By K. Osterwalder

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**Extra info for Mathematical problems in theoretical physics: proceedings of the International Conference on Mathematical Physics held in Lausanne, Switzerland, August 20-25, 1979 **

**Sample text**

Cn f dnx ~V(y)~ Ix-yl In [4] the same is proved (15) for the Cwiekel- Iv(x)I n/2 n ~ 3 dimensions. (16) The reason is that (apart from the value of c ) (16) is a classical phase volume and thus independent of n a(x). An immediate consequence of these estimates was noted in [4,10]: Theorem 5 extends to matter FrShlieh The Lieb-Thirring estimate of the stability of matter in arbitrary magnetic fields, [65] and by a remark of (given in [4]) also to matter coupled to a quantized radiation field. The stability (homogeneous of any eigenvalue field) results E(0) < 0 of from the continuity H(0) for properties a(x) i = ~ B A x of the re - 31 solvent as B ÷ 0 [4]: (z - H(a)) -I = (z - H o ( a ) ) - I + (z - H(a)) -I V(z - H0(a))-i strongly (not norm) continuous as B + 0 for z around in the r e s o l v e n t E(0) norm continuous as B ÷ 0 set of H(0).

SCHECTER, A new criterion for scattering theory, Duke Math. , 444, 863 (1977). B. Phys. 5__33,151 (1977). T. Phys. 67, 85 (1979). J. KUPSH,W. Phys. 2, 147 (1966). M. REED, B. SIMON, Scattering Theory (Methods of Modern ~4athematical Physics, vol. I I I ) , Academic Press (1979). L. HORMANDER,The existence of wave-operators in scattering theory, Math. Z. 146, 69 (1976). T. Math. Soc. Japan 9, 239 (1957). B. PEARSON, A generalisation of Birman's trace theorem, J. Anal. 28, 182 (1978). D. DOLLARD, Interpretation of Kato's invariance principle in scattering theory, J.

May h a v e V = -i/r magnetic to m ~ 0 is p r o v e d field if Another , and H(0,1) peculiar and may in can produce infihas no feature be d e g e n e r a t e . [7]. By the extra bound same states [3,7]. it a l s o follows V = -i/r that diverges the ground state as B ÷ ~ , since binding the energy Coulomb poten- 33 tial is c a t a s t r o p h i c in one dimension. e. a w e a k c o u p l i n g p r o b l e m for asymptotic B -I/2) B ÷ ~ . This is used to derive the law [2,6]: B - E0(B) = [~ ~n B - £n ~n B - 0,05796 in extreme m a g n e t i c fields + O (,~n ~n B)]2 £n B (B > > i ~ 109Gauss) We end with the remark that the 2-particle reduce to the 1-particle p r o b l e m geneous magnetic Zeeman p r o b l e m does not (19).