Related papers: A Gutzwiller type trace formula for the magnetic D…
We study the resolvent kernel (Green's function) of magnetic Dirac operators on a half-plane with boundary conditions interpolating between infinite mass and zigzag cases, excluding the latter. We show that these kernels have all the…
We extend the Gutzwiller trace formula to systems of noninteracting identical particles. The standard relation for isolated orbits does not apply since the energy of each particle is separately conserved causing the periodic orbits to occur…
We study the dependence of the eta invariant $\eta_D$ on the spin structure, where $D$ is a twisted Dirac operator on a (4k+3)-dimensional spin manifold. The difference between the eta invariants for two spin structures related by a…
It is introduced and studied para-Ricci-like solitons with potential Reeb vector field on almost paracontact almost paracomplex Riemannian manifolds. The special cases of para-Einstein-like, para-Sasaki-like and having a torse-forming Reeb…
Simple unsmoothed formulas to compute the Riemann zeta function, and Dirichlet $L$-functions to a power-full modulus, are derived by elementary means (Taylor expansions and the geometric series). The formulas enable square-root of the…
Let $G$ be a compact Lie group acting on a compact complex manifold $M$. We prove a trace density formula for the $G$-Lefschetz number of a differential operator on $M$. We generalize Engeli and Felder's recent results to orbifolds.
We show that whenever a Hamiltonian diffeomorphism or a Reeb flow has a finite number of periodic orbits, the mean indices of these orbits must satisfy a resonance relation, provided that the ambient manifold meets some natural…
We study two special cases of the equivariant index defined in part I of this series. We apply this index to deformations of Spin$^c$-Dirac operators, invariant under actions by possibly noncompact groups, with possibly noncompact orbit…
In this paper we provide a means to approximate Dirac operators with magnetic fields supported on links in $\mathbb{S}^3$ (and $\mathbb{R}^3$) by Dirac operators with smooth magnetic fields. We then proceed to prove that under certain…
We deduce trace properties for modulation spaces of Gelfand-Shilov distributions. We use these properties to show that pseudo-differential operators with amplitudes in suitable modulation spaces, agree with pseudo-differential operators of…
We study the existence of multiple closed Reeb orbits on some contact manifolds by means of $S^1$-equivariant symplectic homology and the index iteration formula. It is proved that a certain class of contact manifolds which admit…
We consider linear star products on $R^d$ of Lie algebra type. First we derive the closed formula for the polydifferential representation of the corresponding Lie algebra generators. Using this representation we define the Weyl star product…
We generalize the transgression formula for the eta form of Bismut, Cheeger and Berline, Getzler, Vergne for vertical Dirac operators on a fibre bundle with odd dimensional fibres where the Dirac operators have locally at most one…
We consider the magnetic Dirac operator on a curved strip whose boundary carries the infinite mass boundary condition. When the magnetic field is large, we provide the reader with accurate estimates of the essential and discrete spectra. In…
In this note, we prove lower and upper bounds for Dirac operators of submanifolds in certain ambient manifolds in terms of conformal and extrinsic quantities.
We formulate, for any Lie group G acting isometrically on a manifold M, the general notion of a G-equivariant elliptic operator that is invertible outside of a G-cocompact subset of M. We prove a version of the Rellich lemma for this…
Let (M,g) be a compact Riemannian spin manifold. The Atiyah-Singer index theorem yields a lower bound for the dimension of the kernel of the Dirac operator. We prove that this bound can be attained by changing the Riemannian metric g on an…
In these survey lectures, we investigate the geometric and analytic properties of transverse Dirac operators. In particular, we define a transverse Dirac operator associated to a distribution that is essentially self-adjoint (Prokhorenkov-R…
We prove the mathematically rigorous (semi-)classical limit $\hbar \to 0$ of the Dirac equation with time-dependent external electromagnetic field to relativistic Vlasov equations with Lorentz force for electrons and positrons. In this…
The spectral eta-invariant of a self-adjoint elliptic differential operator on a closed manifold is rigid, provided that the parity of the order is opposite to the parity of dimension of the manifold. The paper deals with the calculation of…