Asymptotic Analysis - Volume 111, issue 2

Authors: Leugering, Günter | Nazarov, Sergei A. | Taskinen, Jari

Article Type: Research Article

Abstract: We develop and investigate radiation conditions at infinity for composite piezo-elastic waveguides. The approach is based on the Mandelstam radiation principle according to which the energy flux at infinity is directed away from the source and which implies constraints on the (sign of the) group velocities. On the other side, the Sommerfeld radiation condition implies limitations on the wave phase velocity and is, in fact, not applicable in the context of piezo-elastic wave guides. We analyze the passage to the limit when the piezo-electric moduli tend to zero in certain regions yielding purely elastic inclusions there. We provide a number …of examples, e.g. elastic and acoustic waveguides as well as purely elastic insulating and conducting inclusions. Show more

Keywords: Piezo-electricity, piezo-elasticity, waveguide, asymptotic decomposition, detached asymptotics, Umov–Poynting vector, Mandelstam principle, Sommerfeld principle, limited absorption principle

DOI: 10.3233/ASY-181487

Citation: Asymptotic Analysis, vol. 111, no. 2, pp. 69-111, 2019

Authors: Sambou, Diomba

Article Type: Research Article

Abstract: We consider Dirac, Pauli and Schrödinger quantum Hamiltonians with constant magnetic fields of full rank in L 2 ( R 2 d ) , d ⩾ 1 , perturbed by non-self-adjoint (matrix-valued) potentials. On the one hand, we show the existence of non-self-adjoint perturbations, generating near each point of the essential spectrum of the operators, infinitely many (complex) eigenvalues. On the other hand, we give asymptotic behaviours of the number of the (complex) eigenvalues. In particular, for compactly supported potentials, our results establish non-self-adjoint extensions of Raikov–Warzel [Rev. in Math. …Physics 14 (2002 ), 1051–1072] and Melgaard–Rozenblum [Commun. PDE. 28 (2003 ), 697–736] results. So, we show how the (complex) eigenvalues converge to the points of the essential spectrum asymptotically, i.e., up to a multiplicative explicit constant, as 1 d ! ( | ln r | ln | ln r | ) d , r ↘ 0 , in small annulus of radius r > 0 around the points of the essential spectrum. Show more

Keywords: Quantum magnetic Hamiltonians of full rank, non-self-adjoint (matrix-valued) perturbations, complex eigenvalues, Lieb–Thirring inequalities

DOI: 10.3233/ASY-181491

Citation: Asymptotic Analysis, vol. 111, no. 2, pp. 113-136, 2019