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  • 1
    Electronic Resource
    Electronic Resource
    General lower bounds for resonances in one dimension (1982)
    Springer
    Communications in mathematical physics 86 (1982), S. 221-225 
    Details
    ISSN: 1432-0916
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract Lower bounds are derived for the magnitude of the imaginary parts of the resonance eigenvalues of a Schrödinger operator $$ - \frac{{d^2 }}{{dx^2 }} + V(x)$$ on the line, depending only on the support and bounds ofV and on the real part of the resonance eigenvalue. For example, if the resonance eigenvalue is denotedE +iɛ, then there existC and ℓ0 depending only on ‖E‖∞ andE such that if the support ofV is contained in an interval of length ℓ 〉 ℓ0, then $$\left| \varepsilon \right| 〉 \frac{{m^3 \sqrt E }}{{(m + \sqrt E )^2 }}\exp ( - m\ell )(1 - C\ell ^{ - 1} ),$$ wherem‖V(x)−E− ∞ 1/2 .
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01206011
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  • 2
    Electronic Resource
    Electronic Resource
    Maximal and minimal eigenvalues and their associated nonlinear equations (1987)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 28 (1987), S. 1770-1786 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: The spectral theory of uniformly elliptic operators A under perturbations V giving rise to operators of the form HV=A+V(x) on a bounded or unbounded region, such as Schrödinger operators, are considered. Suppose that (parallel)V(parallel)p is constrained, but V is otherwise unspecified. The theory of the potentials V that maximize or minimize the eigenvalues of HV is presented. The optimizing potentials are typically determined by equations of the form −Δu+W(x)u=±cuα+Λu. The optimization of eigenvalues also turns out to be related to the determination of the best constants in Sobolev's inequality, and, in its one-dimensional simplification, to a classical oscillator problem with "instanton'' properties.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.527488
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  • 3
    Electronic Resource
    Electronic Resource
    On the asymptotic distribution of eigenvalues of banded matrices (1988)
    Springer
    Constructive approximation 4 (1988), S. 403-417 
    Details
    ISSN: 1432-0940
    Keywords: 42C05 ; 35P20 ; Orthogonal polynomials ; Asymptotic distribution of eigenvalues ; Density of states ; Recurrence coefficients
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We consider the abstract measures, known as thedensity- of- states measures, associated with the asymptotic distribution of eigenvalues of infinite banded Hermitian matrices. Two widely used definitions of these measures are shown to be equivalent, even in the unbounded case, and we prove that the density of states is invariant under certain, possibly unbounded, perturbations. Also considered are measures associated with the asymptotic distribution of eigenvalues of rescaled unbounded matrices. These measures are associated with the so-called contracted spectrum when the matrices are tridiagonal. Finally, we produce several examples clarifying the nature of the density of states.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02075470
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  • 4
    Electronic Resource
    Electronic Resource
    Perturbation theory for shape resonances and large barrier potentials (1982)
    Springer
    Communications in mathematical physics 83 (1982), S. 151-170 
    Details
    ISSN: 1432-0916
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract We develop a systematic perturbation and resonance theory for the one-dimensional Schrödinger equation of the form $$( - d^2 /dx^2 + U(x) + \lambda V(x) - E)\psi (x) = 0,0 \leqq x〈 \infty ,$$ where the barrier potentialV(x) is supported only wherex≧1 and is non-negative there, and λ is a real parameter tending to infinity. We prove that every λ=∞ eigenvalue turns into a resonance or an eigenvalue for finite λ.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01976039
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  • 5
    Electronic Resource
    Electronic Resource
    On the rate of asymptotic eigenvalue degeneracy (1978)
    Springer
    Communications in mathematical physics 60 (1978), S. 73-95 
    Details
    ISSN: 1432-0916
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract The gap between asymptotically degenerate eigenvalues of one-dimensional Schrödinger operators is estimated. The procedure is illustrated for two examples, one where the solutions of Schrödinger's equation are explicitly known and one where they are not. For the latter case a comparison theorem for ordinary differential equations is required. An incidental result is that a semiclassical (W-K-B) method gives a much better approximation to the logarithmic derivative of a wave-function than to the wave-function itself; explicit error-bounds for the logarithmic derivative are given.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01609474
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  • 6
    Electronic Resource
    Electronic Resource
    Double Wells (1980)
    Springer
    Communications in mathematical physics 75 (1980), S. 239-261 
    Details
    ISSN: 1432-0916
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract Schrödinger operators with interactions symmetric about a plane (double-well potentials) occur in several branches of physics, such as chemistry and quantum field theory. They commonly exhibit asymptotic eigenvalue degeneracy, i.e., pairs of eigenvalues coalesce as the potential wells get farther apart. After a sketch of the theory of double wells, it is shown that the problem of estimating the gap between two such eigenvalues is reducible to finding asymptotics of eigenfunctions. For several examples and classes of potentials the gap is estimated or bounded above and below. The general case is fullyn-dimensional.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01212711
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  • 7
    Electronic Resource
    Electronic Resource
    Estimating tunneling phenomenaSupported by U.S. NSF Grant No. 7926408. (1982)
    New York, NY : Wiley-Blackwell
    International Journal of Quantum Chemistry 21 (1982), S. 199-207 
    Details
    ISSN: 0020-7608
    Keywords: Computational Chemistry and Molecular Modeling ; Atomic, Molecular and Optical Physics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Chemistry and Pharmacology
    Notes: Several tunneling phenomena are surveyed. A unified, rigorous treatment of them can be based on a simple technique of integration by parts coupled with growth estimates of eigenfunctions, using, for example, WKB approximations. In particular, the analysis of Harrell and Simon of the resonance widths in the hydrogen Stark effect can be simplified.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/qua.560210117
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    Paper (German National Licenses)
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