On the index of elliptic partial differential operators inR n

Summary Let A= be an elliptic differential operator inR _u, If, for |α|=l, the coefficients a_α are ? nearly constant ? and, for |α|<l, they tend to zero at infinity with a certain swiftness, it is proved that A is a Fredholm operator with index_x(A)=0 between a suitable weighted Sobolev space M contained in W^l,p (R n) and L^p(R ⁿ, (1+|x|)^lp)== . It is shown, by counterexamples, that the above result, holds only if n>l, p>n/(n−l) and that isomorphism results can be obtained, in general only if the coefficients a_α(|α|<l) are assumed to be ? sufficiently small ? also on compact sets. Then a Sturm-Liouville type problem is studied and a class of negative and falling off at infinity potentials V(x) is constructed in such a way that the Schr?dinger operator H=−Δ+V(x), in L²(R ⁿ), has a zero eigenvalue.

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Sunto Sia un operatore differenziale ellittico inR ⁿ. Se, per |α|=l, i coefficienti a_α sono ? quasi costanti ? e, per |α|<l, tendono a zero all'infinito con una certa rapidità, si dimostra che A è un operatore di Fredholm con indice_X(A)=0 tra un opportuno spazio di Sobolev con peso M contenuto in W^l,p(R ⁿ) ed L^p(R ⁿ, (1+|x|)^lp)== . Si prova, mediante controesempi, che tale risultato è valido solo se n>l, p>n/(n−l) e che teoremi di isomorfismo si possono ottenere, in generale, solo se si assume che i coefficienti a_α (|α|<l) sono ? sufficientemente piccoli ? anche su insiemi compatti. Si studia quindi un problema del tipo Sturm-Liouville e si costruisce una classe di potenziali V(x) negativi e convergenti a zero all'infinito, tali che l'operatore di Schr?dinger H=−δ+V(x) in L²(R ⁿ) abbia un autovalore nullo.

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Entrata in Redazione il 10 agosto 1977.

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Work supported by C.N.R. (G.N.A.F.A.).     

Applications of conditional power function of two-sample permutation test

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Experimental implementation of a concatenated quantum error-correcting code

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QoS guarantee in telecommunication networks: technologies and solutions

The aim of this paper is to show the effect of the presence of specific management functions within a network that offers quality of service (QoS). The objective is not privileging a particular technology but to highlight the importance to know which control functions a solution may use, which performance limits the functions have and what can be a realistic user expectation. The paper focuses on the meaning of QoS and on the applications requiring quality, then describes QoS solutions including transport technologies, QoS‐oriented technologies, parameters and management functions. In more detail, the effect on QoS provision of the following issues is investigated and discussed concerning the possibility (or not) to aggregate and differentiate traffic, the implementation of call admission control and of traffic filtering to limit flows to their committed rates. Again, the conclusions should not be considered a merit mark about technology, but only an investigation about: what users and customers should expect by the technologies using specific control functions evidencing that the real limitations are not imposed by a specific technology, whose features may be changed and extended, but by the application of control functions that can guarantee requirements' matching. Copyright © 2004 John Wiley & Sons, Ltd.     

Vibrational spectrum and assignment of 4-cyclopentene-1,3-dione

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Simulation and experimental results on the forward J–V characteristic of Al implanted 4H–SiC p–i–n diodes

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