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  • 1
    Book
    Book
    Semigroups of linear operators and applications (1985)
    New York u.a. :Oxford University Press,
    Details
    Title: Semigroups of linear operators and applications
    Author: Goldstein, Jerome A.
    Publisher: New York u.a. :Oxford University Press,
    Year of publication: 1985
    Pages: 245 S.
    Series Statement: Oxford mathematical monographs
    Type of Medium: Book
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  • 2
    Book
    Book
    Semigroups of linear operators and applications (2017)
    New York :Oxford University Press,
    Details
    Title: Semigroups of linear operators and applications
    Author: Goldstein, Jerome A.
    Edition: 2. Ed.
    Publisher: New York :Oxford University Press,
    Year of publication: 2017
    Pages: 245 S.
    Series Statement: Oxford mathematical monographs
    Type of Medium: Book
    URL: http://search.ebscohost.com/login.aspx?direct=true&db=nlebk&AN=1521108&site=ehost-live
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  • 3
    Electronic Resource
    Electronic Resource
    Thomas–Fermi theory with an external magnetic field (1991)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 32 (1991), S. 2907-2917 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: Of concern is a rigorous Thomas–Fermi theory of ground state electron densities for quantum mechanical systems in an external magnetic field. The energy functional takes the form E(ρ1,ρ2)=∑2i=1ci ∫R3ρi (x)5/3 dx + (1)/(2) ∫R3∫R3[ρ(x)ρ(y)/||x−y||]dx dy +∫R3V(x)ρ(x)dx +∫R3(B(x)(ρ1(x)−ρ2(x))dx; here ci is a positive constant, ρ1 [resp. ρ2] is the density of spin-up [resp. spin-down] electrons, ρ=ρ1+ρ2 is the total electron density, V is a given potential (typically a Coulomb potential describing electron–nuclear attraction), and B describes the effect of the external magnetic field. Let Ni=∫R3ρi(x)dx be the number of spin-up and spin-down electrons for i=1,2, and let N=N1+N2 be the total number of electrons. Specifying N and minimizing E(ρ1,ρ2) generally leads to a spin polarized system. For example, if B≤0 and B(large-closed-square)0, then ρ1≥ρ2 and N1〉N2. This and a number of related results are proved.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.529084
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  • 4
    Electronic Resource
    Electronic Resource
    Spin-polarized Thomas–Fermi theory (1988)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 29 (1988), S. 709-716 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: Of concern is a rigorous Thomas–Fermi theory of electron densities for spin-polarized quantum-mechanical systems. The number N↑, N↓ of spin-up and spin-down electrons are specified in advance, and one seeks to minimize the energy functional E(ρ↑,ρ↓) =c1∫R3(ρ↑(x)5/3 +ρ↓(x)5/3)dx +c2∫R3∫R3[ρ(x)ρ(y)/||x −y||]dx dy +∫R3V(x)ρ(x)dx, where c1, c2 are given positive constants, ρ↑ and ρ↓ are non-negative functions, ρ=ρ↑ +ρ↓ is the total electron density, ∫R3ρ↑(x)dx =N↑, ∫R3ρ↓(x)dx =N↓, and V is a given potential. These results are analogous to the classical rigorous (spin-unpolarized) Thomas–Fermi theory developed by Lieb and Simon [Phys. Rev. Lett. 33, 681 (1973)] and by Bénilan and Brezis ("The Thomas–Fermi problem,'' in preparation).
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.528011
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  • 5
    Electronic Resource
    Electronic Resource
    A rigorous modified Thomas–Fermi theory for atomic systems (1987)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 28 (1987), S. 1198-1202 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: Recently Parr and Ghosh [Proc. Natl. Acad. Sci. USA 83, 3577 (1986)] proposed a variant of the classical Thomas–Fermi theory of electrons in an atom. They produced a continuous electron density by introducing the constraint that the integral ∫R3 e−2k||x|| Δρ(x)dx exists, where k is determined by the nuclear cusp condition. Their results give improved calculations of ground state electron densities and energies. The present paper provides a rigorous mathematical foundation for the work of Parr and Ghosh and converts their results into theorems. Some generalizations are also obtained.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.527567
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  • 6
    Electronic Resource
    Electronic Resource
    Bound states and scattered states for contraction semigroups (1985)
    Springer
    Acta applicandae mathematicae 4 (1985), S. 93-98 
    Details
    ISSN: 1572-9036
    Keywords: Primary ; 47D05 ; secondary ; 34G10 ; 35P25 ; 47D45 ; RAGE theorem ; contraction semigroups ; asymptotics ; bound state ; scattered state ; mean ergodic theorem ; Wiener's theorem ; almost convergence
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract LetA generate a strongly continuous contraction semigroup {T(t)} on a Hilbert space and letL be a bounded operator. IfL(ζI−A)−1 is compact, then the Cesàro limit of ‖LT(t)f‖2 (ast→∞) is computed for all vectorsf. This limit is interpreted in terms of bound and scattered states in the context of quantum mechanical and classical wave propagation problems.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02293492
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  • 7
    Electronic Resource
    Electronic Resource
    Book reviews (1995)
    Springer
    Zeitschrift für angewandte Mathematik und Physik 46 (1995), S. 475-478 
    Details
    ISSN: 1420-9039
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01003563
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  • 8
    Electronic Resource
    Electronic Resource
    Regularized semigroups, iterated Cauchy problems and equipartition of energy (1993)
    Springer
    Monatshefte für Mathematik 115 (1993), S. 47-66 
    Details
    ISSN: 1436-5081
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The iterated Cauchy problem under consideration is $$\Pi _{k = 1}^n (d/dt - A_k )u(t) = 0(t \geqslant 0).(*)$$ Here {A 1,..., An} are unbounded linear operators on a Banach space. The initial value problem for (*) is governed by a semigroup of some sort. When eachA k is a (C 0) semigroup generator, this semigroup is of class (C 0) and was studied by J. T. Sandefur [26]. This result is extended to the case when eachA k generates aC-regularized semigroup (withC independent ofk). This means one can solveu′=Au, u(0)=f∈C (Dom (A)) and getu(t)→0 wheneverC −1f→0; hereC is bounded and injective. When theA k are commuting generator withA k-Aj injective fork≠j, then the Goldstein-Sandefur d'Alembert formula [19] is extended, viz. solutions of (*) (with suitable restrictions on the initial data) are of the form $$u = \sum\nolimits_{i = 1}^n {u_i } $$ whereu i is a solution ofu′ i=Aiui. Examples and applications are given. Included among the examples is the establishment of a form of equipartition of energy for the Laplace equation; equipartition of energy is wellknown for the wave equation. A final section of the paper deals with the absence of necessary conditions for equipartition of energy.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01311210
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  • 9
    Electronic Resource
    Electronic Resource
    Some developments in semigroups of operators since Hille-Phillips (1981)
    Springer
    Integral equations and operator theory 4 (1981), S. 350-365 
    Details
    ISSN: 1420-8989
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01697971
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  • 10
    Electronic Resource
    Electronic Resource
    On the convergence and approximation of cosine functions (1973)
    Springer
    Aequationes mathematicae 9 (1973), S. 309-309 
    Details
    ISSN: 1420-8903
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01832655
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