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020 _a9783031490354
_9978-3-031-49035-4
024 7 _a10.1007/978-3-031-49035-4
_2doi
050 4 _aQC173.96-174.52
072 7 _aPHQ
_2bicssc
072 7 _aSCI057000
_2bisacsh
072 7 _aPHQ
_2thema
100 1 _aVeliev, Oktay.
_eauthor.
_4aut
_4http://id.loc.gov/vocabulary/relators/aut
_971871
245 1 0 _aMultidimensional Periodic Schrödinger Operator
_h[electronic resource] :
_bPerturbation Theories for High Energy Regions and Their Applications /
_cby Oktay Veliev.
250 _a3rd ed. 2024.
264 1 _aCham :
_bSpringer International Publishing :
_bImprint: Springer,
_c2024.
300 _aXIII, 411 p. 1 illus.
_bonline resource.
336 _atext
_btxt
_2rdacontent
337 _acomputer
_bc
_2rdamedia
338 _aonline resource
_bcr
_2rdacarrier
347 _atext file
_bPDF
_2rda
490 1 _aSpringer Tracts in Modern Physics,
_x1615-0430 ;
_v291
505 0 _aPreliminary Facts -- From One-dimensional to Multidimensional -- Asymptotic Formulas for the Bloch Eigenvalues and Bloch Functions -- Constructive Determination of the Spectral Invariants -- Periodic Potential from the Spectral Invariants -- Conclusions and Some Generalization.
520 _aThis book describes the direct and inverse problems of the multidimensional Schrödinger operator with a periodic potential, a topic that is especially important in perturbation theory, constructive determination of spectral invariants and finding the periodic potential from the given Bloch eigenvalues. It provides a detailed derivation of the asymptotic formulas for Bloch eigenvalues and Bloch functions in arbitrary dimensions while constructing and estimating the measure of the iso-energetic surfaces in the high-energy regime. Moreover, it presents a unique method proving the validity of the Bethe–Sommerfeld conjecture for arbitrary dimensions and arbitrary lattices. Using the perturbation theory constructed, it determines the spectral invariants of the multidimensional operator from the given Bloch eigenvalues. Some of these invariants are explicitly expressed by the Fourier coefficients of the potential, making it possible to determine the potential constructively using Bloch eigenvalues as input data. Lastly, the book presents an algorithm for the unique determination of the potential. This updated and significantly expanded third edition features an extension of this framework to all dimensions, offering a now complete theory of self-adjoint Schrödinger operators within periodic potentials. Drawing from recent advancements in mathematical analysis, this edition delves even deeper into the intricacies of the subject. It explores the connections between the multidimensional Schrödinger operator, periodic potentials, and other fundamental areas of mathematical physics. The book's comprehensive approach equips both students and researchers with the tools to tackle complex problems and contribute to the ongoing exploration of quantum phenomena.
650 0 _aQuantum physics.
_966608
650 0 _aOperator theory.
650 0 _aDifferential equations.
650 0 _aCondensed matter.
_966773
650 0 _aMathematical physics.
650 1 4 _aQuantum Physics.
_966608
650 2 4 _aOperator Theory.
650 2 4 _aDifferential Equations.
650 2 4 _aCondensed Matter Physics.
_971872
650 2 4 _aMathematical Physics.
710 2 _aSpringerLink (Online service)
773 0 _tSpringer Nature eBook
776 0 8 _iPrinted edition:
_z9783031490347
776 0 8 _iPrinted edition:
_z9783031490361
776 0 8 _iPrinted edition:
_z9783031490378
830 0 _aSpringer Tracts in Modern Physics,
_x1615-0430 ;
_v291
_971873
856 4 0 _uhttps://doi.org/10.1007/978-3-031-49035-4
912 _aZDB-2-PHA
912 _aZDB-2-SXP
999 _c57920
_d57920