[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-81565-en":3,"doc-seo-81565-105":29,"detail-sidebar-cat-0-en-105":90},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":4,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},81565,549758146520,"Patrick","https://ap-avatar.wpscdn.com/avatar/80002397d8c0411e94?_k=1775819394049821470",8,"Research & Report","Bifurcations in Interior Transmission Eigenvalues: Theory and Computation","Interior transmission eigenvalue (ITP) problems underpin inverse scattering theory and the spectral study of inhomogeneous media. Although the PDE formulation depends smoothly on the refractive index, the parameter-to-eigenpair map can be non-smooth and exhibit bifurcations. This work develops a theoretical framework giving sufficient conditions for such behavior on general domains, then refines the characterization for radially symmetric geometries. A parametric discrete nonlinear eigenproblem with a match-based adaptive contour eigensolver enables efficient tracking of eigenvalue trajectories. Numerics confirm predictions and uncover new non-smooth effects.","arXiv :2511 . 11016v3 [math .NA] 10 Jul 2026  \nBifurcations in Interior Transmission Eigenvalues: Theory and Computation  \nDavide Pradovera  \nStockholm University, Department of Mathematics,  \nRoslagsvägen 26, 11419 Stockholm, Sweden  \n[davide.pradovera@math.su.se](davide.pradovera@math.su.se)  \nAlessandro Borghi  \nTechnical University Berlin, Institute of Mathematics,  \nStraße des 17 . Juni 136, 10623 Berlin, Germany  \n[borghi@tu-berlin.de](borghi@tu-berlin.de)  \nLukas Pieronek  \nIndependent researcher  \n[pieronek.lukas@gmail.com](pieronek.lukas@gmail.com)  \nAndreas Kleefeld  \nForschungszentrum Jülich GmbH,  \nJülich Supercomputing Centre, Wilhelm-Johnen-Str. ,  \n52425 Jülich, Germany  \nUniversity of Applied Sciences Aachen,  \nFaculty of Medical Engineering and Technomathematics,  \nHeinrich-Mußmann-Str. 1, 52428 Jülich, Germany  \n[a.kleefeld@fz-juelich.de](a.kleefeld@fz-juelich.de)  \nAbstract  \nThe interior transmission eigenvalue problem (ITP) plays a central role in inverse scattering theory and in the spectral analysis of inhomogeneous media. Despite its smooth dependence on the refractive index at the PDE level, the corresponding spectral map from material parameters to eigenpairs may exhibit non-smooth or bifurcating behavior. In this work, we develop a theoretical framework identifying sufficient conditions for such non-smooth spectral behavior in the ITP on general domains. We further specialize our analysis to some radially symmetric geometries, enabling a more precise characterization of bifurcations in the spectrum. Computationally, we formulate the ITP as a parametric, discrete, nonlinear eigenproblem and use a match-based adaptive contour eigensolver to accurately and efficiently track eigenvalue trajectories under parameter variation. Numerical experiments confirm the theoretical predictions and reveal novel non-smooth spectral effects.  \nKeywords: Interior transmission eigenvalues; nonlinear eigenvalue problems; spectral bifurcation; parameter-dependent PDEs; contour-integral methods; numerical eigensolvers.  \nMSC2020: 35P30; 65H17; 47J10; 78A46; 37G10 .  \n1 Introduction  \nThe interior transmission eigenvalue problem (ITP) arises in inverse acoustic scattering theory and mathematical physics. Its importance lies in its role in understanding how time-harmonic acoustic waves interact with inhomogeneous media, particularly for reconstructing properties of objects from scattered wave data in inverse problems [5, 6] . The ITP first appeared in 1986, when Kirsch studied the denseness property of the far-field operator [24] . This was later followed by the work of Colton and Monk [15], who studied it in connection with the linear sampling method. These seminal works established the ITP as a central concept in inverse scattering problems, and interior transmission eigenvalues (the “solutions” of the ITP) have since played a crucial role in algorithms for such problems. In particular, ITP eigenvalues encode valuable information about the refractive index of a medium and its geometry.  \nThe ITP is computationally challenging due to the mixed boundary conditions and the coupling between different wave fields. This is easily seen in the strong form of the ITP:  \n(∆v+wv0 κ2 nw inonD, . (1)  \nHere and throughout, the scatterer D is a given bounded domain whose boundary ∂D has outward-pointing normal ν, and 0 \u003C n ≢ 1 is the refractive index. We call the nonzero wavenumber κ ∈ C0 := C \\ {0} an interior transmission eigenvalue (ITE) if there exist nontrivial eigenfunctions v and w solving (1) . See Section 2 for more details on the problem and on the function spaces where eigenfunctions are sought.  \nThe discreteness [12,17,20,24,38] and existence [10,32] of real ITEs have been studied in great detail. However, the problem is not self-adjoint and therefore non-real eigenvalues may exist. For some special geometries such as spherically stratified media, proofs of the existence of non-real ITEs have been developed [13, 14, 16, 31,","cbCaioEtWFN2xR62","https://ap.wps.com/l/cbCaioEtWFN2xR62","pdf",1318051,1,26,"English","en",105,"# Introduction\n## Contribution of the paper","[{\"question\":\"What is the interior transmission eigenvalue problem (ITP) used for?\",\"answer\":\"The ITP is central to inverse acoustic scattering and mathematical physics, helping connect wave scattering data to properties of inhomogeneous media such as refractive index and geometry.\"},{\"question\":\"Why can eigenvalues exhibit non-smooth or bifurcating behavior even when the PDE depends smoothly on refractive index?\",\"answer\":\"The paper explains that the smooth dependence at the PDE level does not guarantee smoothness of the spectral map from material parameters to eigenpairs, which can develop singularities or bifurcations.\"},{\"question\":\"How does the paper compute and track eigenvalue trajectories when parameters vary?\",\"answer\":\"It formulates the ITP as a parametric, discrete, nonlinear eigenproblem and applies a match-based adaptive contour eigensolver to accurately and efficiently follow eigenvalue trajectories under parameter changes.\"}]",1784174352,66,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":85,"head_meta":87,"extra_data":89,"updated_unix":27},"bifurcations-in-interior-transmission-eigenvalues-theory-and-computation","",{"@graph":35,"@context":84},[36,53,67],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/bifurcations-in-interior-transmission-eigenvalues-theory-and-computation/81565/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":61,"encodingFormat":60,"isAccessibleForFree":62,"interactionStatistic":63},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-16",true,{"@type":64,"interactionType":65,"userInteractionCount":4},"InteractionCounter",{"@type":66},"ViewAction",{"@type":68,"mainEntity":69},"FAQPage",[70,76,80],{"name":71,"@type":72,"acceptedAnswer":73},"What is the interior transmission eigenvalue problem (ITP) used for?","Question",{"text":74,"@type":75},"The ITP is central to inverse acoustic scattering and mathematical physics, helping connect wave scattering data to properties of inhomogeneous media such as refractive index and geometry.","Answer",{"name":77,"@type":72,"acceptedAnswer":78},"Why can eigenvalues exhibit non-smooth or bifurcating behavior even when the PDE depends smoothly on refractive index?",{"text":79,"@type":75},"The paper explains that the smooth dependence at the PDE level does not guarantee smoothness of the spectral map from material parameters to eigenpairs, which can develop singularities or bifurcations.",{"name":81,"@type":72,"acceptedAnswer":82},"How does the paper compute and track eigenvalue trajectories when parameters vary?",{"text":83,"@type":75},"It formulates the ITP as a parametric, discrete, nonlinear eigenproblem and applies a match-based adaptive contour eigensolver to accurately and efficiently follow eigenvalue trajectories under parameter changes.","https://schema.org",{"og:url":51,"og:type":86,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":88,"canonical":51},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":91},[92,96,100,104,109,114,119,122,127,130,134],{"id":20,"doc_module":4,"doc_module_name":45,"category_name":93,"show_sort_weight":94,"slug":95},"Story & Novel",90,"story-novel",{"id":46,"doc_module":4,"doc_module_name":45,"category_name":97,"show_sort_weight":98,"slug":99},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":45,"category_name":101,"show_sort_weight":102,"slug":103},"Exam",70,"exam",{"id":105,"doc_module":4,"doc_module_name":45,"category_name":106,"show_sort_weight":107,"slug":108},5,"Comic",60,"comic",{"id":110,"doc_module":4,"doc_module_name":45,"category_name":111,"show_sort_weight":112,"slug":113},6,"Technology",50,"technology",{"id":115,"doc_module":4,"doc_module_name":45,"category_name":116,"show_sort_weight":117,"slug":118},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":45,"category_name":12,"show_sort_weight":120,"slug":121},30,"research-report",{"id":123,"doc_module":4,"doc_module_name":45,"category_name":124,"show_sort_weight":125,"slug":126},9,"Religion & Spirituality",20,"religion-spirituality",{"id":125,"doc_module":4,"doc_module_name":45,"category_name":128,"show_sort_weight":125,"slug":129},"World Cup","world-cup",{"id":131,"doc_module":4,"doc_module_name":45,"category_name":132,"show_sort_weight":131,"slug":133},10,"Lifestyle","lifestyle",{"id":135,"doc_module":4,"doc_module_name":45,"category_name":136,"show_sort_weight":105,"slug":137},19,"General","general"]