Lieb–Thirring type inequalities for non-self-adjoint perturbations of magnetic Schrödinger operators

Abstract

Let H := H 0 + V and H ⊥ := H 0,⊥ + V be respectively perturbations of the free Schrödinger operators H 0 on L 2 R 2d+1 and H 0,⊥ on L 2 R 2d , d ≥ 1 with constant magnetic field of strength b > 0, and V is a complex relatively compact perturbation. We prove Lieb-Thirring type inequalities for the discrete spectrum of H and H ⊥ . In particular, these estimates give a priori information on the distribution of the discrete eigenvalues around the Landau levels of the operator, and describe how fast sequences of eigenvalues converge.

Notes on the Cauchy problem for the self-adjoint and non-self-adjoint Schrödinger equations with polynomially growing potentials

“…As for the non-self-adjoint Schrödinger equations, there are many papers on the spectral analysis (cf. [4], [17]). The authors don't know the results related to our results.…”

Some Remarks on Upper Bounds for Weierstrass Primary Factors and Their Application in Spectral Theory

“…where R p is an explicitly known constant and a n (K) denotes the nth approximation number of K. For other appearances of Γ p in eigenvalue estimates (sometimes with a different notation), see, e.g. [4,8,12,25,22,9,11,16,32,13,24,14,15,17,23]. The results from below will allow us to compute the Γ p 's numerically (apparently, this has not been done before).…”

Very weak solutions of wave equation for Landau Hamiltonian with irregular electromagnetic field

“…We can also mention papers [19,26], where the authors investigated properties of eigenfunctions of perturbed Hamiltonians, and in [20,23,24,[30][31][32]37] asymptotics of the eigenvalues for perturbed Landau Hamiltonians were described.…”