In 2 , the following extension of the higher order Rellich inequality / AV(x) 2>7(n,a,i) / /(x) 2- (1) JRn lxl JRn lxl was proven by W. Allegretto for all / G C£ (Rn {0}). The constant 7 is calculated explicitly by the author for all n > 2, a > 0 and j 6 N, giving the value of the constant in the previously unknown case n < a + 4j. Hence proving that 7 is equal to zero if and only if n < a + 4j and n a = 0 (mod 2). In this problematic case, the author finds that the higher order Rellich inequality (1) can be forced to be non-trivial if further restrictions are placed on the function in n_1. An alternative method to restricting the functional class is to look at the Rellich type inequality / AA/(x) 2 >*(n>a,4) / l/(x) 2 (2) JRn lxl JRn lxl found by W.D. Evans and R.T. Lewis in 15 for n = 2,3,4. The magnetic Laplacian is of the form Aa = (V zA)2 where in spherical coordinates H*(*i)ei ifn = 2, with e L (0,27r) and (0) = (2r). The potential A is of Aharonov- Bohm type and the constant $ is dependant upon the distance of the magnetic flux to the integers Z. By finding the discrete spectrum of the Friedrichs extension of A a in L2(Sn_1), the author is able to extend the Rellich type inequality (2) to all n > 2 and a > 0. Consequently, the higher order Rellich type inequality / Ai/(x) a>n(n,a,*,j)/ l/(x) 2 (4) JRn Ix x j can be constructed. The inequality (4) is shown to be non-trivial for all n < a + and n a = 0 (mod 2), the previously problematic case. The Rellich type inequality (4) enables an analysis of the spectral properties of perturbations of the magnetic operator AA to be undertaken in L2(IRn), n > 2. Furthermore, a CLR type bound for the number of negative eigenvalues of the operator AA can be found in L2(R8), a space in which there is no CLR bound for the operator A4.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:584127 |
| Date | January 2007 |
| Creators | Thomas, James Christian |
| Publisher | Cardiff University |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
| Source | http://orca.cf.ac.uk/55052/ |
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