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The journal Asymptotic Analysis fulfills a twofold function. It aims at publishing original mathematical results in the asymptotic theory of problems affected by the presence of small or large parameters on the one hand, and at giving specific indications of their possible applications to different fields of natural sciences on the other hand.
Asymptotic Analysis thus provides mathematicians with a concentrated source of newly acquired information which they may need in the analysis of asymptotic problems.
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
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
Citation: Asymptotic Analysis, vol. 111, no. 2, pp. 113-136, 2019
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