{"title":"Surface Waves on Infinite Boundaries","authors":"Dmitrii Yafaev","doi":"10.1134/S1234567825030115","DOIUrl":"10.1134/S1234567825030115","url":null,"abstract":"<p> We develop scattering theory for the Laplace operator in the half-space with Robin type boundary conditions on the boundary plane. In particular, we show that, in addition to usual space waves living in cones and described by standard wave operators, surface waves may arise in this problem. They are localized in parabolic neighbourhoods of the boundary. We find conditions on the boundary coefficient ensuring the existence of surface waves. Several open problems are formulated. </p>","PeriodicalId":575,"journal":{"name":"Functional Analysis and Its Applications","volume":"59 3","pages":"366 - 389"},"PeriodicalIF":0.7,"publicationDate":"2025-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1134/S1234567825030115.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145316339","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"New Remarks on the Scattering for a Perturbed Polyharmonic Operator","authors":"Grigori Rozenblum","doi":"10.1134/S1234567825030085","DOIUrl":"10.1134/S1234567825030085","url":null,"abstract":"<p> We obtain sufficient conditions for the perturbation of the power of the resolvent of the polyharmonic operator under a perturbation by a highly singular potential to belong to Schatten classes. </p>","PeriodicalId":575,"journal":{"name":"Functional Analysis and Its Applications","volume":"59 3","pages":"321 - 329"},"PeriodicalIF":0.7,"publicationDate":"2025-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145316155","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"A Generalized Birman–Schwinger Principle and Applications to One-Dimensional Schrödinger Operators with Distributional Potentials","authors":"Fritz Gesztesy, Roger Nichols","doi":"10.1134/S1234567825030024","DOIUrl":"10.1134/S1234567825030024","url":null,"abstract":"<p> Given a self-adjoint operator <span>(H_0)</span> bounded from below in a complex, separable Hilbert space <span>(mathcal H)</span>, the corresponding scale of spaces <span>(mathcal H_{+1}(H_0) subset mathcal H subset mathcal H_{-1}(H_0)=[mathcal H_{+1}(H_0)]^*)</span>, and a fixed <span>(Vin mathcal B(mathcal H_{+1}(H_0),mathcal H_{-1}(H_0)))</span>, we define the operator-valued map <span>(A_V(,cdot,)colon rho(H_0)to mathcal B(mathcal H))</span> by </p><p> where <span>(rho(H_0))</span> denotes the resolvent set of <span>(H_0)</span>. Assuming that <span>(A_V(z))</span> is compact for some <span>(z=z_0in rho(H_0))</span> and has norm strictly less than one for some <span>(z=E_0in (-infty,0))</span>, we employ an abstract version of Tiktopoulos’ formula to define an operator <span>(H)</span> in <span>(mathcal H)</span> that is formally realized as the sum of <span>(H_0)</span> and <span>(V)</span>. We then establish a Birman–Schwinger principle for <span>(H)</span> in which <span>(A_V(,cdot,))</span> plays the role of the Birman–Schwinger operator: <span>(lambda_0in rho(H_0))</span> is an eigenvalue of <span>(H)</span> if and only if <span>(1)</span> is an eigenvalue of <span>(A_V(lambda_0))</span>. Furthermore, the geometric (but not necessarily the algebraic) multiplicities of <span>(lambda_0)</span> and <span>(1)</span> as eigenvalues of <span>(H)</span> and <span>(A_V(lambda_0))</span>, respectively, coincide. </p><p> As a concrete application, we consider one-dimensional Schrödinger operators with <span>(H^{-1}(mathbb{R}))</span> distributional potentials. </p>","PeriodicalId":575,"journal":{"name":"Functional Analysis and Its Applications","volume":"59 3","pages":"224 - 250"},"PeriodicalIF":0.7,"publicationDate":"2025-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145316345","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Elena Zhizhina, Andrey Piatnitski, Vladimir Sloushch, Tatiana Suslina
{"title":"Homogenization of the Lévy-type Operators","authors":"Elena Zhizhina, Andrey Piatnitski, Vladimir Sloushch, Tatiana Suslina","doi":"10.1134/S1234567825030036","DOIUrl":"10.1134/S1234567825030036","url":null,"abstract":"<p> In <span>(L_2(mathbb R^d))</span>, we consider a selfadjoint operator <span>({mathbb A}_varepsilon)</span>, <span>(varepsilon >0)</span>, of the form </p><p> where <span>(0< alpha < 2)</span>. It is assumed that a function <span>(mu(mathbf{x},mathbf{y}))</span> is bounded, positive definite, periodic in each variable, and is such that <span>(mu(mathbf{x},mathbf{y})=mu(mathbf{y},mathbf{x}))</span>. A rigorous definition of the operator <span>({mathbb A}_varepsilon)</span> is given in terms of the corresponding quadratic form. It is proved that the resolvent <span>(({mathbb A}_varepsilon+I)^{-1})</span> converges in the operator norm on <span>(L_2(mathbb R^d))</span> to the operator <span>(({mathbb A}^0+I)^{-1})</span> as <span>(varepsilonto 0)</span>. Here, <span>({mathbb A}^0)</span> is an effective operator of the same form with the constant coefficient <span>(mu^0)</span> equal to the mean value of <span>(mu(mathbf{x},mathbf{y}))</span>. We obtain an error estimate of order <span>(O(varepsilon^alpha))</span> for <span>(0< alpha < 1)</span>, <span>(O(varepsilon (1+| operatorname{ln} varepsilon|)^2))</span> for <span>( alpha=1)</span>, and <span>(O(varepsilon^{2- alpha}))</span> for <span>(1< alpha < 2)</span>. In the case where <span>(1< alpha < 2)</span>, the result is refined by taking the correctors into account. </p>","PeriodicalId":575,"journal":{"name":"Functional Analysis and Its Applications","volume":"59 3","pages":"251 - 257"},"PeriodicalIF":0.7,"publicationDate":"2025-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145316346","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Perron’s Method in the Dirichlet Problem for the Soft Laplacian on a Stratified Set","authors":"N. S. Dairbekov, O. M. Penkin, D. V. Savasteev","doi":"10.1134/S1064562424602658","DOIUrl":"10.1134/S1064562424602658","url":null,"abstract":"<p>The solvability of the Dirichlet problem for the soft Laplacian on a stratified set is proved using a modification of the well-known Perron method.</p>","PeriodicalId":531,"journal":{"name":"Doklady Mathematics","volume":"111 1","pages":"16 - 19"},"PeriodicalIF":0.6,"publicationDate":"2025-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145316394","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On the Zaremba Problem for Inhomogeneous p-Laplace Equation with Drift","authors":"Yu. A. Alkhutov, M. D. Surnachev, A. G. Chechkina","doi":"10.1134/S1064562424602749","DOIUrl":"10.1134/S1064562424602749","url":null,"abstract":"<p>A higher integrability of the gradient of a solution to the Zaremba problem in a bounded strictly Lipschitz domain is proved for an inhomogeneous <i>p</i>-Laplace equation with lower terms.</p>","PeriodicalId":531,"journal":{"name":"Doklady Mathematics","volume":"111 1","pages":"1 - 5"},"PeriodicalIF":0.6,"publicationDate":"2025-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145316134","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On the Extraction of Random Bit Sequences in Quantum Random Number Generators with Several Independent Markov Sources","authors":"I. M. Arbekov, S. N. Molotkov","doi":"10.1134/S1064562424602701","DOIUrl":"10.1134/S1064562424602701","url":null,"abstract":"<p>The paper presents a method for extracting provably random bit sequences from several independent Markov chain trajectories, each having an arbitrary finite order. In implementing quantum random number generators, the combined use of several trajectories makes it possible to significantly increase the speed of generating output bit sequences.</p>","PeriodicalId":531,"journal":{"name":"Doklady Mathematics","volume":"111 1","pages":"6 - 15"},"PeriodicalIF":0.6,"publicationDate":"2025-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145316135","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On Some Number Theoretic Sum","authors":"V. V. Iudelevich","doi":"10.1134/S1064562424601586","DOIUrl":"10.1134/S1064562424601586","url":null,"abstract":"<p>We obtain an asymptotic formula for the sum \u0000 <span>(Q(x) = sumlimits_{substack{ n leqslant x r(n + 1) ne 0 } } frac{{r(n)}}{{r(n + 1)}};;(x to + infty ),)</span> \u0000where <span>(r(n))</span> denotes the number of representations of <i>n</i> as a sum of two squares.</p>","PeriodicalId":531,"journal":{"name":"Doklady Mathematics","volume":"111 1","pages":"25 - 28"},"PeriodicalIF":0.6,"publicationDate":"2025-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145316396","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Unbounded Integral Hankel Operators","authors":"Alexander Pushnitski, Sergei R. Treil","doi":"10.1134/S1234567825030073","DOIUrl":"10.1134/S1234567825030073","url":null,"abstract":"<p> For a wide class of unbounded integral Hankel operators on the positive half-line, we prove essential self-adjointness on the set of smooth compactly supported functions. </p>","PeriodicalId":575,"journal":{"name":"Functional Analysis and Its Applications","volume":"59 3","pages":"297 - 320"},"PeriodicalIF":0.7,"publicationDate":"2025-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145316154","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}