Spectra of magnetic Schrödinger operators

Mikhail Shubin (Northeastern University)


It is well known that the condition V(x)  as x implies that the Schrödinger operator H = -  + V(x) in Rn has a discrete spectrum (K. Friedrichs, 1934). In physical language, this means that if a classical particle cannot escape to infinity (being forced to remain in a potential well), then the corresponding quantum particle is also localized. Similar results about magnetic Schrödinger operators (in Rn or on manifolds) were obtained in my joint work with V. Kondrat'ev and will be explained in the talk. They are formulated in terms of effective potentials which are constructed from both electric and magnetic fields. The most advanced of these results use the Wiener capacity and in case of vanishing magnetic field coincide with the necessary and sufficient conditions given by A. M. Molchanov in 1953.

The spectrum of a magnetic Schrödinger operator can be discrete due to the growth of the electric potential alone, but also due to a regular growth of magnetic field because of noncommutativity of magnetic translations which can be used in an uncertainty principle type argument.  I will explain new conditions for the discreteness of spectrum which combine the influence of electric and magnetic fields by introducting effective potentials.  (These new results are joint with V. Kondrat'ev.)

Open Problems:

The magnetic Schrödinger operator in Rn or on a Riemannian manifold is given by the formula HA,V= dA*dA + V(x), where $dA = d + A is  a deformed de Rham differential, acting from functions to 1-forms, A is a 1-form,  both V and A are real-valued.  The 2-form B = dA is called a magnetic field.

Here are some open problems concerning qualitative behavior of the spectrum of this operator.

1. Establish  simple necessary conditions for the spectrum to be discrete in terms of the electric potential V and magnetic field B.

An example of a necessary condition:  if V  0 and the spectrum of HA,V is discrete, then for any fixed r > 0

as |x| . Here B(x,r) is a ball of the radius r with the center at x.

However this condition is not good because it is reasonable to expect that both V and B enter with the same power.  This motivates the following more concrete question:

1a. Can we replace |B|2 by |B| in the necessary condition above?

2. Find a necessary and sufficient condition for the spectrum of HA,V to be discrete so as to extend the result of A. M. Molchanov (1953) who established such a condition in case when V is semi-bounded below and B = 0 (on Rn).

The Molchanov condition is formulated in terms of the Wiener capacity.  So the following question naturally arises:

2a. Is there a good notion of magnetic capacity?

3. Relate a localization of classical particles moving in an electromagnetic field, with a quantum localization which means a decay of eigenfunctions of HA,V at infinity. For example, the quantum localization may mean the property of the eigenfunctions to be in L2 or, stronger,  the discreteness of spectrum of HA,V.
 

Fall 2000  meeting of the PNGS