Spectral stability of the Coulomb-Dirac Hamiltonian with anomalous magnetic moment David Otieno Ambogo Fachbereich Mathematik Universit¨ at Osnabr ¨ uck COCOA Osnabr¨ uck, Deutschland. June 10th-14th 2013 Abstract By use of asymptotic integration and Pr ¨ ufer angles, we show that the point spectrum of the Coulomb-Dirac oper- ator H 0 is the limit of the point spectrum of the Dirac oper- ator with anomalous magnetic moment H a as a → 0. For negative angular momentum quantum number κ, this holds for all coupling constants c for which H 0 has self-adjoint realisation. For positive κ, there is a region near the origin where the eigenfunctions of H a experience oscillations. Introduction By separation of variables in spherical coordinates, the Dirac operator with a Coulomb potential H 0 = −i α ·∇ + βm − Ze |x| , taking  = c =1 (1) with  α =(α 1 ,α 2 ,α 3 ) and β symmetric 4 × 4 matrices satis- fying the anticommutation relations α i α j + α j α i =2δ ij I ∀ i, j =1, 2, 3 α i β + βα i =0 α 2 = β 2 = I, is unitarily equivalent to a direct sum of one-dimensional Dirac operators on the half line H 0 = −iσ 2 d dr + mσ 3 + κ r σ 1 − Ze r r ∈ (0, ∞) (2) defined on L 2 (0, ∞) 2 with domain D = C ∞ 0 (0, ∞) 2 . The σ i are the Pauli matrices and κ ∈ Z \ 0 is the angular momen- tum quantum number. Z is the atomic number. Pauli suggested a modification of (1) to include the anoma- lous magnetic moment term and equation (2) with this term becomes H a = −iσ 2 d dr + mσ 3 +( κ r + a r 2 )σ 1 + c r r ∈ (0, ∞) (3) defined on L 2 (0, ∞) 2 with c = −Ze. The mathematical in- vestigation of (3) was initiated by Behncke [1], [2] and [3]. He has shown that H a is essentially self-adjoint for a very large class of potentials including the Coulomb potential. This is in marked contrast to the case of H 0 [5] which is essentially self-adjoint on its minimal domain if and only if c 2 <κ 2 − 1 4 .C ∞ 0 (0, ∞) 2 is a common core for both H 0 and H a at least for c 2 <κ 2 − 1 4 and hence H a srs → H 0 as a → 0, meaning the spectrum cannot expand suddenly in the limit though it can contract. The essential spectrum σ ess (H a ) of H a is similar that of H 0 and is (−∞, −m] ∪ [m, ∞). H 0 is known to have infinitely many eigenvalues in the spec- tral gap [−m, m], which accumulate at right end point m. One would therefore expect that the eigenvalues of H a will be perturbations of those of H 0 such that each eigenvalue of H 0 will be the limit of exactly one eigenvalue branch of H a as a → 0. This expectation is partly influenced by the strong resolvent convergence of H a to H 0 . Behncke [3] by decoupling equation (3) has shown stability of the point spectra at least for κ ≥ 3. Kalf and Schmidt [4] ex- tended Behncke’s results to hold for all κ by using asymp- totic analysis of Pruefer and Riccati equations equivalent to the eigenvalue equation of H a . In our study we use the asymptotic integration in conjunction with the Pruefer an- gle to obtain similar results as those of Kalf and Schmidt. Asymptotic Integration Our starting point is the eigenvalue equation (H a − λ)y =0. Written explicitly one obtains y ′ = Ay A =  −( a x 2 + κ x ) − c x + m + λ c x + m − λ a x 2 + κ x  (4) The equivalent Pruefer differential equations are θ ′ (x)= c x − λ +( a x 2 + κ x ) sin 2θ + m cos 2θ (ln ρ) ′ = m sin 2θ − ( a x 2 + κ x ) cos 2θ Using asymptotic integration, we study the behaviour of eigenfunctions near the end points 0 and ∞. Clearly, the eigenvalue equation is singular at these points. For spec- tral convergence and stability, we need the eigenfunctions to be square integrable. Stability here implies that for ǫ> 0 and for some λ a ∈ (λ 0 − ǫ, λ 0 + ǫ), the λ a −eigenfunction y (λ, x) are ǫ-approximate to those of H 0 . If one can show that for sufficiently small a, the equivalent Pruefer angles corresponding to the L 2 solutions of H a converge to those of H 0 , then we are done. To perform asymptotic integra- tion, we need distinct eigenvalues and the corresponding eigenvectors. Behaviour near infinity Here m and λ terms are dominant and the eigenvalues are given as µ a ≈±(m + m − ) 1 2  1+ cλ m + m − x + κ 2 − c 2 2m + m − x 2 + o(x −3 )  (5) The corresponding eigenvectors are  1 b +  and  b − 1  for µ a ≷ 0, respectively with b ± ≈±(m ± ) −1  (m + m − ) 1 2 + cλ x(m + m − ) 1 2 + κ x ± ( m ∓ m ± ) 1 2 c x + o(x −2 )  The diagonalising matrix T is formed by these vectors as its columns and the trans- formation y = Tz yields z ′ = (Λ − T −1 T ′ )z , with Λ= diag(µ, −µ). The correction terms can be easily ob- tained. Further diagonalisation can be carried out, however in our case the first one is sufficient. The eigenfunctions thus have the form y (λ, x) ≈ (  b + r (x))x ± mλ m + m − e ±(m + m − ) 1 2 x with r (x) ≈ o(x −2 ). The L 2 −angle is thus given by tan θ a ≈ b −1 − = −  m − m +  1 2 + a 1 x + a 2 x 2 + a m + x 2 where a 1 = m m 3 2 + m 1 2 − ( κz 1+z − c ) and λ 2 m 2 = (1+ z ) −1 with z = c 2 (n ′ +(κ 2 −c 2 ) 1 2 ) 2 , and a 2 =  m − m +  1 2 (κ 2 −c 2 ) 2m + m − + c 2 λ (m + m − ) 3 2 + cκ m + m − . For a =0, this L 2 -angle has the same value as that of H 0 . This means that θ a → θ 0 as a → 0, giving exact convergence for all κ. Behaviour near zero. Here we introduce a new variable t = x −1 , giving a new system y ′ = Ay, A =  a + κ t c t − t −2 (m + λ) − c t − t −2 (m − λ) −(a + κ t )  (6) Here m and λ terms are small and can be neglected. Thus the eigenvalues are µ a ≈± 1 t [(at + κ) 2 − c 2 ] 1 2 and the corre- sponding eigenvectors are  1 b  and  b 1  for µ a ≷ 0, respectively with b = − (µ a +a+ κ t ) c t . The above diagonalisation procedure can be done again and the eigenfunctions thus takes the form y (λ, x) ≈ (  b + r (x))e  t a µ a ds . The corresponding L 2 -angle is given by tan θ a ≈ [(at + κ) 2 − c 2 ] 1 2 + at + κ −c . (7) Here we distinguish between two cases: κ< 0 and κ> 0. Case κ< 0, a< 0 Close to the origin, the term at + κ is dominant. By assump- tion, |κ| > |c| and hence b → 0. This means that tan θ a → 0 as x → 0 and thus θ a → nπ, n =0, 1, 2, .... If a → 0, we have tan θ a → (κ 2 −c 2 ) 1 2 +κ −c which is equal to the case for H 0 . This is in line with the results obtained by Kalf and Schmidt and we have the following stability result. Proposition 1 For any ǫ> 0 and λ 0 there exists a 0 < 0 and R(ǫ) so that any normalized λ(a)− eigenfunction u of H a with λ(a) − λ 0 ≤ ǫ satisfies ||y || ≤ ǫ uniformly in [λ 0 − ǫ, λ 0 + ǫ] and a 0 <a< 0 Case κ> 0, a< 0 Here we have three regions to consider. The region where κ is dominant, the region where a term is dominant and the the region where the a term is approximately equal to κ. In the latter case, µ a changes sign and the eigenfunctions experiences oscillations. This transition takes place in the interval κ+|c| −a <t< κ−|c| −a . Thus there are values of c where µ a is imaginary. Except for these values of c, the eigen- functions are stable since in the region κ+|c| −a <t< κ−|c| −a , the functions are approximately constant. In the region where κ is dominant, a x 2 is integrable and we have a regular pertur- bation. Near the origin, where the a term is dominant, we can apply asymptotic integration to obtain the behaviour of the eigenfunctions. Conclusion In quantum mechanics, the eigenvalues correspond to the energy levels. The eigenfunctions are the states of the sys- tem at any time t. Stability therefore of the eigenvalues im- plies one has a bound on the energy levels and hence also a bound on the states. Our method, asymptotic integration, can be used in obtaining better estimates of the energy lev- els and thus can provide Physicists with an easier way of obtaining bound states. References [1] H. Behncke, The Dirac equation with an anomalous magnetic moment, Math. Z. 174 (1980), 213-225. [2] H. Behncke, Dirac equation with an anomalous mag- netic moment II, Ordinary and Partial Differential Equa- tions, lecture Notes in Math., Vol. 964, Springer, Berlin and New York, 1982, pp. 77-85. [3] H. Behncke, Spectral properties of the Dirac equation with anomalous magnetic moment, J. Math. Phys. 26 (10) (1985) 25562559 [4] H. Kalf, K.M. Schmidt, Spectral Stability of the Coulomb-Dirac Hamiltonian with Anomalous Magnetic Moment, J. Diff. Eq. 205 (2004), 408-423. [5] Weidmann J., Spectral Theory of Ordinary Differential Operators. Springer Lect. Notes 1258, Springer-Verlag, Berlin, 1987. † Address for correspondence: Fachbereich Mathematik, Universit¨ at Osnabr ¨ uck, 49069 Osnabr ¨ uck, Deutschland