Rupert L. Frank

activity

20102025

most citedEigenvalue bounds for Schrödinger operators with complex potentials

144 citations · 163 across the 8 of their papers we have counts for

collaborators

8 papers

math.AP2025

Openness of shape optimizers in higher-order and non-scalar problems

Rupert L. Frank

We consider the first eigenvalues of the polyharmonic, Lamé and Stokes operators with Dirichlet boundary conditions on sets of given finite measure. It is shown that a quasi-open s…

math.SP2025

Eigenvalue lower bounds through a generalized inradius

Rupert L. Frank, Ari Laptev, Durvudkhan Suragan

Lieb has shown a lower bound on the smallest Dirichlet eigenvalue of the Laplace operator in terms of a generalized inradius. We derive similar bounds for Robin eigenvalues, for ei…

math.AP2025

Explicit minimisers for anisotropic Riesz energies

Rupert L. Frank, Joan Mateu, Maria Giovanna Mora +3

In this paper we describe explicitly the energy minimisers of a class of nonlocal interaction energies where the attraction is quadratic, and the repulsion is Riesz-like and anisot…

math.SP2024

Open problem: Violation of locality for Schrödinger operators with complex potentials

Jean-Claude Cuenin, Rupert L. Frank

We explain in which sense Schrödinger operators with complex potentials appear to violate locality (or Weyl's asymptotics), and we pose three open problems related to this phenomen…

math.AP2024

Minimizing Schrödinger eigenvalues for confining potentials

Rupert L. Frank

We consider the problem of minimizing the lowest eigenvalue of the Schrödinger operator $- Δ + V$ in $L^{2} (R^{d})$ when the integral $\int e^{- t V} d x$ is given for some $t > 0$ . W…

math.SP2015★ 1 cited

Eigenvalue bounds for Schrödinger operators with complex potentials. II

Rupert L. Frank, Barry Simon

Laptev and Safronov conjectured that any non-positive eigenvalue of a Schrödinger operator $- Δ + V$ in $L^{2} (R^{ν})$ with complex potential has absolute value at most a constant…

math.AP2013★ 18 cited

Uniqueness of radial solutions for the fractional Laplacian

Rupert L. Frank, Enno Lenzmann, Luis Silvestre

We prove general uniqueness results for radial solutions of linear and nonlinear equations involving the fractional Laplacian $(- Δ)^{s}$ with $s \in (0, 1)$ for any space dimensions $…

math.SP2010★ 144 cited

Eigenvalue bounds for Schrödinger operators with complex potentials

Rupert L. Frank

We show that the absolute values of non-positive eigenvalues of Schrödinger operators with complex potentials can be bounded in terms of L_p-norms of the potential. This extends an…