Discreteness of Spectrum for the Magnetic Schrödinger Operators

Abstract

We consider a magnetic Schrödinger operator H in R n or on a Riemannian manifold M of bounded geometry. Sufficient conditions for the spectrum of H to be discrete are given in terms of behavior at infinity for some effective potentials V ef f which are expressed through electric and magnetic fields. These conditions can be formulated in the form V ef f (x) → +∞ as x → ∞. They generalize the classical result by K.Friedrichs (1934), and include earlier results of J. Avron, I. Herbst and B. Simon (1978), A. Dufresnoy (1983) and A. which were obtained in the absence of an electric field. More precise sufficient conditions can be formulated in terms of the Wiener capacity and extend earlier work by A.M. Molchanov (1953) and V. Kondrat'ev and M. Shubin (1999) who considered the case of the operator without a magnetic field. These conditions become necessary and sufficient in case there is no magnetic field and the electric potential is semi-bounded below.

Discreteness of Spectrum and Strict Positivity Criteria for Magnetic Schrödinger Operators

“…A simple argument given in [1] (see also Corollary 1.4 in [20]) shows that if H 0,V has a discrete spectrum, then the same is true for H a,V whatever the vector potential a. Therefore the condition (M c ) together with V ≥ 0 is sufficient for the discreteness of spectrum of H a,V .…”

“…The results of [1,6,14], were improved in [20]. In particular, some sufficient conditions for the spectrum of H a,V to be discrete were given.…”

“…However no necessary and sufficient conditions of the discreteness of the spectrum with both fields present were provided in [20]. Here we will give such conditions which actually separate the influence of the electric and magnetic fields.…”

“…Other, more effective sufficient conditions (which do not include µ 0 ) and related results (in particular, asymptotics of eigenvalues under appropriate conditions) can be found in [4,6,8,11,12,13,14,15,20,23,29,32,34].…”

“…So in fact the spectrum depends not on the magnetic potential a itself but on the magnetic field B = da. Remark 1.10 Theorem 1.2 holds on every manifold of bounded geometry, with cubes replaced by balls in the formulation (see [19] and Section 6 in [20] for necessary adjustments which should be done to treat the more general case compared with the case of operators on R n ). However it is not at all clear how to extend Theorem 1.8 to this case.…”