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1. CJM 2007 (vol 59 pp. 393)
Le splitting pour l'opÃ©rateur de Klein--Gordon: une approche heuristique et numÃ©rique Dans cet article on \'etudie la diff\'erence entre les deux
premi\`eres valeurs propres, le splitting, d'un op\'erateur de
Klein--Gordon semi-classique unidimensionnel, dans le cas d'un
potentiel sym\'etrique pr\'esentant un double puits. Dans le cas d'une
petite barri\`ere de potentiel, B. Helffer et B. Parisse ont obtenu
des r\'esultats analogues \`a ceux existant pour l'op\'erateur de
Schr\"odinger. Dans le cas d'une grande barri\`ere de potentiel, on
obtient ici des estimations des tranform\'ees de Fourier des fonctions
propres qui conduisent \`a une conjecture du splitting. Des calculs
num\'eriques viennent appuyer cette conjecture.
Categories:35P05, 34L16, 34E05, 47A10, 47A70 |
2. CJM 2005 (vol 57 pp. 771)
The Resolvent of Closed Extensions of Cone Differential Operators We study closed extensions $\underline A$ of
an elliptic differential operator $A$ on a manifold with conical
singularities, acting as an unbounded operator on a weighted $L_p$-space.
Under suitable conditions we show that the resolvent
$(\lambda-\underline A)^{-1}$ exists
in a sector of the complex plane and decays like $1/|\lambda|$ as
$|\lambda|\to\infty$. Moreover, we determine the structure of the resolvent
with enough precision to guarantee existence and boundedness of imaginary
powers of $\underline A$.
As an application we treat the Laplace--Beltrami operator for a metric with
straight conical degeneracy and describe domains yielding
maximal regularity for the Cauchy problem $\dot{u}-\Delta u=f$, $u(0)=0$.
Keywords:Manifolds with conical singularities, resolvent, maximal regularity Categories:35J70, 47A10, 58J40 |
3. CJM 2003 (vol 55 pp. 449)
Graph Subspaces and the Spectral Shift Function We obtain a new representation for the solution to the operator
Sylvester equation in the form of a Stieltjes operator integral.
We also formulate new sufficient conditions for the strong
solvability of the operator Riccati equation that ensures the
existence of reducing graph subspaces for block operator matrices.
Next, we extend the concept of the Lifshits-Krein spectral shift
function associated with a pair of self-adjoint operators to the
case of pairs of admissible operators that are similar to
self-adjoint operators. Based on this new concept we express the
spectral shift function arising in a perturbation problem for block
operator matrices in terms of the angular operators associated with
the corresponding perturbed and unperturbed eigenspaces.
Categories:47B44, 47A10, 47A20, 47A40 |
4. CJM 2001 (vol 53 pp. 756)
Correction to: Upper Bounds for the Resonance Counting Function of SchrÃ¶dinger Operators in Odd Dimensions |
Correction to: Upper Bounds for the Resonance Counting Function of SchrÃ¶dinger Operators in Odd Dimensions The proof of Lemma~3.4 in [F] relies on the incorrect equality $\mu_j
(AB) = \mu_j (BA)$ for singular values (for a counterexample, see [S,
p.~4]). Thus, Theorem~3.1 as stated has not been proven. However,
with minor changes, we can obtain a bound for the counting function in
terms of the growth of the Fourier transform of $|V|$.
Categories:47A10, 47A40, 81U05 |
5. CJM 2000 (vol 52 pp. 197)
Sublinearity and Other Spectral Conditions on a Semigroup Subadditivity, sublinearity, submultiplicativity, and other
conditions are considered for spectra of pairs of operators on a
Hilbert space. Sublinearity, for example, is a weakening of the
well-known property~$L$ and means $\sigma(A+\lambda B) \subseteq
\sigma(A) + \lambda \sigma(B)$ for all scalars $\lambda$. The
effect of these conditions is examined on commutativity,
reducibility, and triangularizability of multiplicative semigroups
of operators. A sample result is that sublinearity of spectra
implies simultaneous triangularizability for a semigroup of compact
operators.
Categories:47A15, 47D03, 15A30, 20A20, 47A10, 47B10 |
6. CJM 1998 (vol 50 pp. 538)
Upper bounds for the resonance counting function of SchrÃ¶dinger operators in odd dimensions The purpose of this note is to provide a simple proof of the sharp
polynomial upper bound for the resonance counting function of
a Schr\"odinger operator in odd dimensions. At the same time
we generalize the result to the class of super-exponentially
decreasing potentials.
Categories:47A10, 47A40, 81U05 |