On the Number of Bound States for Weak Perturbations of Spin-orbit Hamiltonians

We give a variational proof of the existence of infinitely many bound states below the continuous spectrum for some weak perturbations of a class of spin-orbit Hamiltonians including the Rashba and Dresselhaus Hamiltonians. In the recent paper [3] Chaplik and Magarill have discovered a surprising fact: the Rashba Hamiltonian HR, HR = ( p αR(py + ipx) αR(py − ipx) p ) (αR is the Rashba constant expressing the strength of the spin-orbit coupling [2, 7]) perturbed by a short-range rotationally symmetric negative potential has an infinite number of eigenvalues below the threshold of the continuous spectrum. More precisely, for a rotationally symmetric shallow potential well V (mUR/~ ≪ 1, where m is the effective mass and U and R are the depth and radius of the well, respectively), in [3] a system of equations was derived to be satisfied by the eigenvalues, and it was shown (using the pole approximation for the calculation of some integrals) that the system has an infinite number of solutions below the continuous spectrum of HR + V . In the present note we are going to provide a strict mathematical justification for the existence of infinitely many bound states below the continuous spectrum for short-range perturbations of a much larger class of spin-orbit Hamiltonians, which includes, in particular, the above Rashba Hamiltonian as well as the Dresselhaus Hamiltonian [7], HD = ( p −αD(px + ipy) −αD(px − ipy) p ) (here αD is the Dresselhaus constant). Our technique is elementary and uses the max-min principle in the spirit of [8]. Moreover, the potential well V can be nonsymmetric, and “shallow” in our context means V ∈ L. Denote by H the Hilbert space L(R) ⊗ C of two-dimensional spinors; by F we denote the Fourier transform F : L(R) → L(R); then F2 := F ⊗ 1C2 is the Fourier transform in H. Let H0 be the self-adjoint operator in H whose Fourier transform Ĥ0 := F2H0F 2 is the multiplication by the matrix (1) Ĥ0(p) = ( p A(p) A(p) p ) , p ∈ R , where A is a continuous complex function on R, star ∗ means the complex conjugation and, as usual, p := |p|. It is obvious that H0 is self-adjoint. Clearly, the Rashba and Dresselhaus Hamiltonians have the form (1) with a linear A. In generalizing the linearity we assume (2) lim sup p→∞ |A(p)| p2 < 1