arXiv subjects
Publications and source records attributed to Jochen Bruening.
We prove a formula for the multiplicities of the index of an equivariant transversally elliptic operator on a $G$-manifold. The formula is a sum of integrals over blowups of the strata of the group action and also involves eta invariants of associated elliptic operators. Among the applications, we obtain an index formula for basic Dirac operators on Riemannian foliations, a problem that was open for many years.
In [3] the authors studied the 4-parameter family of isospectral flat 4-tori T^\pm(a,b,c,d) discovered by Conway and Sloane. With a particular method of counting nodal domains they were able to distinguish these tori (numerically) by computing the corresponding nodal sequences relative to a few explicit tuples (a,b,c,d). In this note we confirm the expectation expressed in [3] by proving analytically that their nodal count distinguishes any 4-tuple of distinct positive real numbers.
We give a variational proof of the existence of infinitely many bound states below the continuous spectrum for spin-orbit Hamiltonians (including the Rashba and Dresselhaus cases) perturbed by measure potentials thus extending the results of J.Bruening, V.Geyler, K.Pankrashkin: J. Phys. A 40 (2007) F113--F117.
We give a self-contained presentation of the theory of self-adjoint extensions using the technique of boundary triples. A description of the spectra of self-adjoint extensions in terms of the corresponding Krein maps (Weyl functions) is given. Applications include quantum graphs, point interactions, hybrid spaces, singular perturbations.
We derive explicit expressions for Green functions and some related characteristics of the Rashba and Dresselhaus Hamiltonians with a uniform magnetic field.
We give a variational proof of the existence of infinitely many bound states below the continuous spectrum for some weak perturbations of a class of spin-orbit Hamiltonians including the Rashba and Dresselhaus Hamiltonians.
For Schroedinger operators (including those with magnetic fields) with singular (locally integrable) scalar potentials on manifolds of bounded geometry, we study continuity properties of some related integral kernels: the heat kernel, the Green function, and also kernels of some other functions of the operator. In particular, we show the joint continuity of the heat kernel and the continuity of the Green function outside the diagonal. The proof makes intensive use of the Lippmann-Schwinger equation.
We provide an exhaustive spectral analysis of the two-dimensional periodic square graph lattice with a magnetic field. We show that the spectrum consists of the Dirichlet eigenvalues of the edges and of the preimage of the spectrum of a certain discrete operator under the discriminant (Lyapunov function) of a suitable Kronig-Penney Hamiltonian. In particular, between any two Dirichlet eigenvalues the spectrum is a Cantor set for an irrational flux, and is absolutely continuous and has a band structure for a rational flux. The Dirichlet eigenvalues can be isolated or embedded, subject to the choice of parameters. Conditions for both possibilities are given. We show that generically there are infinitely many gaps in the spectrum, and the Bethe-Sommerfeld conjecture fails in this case.
We investigate the behavior of the Green functions of Schroedinger operators near the diagonal. The only non-trivial cases, where the on-diagonal singularities are non-zero and do not depend on the spectral parameter, are two and three dimensions. In the case of two dimensions, we show that the singularity is independent of both the scalar and the gauge potentials. In dimension three, we obtain conditions for preserving the singularity under perturbations by non-regular potentials. Some examples illustrating dependence of the singularity on general scalar and gauge potentials are presented.
We consider the spectral problem for the two-dimensional Schrödinger operator for a charged particle in strong uniform magnetic and periodic electric fields. The related classical problem is analyzed first by means of the Krylov-Bogoljubov-Alfven and Neishtadt averaging methods. It allows us to show ``almost integrability'' of the the original two-dimensional classical Hamilton system, and to reduce it to a one-dimensional one on the phase space which is a two-dimensional torus. Using the topological methods for integrable Hamiltonian system and elementary facts from the Morse theory, we give a general classification of the classical motion. According this classification the classical motion is separated into different regimes with different topological characteristics (like rotation numbers and Maslov indices). Using these regimes, the semiclassical approximation, the Bohr-Sommerfeld rule and the correspondence principle, we give a general asymptotic description of the (band) spectrum of the original Schrödinger operator and, in particular, estimation for the number of subbands in each Landau band. From this point of view the regimes, are the classical preimages of ``spectral series'' of the Schrödinger operator. We also discuss the relationship between this spectrum and the spectrum of one-dimensional difference operators.