Sharp Spectral Asymptotics for 2-dimensional Schr\"odinger operator with a strong magnetic field. Note about forgotten generic case
I consider magnetic Schr\"odinger operator in dimension $d=2$ assuming that coefficients are smooth and magnetic field is non-degenerating. Then I extend the remainder estimate $O(\mu^{-1}h^{-1}+1)$ derived in \cite{Ivr1} for the case when $V/F$ has no stationary points to the case when it has non-degenerating stationary points. If some of them are saddles and $\mu^3h\ge 2$ then asymptotics contains correction terms of magnitude $\mu^{-1}h^{-1}|\log \mu^3 h|$.
Sharp Spectral Asymptotics for Operators with Irregular Coefficients. I. Pushing the Limits
Abstract We consider operators on compact closed manifolds, with coefficients, first derivatives of which are continuous with continuity modulus and derive semiclassical spectral asymptotics with sharp remainder estimate O(h1−d); for operators with continuity modulus we derive semiclassical spectral asymptotics with the remainder estimate o(h1−d) under standard condition to Hamiltonian flow. For operators with even less regular coefficients we establish less sharp spectral asymptotics. These asymptotics easily yield standard asymptotics with respect to spectral parameter . We will treat operators on manifolds with boundary in the next paper. Preface It is well-known that spectral asymptotics with sharp remainder estimates are derived by means of microlocal analysis. However, one can apply directly microlocal methods only to operators with smooth coefficients. To treat operators with non-smooth coefficients one can approximate them by operators with smooth coefficients and approximation scale ϵ and try to establish spectral asymptotics with remainder estimates, uniform with respect to ϵ. Using this idea, the second author was able in Ref. [3] a to prove that standard sharp spectral asymptotics (including Weyl conjecture) remain true under assumption that first derivatives of coefficients are Hölder continuous (and boundary is that smooth as well). This led to two projects: – to recover whatever possible from Book b under this regularity assumption; – to push regularity condition down, even closer to C1 , while retaining sharp spectral asymptotics. aWrittenin the Spring 1999 when he was Lady Davies Fellow at Technion, Haifa, Israel. bWe use to refer to monograph [2] this way. The second project is more ambitious and this article is the first step in this direction: for manifolds without boundary we prove spectral asymptotics with remainder estimate O(h1−d) provided first derivatives of the coefficients are continuous with continuity modulus c and even o(h1−d) under standard condition to the Hamiltonian flow provided first derivatives of the coefficients are continuous with continuity modulus . Further we recover less sharp spectral asymptotics for operators with less regular coefficients (pushing this condition down in comparison with (3) as well). cWe consider semi-classical spectral asymptotics; standard spectral asymptotics are derived as a simple application. Tight time frame and our wish not to overload a single paper with new ideas force us not to consider boundaries there, but to postpone it to the next article [4]. Definitely, we need to revise the first project as well: – to recover whatever possible from Book under this weaker regularity assumption.
Strong-electric-field eigenvalue asymptotics for the perturbed magnetic Schrödinger operator
We consider the Schrödinger operator with constant full-rank magnetic field, perturbed by an electric potential which decays at infinity, and has a constant sign. We study the asymptotic behaviour for large values of the electric-field coupling constant of the eigenvalues situated in the gaps of the essential spectrum of the unperturbed operator.