlocal (delta)-interaction is positive self-adjoint. Otherwise, (S(z)) is a meromorphic matrix-valued function in (mathbb{C}_{−}) and its properties are closely related to the properties of the corresponding Schrödinger operator. Examples of (S)-matrices are given.
{"title":"On the S-matrix of Schrödinger operator with nonlocal δ-interaction","authors":"A. Główczyk, S. Kużel","doi":"10.7494/OPMATH.2021.41.3.413","DOIUrl":"https://doi.org/10.7494/OPMATH.2021.41.3.413","url":null,"abstract":"Schrödinger operators with nonlocal (delta)-interaction are studied with the use of the Lax-Phillips scattering theory methods. The condition of applicability of the Lax-Phillips approach in terms of non-cyclic functions is established. Two formulas for the (S)-matrix are obtained. The first one deals with the Krein-Naimark resolvent formula and the Weyl-Titchmarsh function, whereas the second one is based on modified reflection and transmission coefficients. The (S)-matrix (S(z)) is analytical in the lower half-plane (mathbb{C}_{−}) when the Schrödinger operator with nonlocal (delta)-interaction is positive self-adjoint. Otherwise, (S(z)) is a meromorphic matrix-valued function in (mathbb{C}_{−}) and its properties are closely related to the properties of the corresponding Schrödinger operator. Examples of (S)-matrices are given.","PeriodicalId":45563,"journal":{"name":"Opuscula Mathematica","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71341921","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}
Pub Date : 2021-01-01DOI: 10.7494/OPMATH.2021.41.3.437
G. Herzog, P. Kunstmann
{"title":"Quadratic inequalities for functionals in l^{∞}","authors":"G. Herzog, P. Kunstmann","doi":"10.7494/OPMATH.2021.41.3.437","DOIUrl":"https://doi.org/10.7494/OPMATH.2021.41.3.437","url":null,"abstract":"","PeriodicalId":45563,"journal":{"name":"Opuscula Mathematica","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71341932","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}
Pub Date : 2021-01-01DOI: 10.7494/opmath.2021.41.4.601
Oleksiy Zelenskiy, V. Darmosiuk, Illia Nalivayko
{"title":"A note on possible density and diameter of counterexamples to the Seymour's second neighborhood conjecture","authors":"Oleksiy Zelenskiy, V. Darmosiuk, Illia Nalivayko","doi":"10.7494/opmath.2021.41.4.601","DOIUrl":"https://doi.org/10.7494/opmath.2021.41.4.601","url":null,"abstract":"","PeriodicalId":45563,"journal":{"name":"Opuscula Mathematica","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71342162","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}
Pub Date : 2021-01-01DOI: 10.7494/opmath.2021.41.5.685
I. Tsyfra
We study the relationship between the solutions of stationary integrable partial and ordinary differential equations and coefficients of the second-order ordinary differential equations invariant with respect to one-parameter Lie group. The classical symmetry method is applied. We prove that if the coefficients of ordinary differential equation satisfy the stationary integrable partial differential equation with two independent variables then the ordinary differential equation is integrable by quadratures. If special solutions of integrable partial differential equations are chosen then the coefficients satisfy the stationary KdV equations. It was shown that the Ermakov equation belong to a class of these equations. In the framework of the approach we obtained the similar results for generalized Riccati equations. By using operator of invariant differentiation we describe a class of higher order ordinary differential equations for which the group-theoretical method enables us to reduce the order of ordinary differential equation.
{"title":"On classical symmetries of ordinary differential equations related to stationary integrable partial differential equations","authors":"I. Tsyfra","doi":"10.7494/opmath.2021.41.5.685","DOIUrl":"https://doi.org/10.7494/opmath.2021.41.5.685","url":null,"abstract":"We study the relationship between the solutions of stationary integrable partial and ordinary differential equations and coefficients of the second-order ordinary differential equations invariant with respect to one-parameter Lie group. The classical symmetry method is applied. We prove that if the coefficients of ordinary differential equation satisfy the stationary integrable partial differential equation with two independent variables then the ordinary differential equation is integrable by quadratures. If special solutions of integrable partial differential equations are chosen then the coefficients satisfy the stationary KdV equations. It was shown that the Ermakov equation belong to a class of these equations. In the framework of the approach we obtained the similar results for generalized Riccati equations. By using operator of invariant differentiation we describe a class of higher order ordinary differential equations for which the group-theoretical method enables us to reduce the order of ordinary differential equation.","PeriodicalId":45563,"journal":{"name":"Opuscula Mathematica","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71342279","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}
Pub Date : 2021-01-01DOI: 10.7494/OPMATH.2021.41.1.5
R. Bentifour, Sofiane El-Hadi Miri
In this paper we deal with the following problem [begin{cases}-sumlimits_{i=1}^{N}left[ left( a+bintlimits_{, Omega }leftvert partial _{i}urightvert ^{p_{i}}dxright) partial _{i}left( leftvert partial _{i}urightvert ^{p_{i}-2}partial _{i}uright) right]=frac{f(x)}{u^{gamma }}pm g(x)u^{q-1} & in Omega, ugeq 0 & in Omega, u=0 & on partial Omega, end{cases}] where (Omega) is a bounded regular domain in (mathbb{R}^{N}). We will assume without loss of generality that (1leq p_{1}leq p_{2}leq ldotsleq p_{N}) and that (f) and (g) are non-negative functions belonging to a suitable Lebesgue space (L^{m}(Omega)), (1lt qlt overline{p}^{ast}), (agt 0), (bgt 0) and (0lt gamma lt 1.)
{"title":"Some existence results for a nonlocal non-isotropic problem","authors":"R. Bentifour, Sofiane El-Hadi Miri","doi":"10.7494/OPMATH.2021.41.1.5","DOIUrl":"https://doi.org/10.7494/OPMATH.2021.41.1.5","url":null,"abstract":"In this paper we deal with the following problem [begin{cases}-sumlimits_{i=1}^{N}left[ left( a+bintlimits_{, Omega }leftvert partial _{i}urightvert ^{p_{i}}dxright) partial _{i}left( leftvert partial _{i}urightvert ^{p_{i}-2}partial _{i}uright) right]=frac{f(x)}{u^{gamma }}pm g(x)u^{q-1} & in Omega, ugeq 0 & in Omega, u=0 & on partial Omega, end{cases}] where (Omega) is a bounded regular domain in (mathbb{R}^{N}). We will assume without loss of generality that (1leq p_{1}leq p_{2}leq ldotsleq p_{N}) and that (f) and (g) are non-negative functions belonging to a suitable Lebesgue space (L^{m}(Omega)), (1lt qlt overline{p}^{ast}), (agt 0), (bgt 0) and (0lt gamma lt 1.)","PeriodicalId":45563,"journal":{"name":"Opuscula Mathematica","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71341683","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}
Pub Date : 2021-01-01DOI: 10.7494/OPMATH.2021.41.1.55
S. Kozerenko
We consider linear and metric self-maps on vertex sets of finite combinatorial trees. Linear maps are maps which preserve intervals between pairs of vertices whereas metric maps are maps which do not increase distances between pairs of vertices. We obtain criteria for a given linear or a metric map to be a positive (negative) under some orientation of the edges in a tree, we characterize trees which admit maps with Markov graphs being paths and prove that the converse of any partial functional digraph is isomorphic to a Markov graph for some suitable map on a tree.
{"title":"More on linear and metric tree maps","authors":"S. Kozerenko","doi":"10.7494/OPMATH.2021.41.1.55","DOIUrl":"https://doi.org/10.7494/OPMATH.2021.41.1.55","url":null,"abstract":"We consider linear and metric self-maps on vertex sets of finite combinatorial trees. Linear maps are maps which preserve intervals between pairs of vertices whereas metric maps are maps which do not increase distances between pairs of vertices. We obtain criteria for a given linear or a metric map to be a positive (negative) under some orientation of the edges in a tree, we characterize trees which admit maps with Markov graphs being paths and prove that the converse of any partial functional digraph is isomorphic to a Markov graph for some suitable map on a tree.","PeriodicalId":45563,"journal":{"name":"Opuscula Mathematica","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71341722","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}
Pub Date : 2021-01-01DOI: 10.7494/opmath.2021.41.4.489
Abdelrachid El Amrouss, O. Hammouti
Let (ninmathbb{N}^{*}), and (Ngeq n) be an integer. We study the spectrum of discrete linear (2n)-th order eigenvalue problems [begin{cases}sum_{k=0}^{n}(-1)^{k}Delta^{2k}u(t-k) = lambda u(t) ,quad & tin[1, N]_{mathbb{Z}}, Delta^{i}u(-(n-1))=Delta^{i}u(N-(n-1)),quad & iin[0, 2n-1]_{mathbb{Z}},end{cases}] where (lambda) is a parameter. As an application of this spectrum result, we show the existence of a solution of discrete nonlinear (2n)-th order problems by applying the variational methods and critical point theory.
{"title":"Spectrum of discrete 2n-th order difference operator with periodic boundary conditions and its applications","authors":"Abdelrachid El Amrouss, O. Hammouti","doi":"10.7494/opmath.2021.41.4.489","DOIUrl":"https://doi.org/10.7494/opmath.2021.41.4.489","url":null,"abstract":"Let (ninmathbb{N}^{*}), and (Ngeq n) be an integer. We study the spectrum of discrete linear (2n)-th order eigenvalue problems [begin{cases}sum_{k=0}^{n}(-1)^{k}Delta^{2k}u(t-k) = lambda u(t) ,quad & tin[1, N]_{mathbb{Z}}, Delta^{i}u(-(n-1))=Delta^{i}u(N-(n-1)),quad & iin[0, 2n-1]_{mathbb{Z}},end{cases}] where (lambda) is a parameter. As an application of this spectrum result, we show the existence of a solution of discrete nonlinear (2n)-th order problems by applying the variational methods and critical point theory.","PeriodicalId":45563,"journal":{"name":"Opuscula Mathematica","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71341544","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}
Pub Date : 2021-01-01DOI: 10.7494/OPMATH.2021.41.1.71
Manabu Naito
We consider the half-linear differential equation of the form [(p(t)|x'|^{alpha}mathrm{sgn} x')' + q(t)|x|^{alpha}mathrm{sgn} x = 0, quad tgeq t_{0},] under the assumption (int_{t_{0}}^{infty}p(s)^{-1/alpha}ds =infty). It is shown that if a certain condition is satisfied, then the above equation has a pair of nonoscillatory solutions with specific asymptotic behavior as (t to infty).
我们在假设(int_{t_{0}}^{infty}p(s)^{-1/alpha}ds =infty)下考虑形式为[(p(t)|x'|^{alpha}mathrm{sgn} x')' + q(t)|x|^{alpha}mathrm{sgn} x = 0, quad tgeq t_{0},]的半线性微分方程。结果表明，如果满足某一条件，则上述方程有一对具有特定渐近行为的非振荡解(t to infty)。
{"title":"Remarks on the existence of nonoscillatory solutions of half-linear ordinary differential equations, I","authors":"Manabu Naito","doi":"10.7494/OPMATH.2021.41.1.71","DOIUrl":"https://doi.org/10.7494/OPMATH.2021.41.1.71","url":null,"abstract":"We consider the half-linear differential equation of the form [(p(t)|x'|^{alpha}mathrm{sgn} x')' + q(t)|x|^{alpha}mathrm{sgn} x = 0, quad tgeq t_{0},] under the assumption (int_{t_{0}}^{infty}p(s)^{-1/alpha}ds =infty). It is shown that if a certain condition is satisfied, then the above equation has a pair of nonoscillatory solutions with specific asymptotic behavior as (t to infty).","PeriodicalId":45563,"journal":{"name":"Opuscula Mathematica","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71341733","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}
Pub Date : 2021-01-01DOI: 10.7494/OPMATH.2021.41.3.301
Mirna Charif, Lech Zielinski
We investigate eigenvalue perturbations for a class of infinite tridiagonal matrices which define unbounded self-adjoint operators with discrete spectrum. In particular we obtain explicit estimates for the convergence radius of the perturbation series and error estimates for the Quantum Rabi Model including the resonance case. We also give expressions for coefficients near resonance in order to evaluate the quality of the rotating wave approximation due to Jaynes and Cummings.
{"title":"Perturbation series for Jacobi matrices and the quantum Rabi model","authors":"Mirna Charif, Lech Zielinski","doi":"10.7494/OPMATH.2021.41.3.301","DOIUrl":"https://doi.org/10.7494/OPMATH.2021.41.3.301","url":null,"abstract":"We investigate eigenvalue perturbations for a class of infinite tridiagonal matrices which define unbounded self-adjoint operators with discrete spectrum. In particular we obtain explicit estimates for the convergence radius of the perturbation series and error estimates for the Quantum Rabi Model including the resonance case. We also give expressions for coefficients near resonance in order to evaluate the quality of the rotating wave approximation due to Jaynes and Cummings.","PeriodicalId":45563,"journal":{"name":"Opuscula Mathematica","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71341851","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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