18 papers
A counterexample to the Kato conjecture for positive commutators
Rupert L. Frank, Paata Ivanisvili
We disprove the conjectural converse to Kato's positivity criterion for commutators of functions of the canonical position and momentum operators and by showing that the op…
Counterexamples to the Landis conjecture in dimensions three and higher
Rupert L. Frank, Paata Ivanisvili
In every dimension three and higher we construct a nonzero function that satisfies the stationary Schrödinger equation with bounded, real-valued potential and decays like $\exp(-c…
The Winding Number at the Critical Hölder Exponent 1/3: Failure of Universal Fourier Summation
Rupert L. Frank, Paata Ivanisvili
The degree (or winding number) of a sufficiently regular map is given in terms of its Fourier coefficients by $$ \operatorname{deg} f = \sum_{n\in\…
Pleijel's theorem for a class of degenerate elliptic operators
Rupert L. Frank, Bernard Helffer
We prove an asymptotic upper bound on the number of nodal domains of eigenfunctions of a class of degenerate elliptic operators. Our proof yields the same constant as in Pleijel's…
The distance between homotopy classes of Sobolev maps on spheres
Rupert L. Frank, Paata Ivanisvili
We consider self-maps of a sphere in the critical Sobolev space with a given Brouwer degree. Our main result is that the (directed) distance between maps of different degrees is eq…
Minimizers for Coulomb gases constrained to a halfspace
Rupert L. Frank, Paata Ivanisvili, Clara Torres-Latorre
We consider a family of optimization problems, based on a mean-field description of particles interacting through Coulomb forces in a quadratic trap. In addition, the particles are…