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Asymptotics of the infimum of the spectrum of Schrödinger operators with magnetic fields. (English) Zbl 0928.35032
Let \(b\) be a 1-form on a domain \(D\) in \(\mathbb{R}^d\) given a Riemannian metric and let \(L(b)\) be a Schrödinger operator on \(D\) with magnetic field \(db\) and Dirichlet boundary conditions. Lower bounds are given for the infimum of the spectrum of \(L(\xi b)\), \(\sigma(\xi b)\), as the real parameter \(\xi\) tends to infinity. This yields an upper bound for \(| I(\xi) |\) as \(\xi\to 0\), where \(I(\xi)\) is the stochastic oscillatory integral \[ I(\xi): =E\left[\exp-i\xi \int^t_0 b\bigl(X(s,x) \bigr)\cdot dX(s,x)\mid X(t,x)= y\right], \] where \(X(s,x)\) is the absorbing barrier Brownian motion on \(D\) and \(E[\cdot\mid\cdot]\) the conditional expectation. Results are then obtained on the existence and regularity of the density of the conditional probability with respect to Lebesgue measure: \[ P\left(\int^t_0 b\bigl(X(s,x) \bigr)\cdot dX(s,x)\in d\lambda\mid X(t,x)= y\right)/d \lambda. \] The same problem is considered for the operator \(L^N(b)\) defined by Neumann boundary conditions, in the case when \(D\) is a half-space. In both Dirichlet and Neumann cases, the results are extended to suitable Riemannian manifolds. A lower bound of the spectrum for a uniform magnetic field in the Neumann problem is obtained and the transverse analyticity of the reflecting barrier Brownian motion proved. Problems in which the magnetic field \(db\) degenerates (finitely or infinitely) on submanifolds are also investigated.

MSC:
35J10 Schrödinger operator, Schrödinger equation
35P15 Estimates of eigenvalues in context of PDEs
35Q55 NLS equations (nonlinear Schrödinger equations)
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