LEADER 03770nam 2200457 450 001 9910483219103321 005 20210225091549.0 010 $a981-15-8199-1 024 7 $a10.1007/978-981-15-8199-1 035 $a(CKB)4100000011469543 035 $a(MiAaPQ)EBC6357794 035 $a(DE-He213)978-981-15-8199-1 035 $a(PPN)250219085 035 $a(EXLCZ)994100000011469543 100 $a20210225d2020 uy 0 101 0 $aeng 135 $aurnn|008mamaa 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 10$aInverse spectral and scattering theory $ean introduction /$fHiroshi Isozaki 205 $a1st ed. 2020. 210 1$aGateway East, Singapore :$cSpringer,$d[2020] 210 4$d©2020 215 $a1 online resource (XII, 130 p. 2 illus.) 225 1 $aSpringerBriefs in mathematical physics ;$vVolume 38 311 $a981-15-8198-3 327 $aChapter 1: One-dimensional inverse problems -- Chapter 2: Multi-dimensional inverse boundary value problems -- Chapter 3: Multi-dimensional Gel?fand-Levitan theory -- Chapter 4: Boundary control method -- Chapter 5: Other topics -- Index. 330 $aThe aim of this book is to provide basic knowledge of the inverse problems arising in various areas in mathematics, physics, engineering, and medical science. These practical problems boil down to the mathematical question in which one tries to recover the operator (coefficients) or the domain (manifolds) from spectral data. The characteristic properties of the operators in question are often reduced to those of Schrödinger operators. We start from the 1-dimensional theory to observe the main features of inverse spectral problems and then proceed to multi-dimensions. The first milestone is the Borg?Levinson theorem in the inverse Dirichlet problem in a bounded domain elucidating basic motivation of the inverse problem as well as the difference between 1-dimension and multi-dimension. The main theme is the inverse scattering, in which the spectral data is Heisenberg?s S-matrix defined through the observation of the asymptotic behavior at infinity of solutions. Significant progress has been made in the past 30 years by using the Faddeev?Green function or the complex geometrical optics solution by Sylvester and Uhlmann, which made it possible to reconstruct the potential from the S-matrix of one fixed energy. One can also prove the equivalence of the knowledge of S-matrix and that of the Dirichlet-to-Neumann map for boundary value problems in bounded domains. We apply this idea also to the Dirac equation, the Maxwell equation, and discrete Schrödinger operators on perturbed lattices. Our final topic is the boundary control method introduced by Belishev and Kurylev, which is for the moment the only systematic method for the reconstruction of the Riemannian metric from the boundary observation, which we apply to the inverse scattering on non-compact manifolds. We stress that this book focuses on the lucid exposition of these problems and mathematical backgrounds by explaining the basic knowledge of functional analysis and spectral theory, omitting the technical details in order to make the book accessible to graduate students as an introduction to partial differential equations (PDEs) and functional analysis. . 410 0$aSpringerBriefs in mathematical physics ;$vVolume 38. 606 $aSpectral theory (Mathematics) 615 0$aSpectral theory (Mathematics) 676 $a515.7222 700 $aIsozaki$b Hiroji$f1950-$0283875 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910483219103321 996 $aInverse spectral and scattering theory$92185777 997 $aUNINA