Eigenvalue asymptotic of Robin Laplace operators on two-dimensional domains with cusps

We consider Robin Laplace operators on a class of two-dimensional domains with cusps. Our main results include the formula for the asymptotic distribution of the eigenvalues of such operators. In particular, we show how the eigenvalue asymptotic depends on the geometry of the cusp and on the boundary conditions

Large time behavior of the heat kernel of two-dimensional magnetic Schroedinger operators

We study the heat semigroup generated by two-dimensional Schroedinger operators with compactly supported magnetic field. We show that if the field is radial, then the large time behavior of the associated heat kernel is determined by its total flux. We also establish some on-diagonal heat kernel estimates and discuss their applications for solutions to the heat equation. An exact formula for the heat kernel, and for its large time asymptotic, is derived in the case of the Aharonov-Bohm magnetic field

Eigenvalue bounds for two-dimensional magnetic Schroedinger operators

We prove that the number of negative eigenvalues of two-dimensional magnetic Schroedinger operators is bounded from above by the strength of the corresponding electric potential. Such estimates fail in the absence of a magnetic field. We also show how the corresponding upper bounds depend on the properties of the magnetic field and discuss their connection with Hardy-type inequalities

The Prague School

The name the “Prague school of Brentano” refers to three generations of thinkers who temporarily or permanently lived in Prague, bound together by teacher/student relationships, and who accepted the main views of Franz Brentano’s philosophy. This chapter discusses central aspects of the philosophical work done in the School