Research Interests:
Research Interests:
Research Interests:
Research Interests:
The Hodge–de Rham Laplacian and <mml:math altimg="si1.gif" display="inline" overflow="scroll" xmlns:xocs="http://www.elsevier.com/xml/xocs/dtd" xmlns:xs="http://www.w3.org/2001/XMLSchema" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns="http://www.elsevier.com/xml/ja/dtd" xmlns:ja="ht...more
Research Interests:
Given a domain ${\rm\Omega}$ of a complete Riemannian manifold ${\mathcal{M}}$, define ${\mathcal{A}}$ to be the Laplacian with Neumann boundary condition on ${\rm\Omega}$. We prove that, under appropriate conditions, the corresponding... more
Given a domain ${\rm\Omega}$ of a complete Riemannian manifold ${\mathcal{M}}$, define ${\mathcal{A}}$ to be the Laplacian with Neumann boundary condition on ${\rm\Omega}$. We prove that, under appropriate conditions, the corresponding heat kernel satisfies the Gaussian upper bound $$\begin{eqnarray}h(t,x,y)\leq \frac{C}{[V_{{\rm\Omega}}(x,\sqrt{t})V_{{\rm\Omega}}(y,\sqrt{t})]^{1/2}}\biggl(1+\frac{d^{2}(x,y)}{4t}\biggr)^{{\it\delta}}e^{-d^{2}(x,y)/4t}\quad \text{for}~t>0,~x,y\in {\rm\Omega}.\end{eqnarray}$$ Here $d$ is the geodesic distance on ${\mathcal{M}}$, $V_{{\rm\Omega}}(x,r)$ is the Riemannian volume of $B(x,r)\cap {\rm\Omega}$, where $B(x,r)$ is the geodesic ball of centre $x$ and radius $r$, and ${\it\delta}$ is a constant related to the doubling property of ${\rm\Omega}$. As a consequence we obtain analyticity of the semigroup $e^{-t{\mathcal{A}}}$ on $L^{p}({\rm\Omega})$ for all $p\in [1,\infty )$ as well as a spectral multiplier result.
Research Interests:
Research Interests:
The diamagnetic inequality is established for the Schrö-dinger operator H (d) 0 in L 2 (R d), d = 2, 3, describing a particle moving in a magnetic field generated by finitely or infinitely many Aharonov-Bohm solenoids located at the... more
The diamagnetic inequality is established for the Schrö-dinger operator H (d) 0 in L 2 (R d), d = 2, 3, describing a particle moving in a magnetic field generated by finitely or infinitely many Aharonov-Bohm solenoids located at the points of a discrete set in R 2 , e.g., a lattice. This fact is used to prove the Lieb-Thirring in-equality as well as CLR-type eigenvalue estimates for the perturbed Schrödinger operator H (d) 0 − V , using new Hardy type inequalities. Large coupling constant eigenvalue asymptotic formulas for the perturbed operators are also proved.
Research Interests:
Research Interests:
Research Interests:
Research Interests:
Research Interests:
Research Interests:
Research Interests:
Research Interests:
Research Interests:
For general nonsymmetric operators A, we prove that the moment of order γ≥1 of negative real parts of its eigenvalues is bounded by the moment of order γ of negative eigenvalues of its symmetric part H=12[A+A∗]. As an application, we... more
For general nonsymmetric operators A, we prove that the moment of order γ≥1 of negative real parts of its eigenvalues is bounded by the moment of order γ of negative eigenvalues of its symmetric part H=12[A+A∗]. As an application, we obtain Lieb–Thirring estimates for non-self-adjoint Schrödinger operators. In particular, we recover recent results by Frank et al. [Lett. Math. Phys. 77, 309 (2006)]. We also discuss moment of resonances of Schrödinger self-adjoint operators.
Research Interests:
Research Interests:
Research Interests:
Research Interests:
Research Interests:
Research Interests:
Research Interests:
Research Interests:
Research Interests:
Research Interests:
Research Interests:
Research Interests:
Research Interests:
Research Interests:
We consider a non-autonomous evolutionary problem \[ \dot{u} (t)+\A(t)u(t)=f(t), \quad u(0)=u_0 \] where the operator $\A(t):V\to V^\prime$ is associated with a form $\fra(t,.,.):V\times V \to \R$ and $u_0\in V$. Our main concern is to... more
We consider a non-autonomous evolutionary problem \[ \dot{u} (t)+\A(t)u(t)=f(t), \quad u(0)=u_0 \] where the operator $\A(t):V\to V^\prime$ is associated with a form $\fra(t,.,.):V\times V \to \R$ and $u_0\in V$. Our main concern is to prove well-posedness with maximal regularity which means the following. Given a Hilbert space $H$ such that $V$ is continuously and densely embedded into $H$ and given $f\in L^2(0,T;H)$ we are interested in solutions $u \in H^1(0,T;H)\cap L^2(0,T;V)$. We do prove well-posedness in this sense whenever the form is piecewise Lipschitz-continuous and symmetric. Moreover, we show that each solution is in $C([0,T];V)$. We apply the results to non-autonomous Robin-boundary conditions and also use maximal regularity to solve a quasilinear problem.