The uniqueness of the solution of the Schrodinger equation with discontinuous coefficients

Consider the Schroeodinger equation: - Du(x) - l(x)u + s(x)u = 0, where D is the Laplacian, l(x) > 0 and s(x) is dominated by l(x). We shall extend the celebrated Kato's result on the asymptotic behavior of the solution to the case where l(x) has unbounded discontinuity. The result will be used to establish the limiting absorption principle for a class of reduced wave operators with discontinuous coefficients.Comment: 29 (twenty-nine) pages; no figures; to appear in Reviews of Mathematical Physic

Spectral properties of Schr\"{o}dinger-type operators and large-time behavior of the solutions to the corresponding wave equation

Let $L$ be a linear, closed, densely defined in a Hilbert space operator, not necessarily selfadjoint. Consider the corresponding wave equations &(1) \quad \ddot{w}+ Lw=0, \quad w(0)=0,\quad \dot{w}(0)=f, \quad \dot{w}=\frac{dw}{dt}, \quad f \in H. &(2) \quad \ddot{u}+Lu=f e^{-ikt}, \quad u(0)=0, \quad \dot{u}(0)=0, where $k>0$ is a constant. Necessary and sufficient conditions are given for the operator $L$ not to have eigenvalues in the half-plane Re$z<0$ and not to have a positive eigenvalue at a given point $k_d^2 >0$. These conditions are given in terms of the large-time behavior of the solutions to problem (1) for generic $f$. Sufficient conditions are given for the validity of a version of the limiting amplitude principle for the operator $L$. A relation between the limiting amplitude principle and the limiting absorption principle is established

On the asymptotic behaviour of solutions to the fractional porous medium equation with variable density

We are concerned with the long time behaviour of solutions to the fractional porous medium equation with a variable spatial density. We prove that if the density decays slowly at infinity, then the solution approaches the Barenblatt-type solution of a proper singular fractional problem. If, on the contrary, the density decays rapidly at infinity, we show that the minimal solution multiplied by a suitable power of the time variable converges to the minimal solution of a certain fractional sublinear elliptic equation.Comment: To appear in DCDS-

Potentials of Gaussians and approximate wavelets

We derive new formulas for harmonic, diffraction, elastic, and hydrodynamic potentials acting on anisotropic Gaussians and approximate wavelets. These formulas can be used to construct accurate cubature formulas for these potentials

The radial curvature of an end that makes eigenvalues vanish in the essential spectrum II

Under the quadratic-decay-conditions of the radial curvatures of an end, we shall derive growth estimates of solutions to the eigenvalue equation and show the absence of eigenvalues.Comment: "$\ge$" in the conditions $(*_4)$ and $(*_5)$ should be replaced by "$>$". $\gamma \ge \frac{n-1}{2}(b-a)$ in the conclusion of Theorem 1.3 should be replaced by $\gamma > \frac{n-1}{2}(b-a)$; trivial miss-calculatio

A radiation condition for uniqueness in a wave propagation problem for 2-D open waveguides

We study the uniqueness of solutions of Helmholtz equation for a problem that concerns wave propagation in waveguides. The classical radiation condition does not apply to our problem because the inhomogeneity of the index of refraction extends to infinity in one direction. Also, because of the presence of a waveguide, some waves propagate in one direction with different propagation constants and without decaying in amplitude. Our main result provides an explicit condition for uniqueness which takes into account the physically significant components, corresponding to guided and non-guided waves; this condition reduces to the classical Sommerfeld-Rellich condition in the relevant cases. Finally, we also show that our condition is satisfied by a solution, already present in literature, of the problem under consideration

Edge restenosis: impact of low dose irradiation on cell proliferation and ICAM-1 expression

BACKGROUND: Low dose irradiation (LDI) of uninjured segments is the consequence of the suggestion of many authors to extend the irradiation area in vascular brachytherapy to minimize the edge effect. Atherosclerosis is a general disease and the uninjured segment close to the intervention area is often atherosclerotic as well, consisting of neointimal smooth muscle cells (SMC) and quiescent monocytes (MC). The current study imitates this complex situation in vitro and investigates the effect of LDI on proliferation of SMC and expression of intercellular adhesion molecule-1 (ICAM-1) in MC. METHODS: Plaque tissue from advanced primary stenosing lesions of human coronary arteries (9 patients, age: 61 ± 7 years) was extracted by local or extensive thrombendarterectomy. SMC were isolated and identified by positive reaction with smooth muscle α-actin. MC were isolated from buffy coat leukocytes using the MACS cell isolation kit. For identification of MC flow-cytometry analysis of FITC-conjugated CD68 and CD14 (FACScan) was applied. SMC and MC were irradiated using megavoltage photon irradiation (CLINAC2300 C/D, VARIAN, USA) of 6 mV at a focus-surface distance of 100 cm and a dose rate of 6 Gy min(-1 )with single doses of 1 Gy, 4 Gy, and 10 Gy. The effect on proliferation of SMC was analysed at day 10, 15, and 20. Secondly, total RNA of MC was isolated 1 h, 2 h, 3 h, and 4 h after irradiation and 5 μg of RNA was used in standard Northern blot analysis with ICAM-1 cDNA-probes. RESULTS: Both inhibitory and stimulatory effects were detected after irradiation of SMC with a dose of 1 Gy. At day 10 and 15 a significant antiproliferative effect was found; at day 20 after irradiation cell proliferation was significantly stimulated. Irradiation with 4 Gy and 10 Gy caused dose dependent inhibitory effects at day 10, 15, and 20. Expression of ICAM-1 in human MC was neihter inhibited nor stimulated by LDI. CONCLUSION: Thus, the stimulatory effect of LDI on SMC proliferation at day 20 days after irradiation may be the in vitro equivalent of a beginning edge effect. Extending the irradiation area in vascular brachytherapy in vivo may therefore merely postpone and not inhibit the edge effect. The data do not indicate that expression of ICAM-1 in quiescent MC is involved in the process

Mathematical Models of Incompressible Fluids as Singular Limits of Complete Fluid Systems

A rigorous justification of several well-known mathematical models of incompressible fluid flows can be given in terms of singular limits of the scaled Navier-Stokes-Fourier system, where some of the characteristic numbers become small or large enough. We discuss the problem in the framework of global-in-time solutions for both the primitive and the target system. © 2010 Springer Basel AG

Eigenvalue problems associated with Korn's inequalities

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/46189/1/205_2004_Article_BF00251798.pd