Joachim Stubbe

A global attracting set for the Kuramoto-Sivashinsky equation

AbstractNew bounds are given for the L2-norm of the solution of the Kuramoto-Sivashinsky equation

BOUND STATES FOR SCHRÖDINGER HAMILTONIANS: PHASE SPACE METHODS AND APPLICATIONS

\partial _t U(x,t) = - (\partial _x^2 + \partial _x^4 )U(x,t) - U(x,t)\partial _x U(x,t)

Trace identities for commutators, with applications to the distribution of eigenvalues

, for initial data which are periodic with periodL. There is no requirement on the antisymmetry of the initial data. The result is

Generalization of the Calogero–Cohn bound on the number of bound states

\mathop {\lim \sup }\limits_{t \to \infty } \left\| {U( \cdot ,t)} \right\|_2 \leqslant const. L^{8/5}

Universal monotonicity of eigenvalue moments and sharp Lieb-Thirring inequalities

.

Order of energy levels in relativistic one-electron models

In this article, we prove and exploit a trace identity for the spectra of Schrodinger operators and similar operators. This identity leads to universal bounds on the spectra, which apply to low-lying eigenvalues, eigenvalue asymptotics, and to partition functions (traces of heat operators). In many cases they are sharp in the sense that there are specific examples for which the inequalities are saturated. Special cases corresponding to known inequalities include those of Hile and Protter and of Yang. Introduction In this article, we prove and exploit an identity for the spectra of self-adjoint operators H modeled on the Dirichlet Laplacian or, more generally, on Schrodinger operators of the form (p−A(x))2 + V (x), (1) where p = 1i ∇ is the usual momentum operator in convenient units and A(x) is the magnetic vector potential. We recover and extend several known inequalities involving sums, differences, and ratios of eigenvalues. Let λj , j = 1, . . . , denote the ordered eigenvalues of the Dirichlet Laplacian on a bounded d-dimensional domain with zero Dirichlet boundary conditions, and recall that Hile and Protter [HiPr80] proved that: d 4 ≤ 1 n n ∑ j=1 λj λn+1 − λj , (2) thereby extending an earlier inequality of Payne, Polya, and Weinberger [PaPoWe56]. In the last few years it has become clear that these and many similar relationships can be realized as special cases of abstract variational bounds involving the interplay among commutators of −∇2, a Cartesian coordinate xj , and the corresponding derivative ∂/∂xj. For this analysis and various extensions of (2) see [Ha88], [Ho90], [Ha93], [HaMi95]. While inequalities of this type have been fairly sharp for low-lying eigenvalues, they have been mostly disappointing for higher eigenvalues. Yang [Ya91], however, Received by the editors September 28, 1995. 1991 Mathematics Subject Classification. Primary 35J10, 35J25, 58G25.