We consider the Dirichlet Laplacian for astrip in \mathbb{R}^2 with one straight boundary and a width \mathbb{R}^2 , where \mathbb{R}^2 is a smooth function of acompact support with a length 2<i>b</i>. We show that in the criticalcase, \mathbb{R}^2 , the operator has nobound statesfor small \mathbb{R}^2 .On the otherhand, a weakly bound state existsprovided \mathbb{R}^2 . In thatcase, there are positive <i>c</i> ₁,<i>c</i> ₂ suchthat the corresponding eigenvalue satisfies \mathbb{R}^2 for all \mathbb{R}^2 sufficiently small.

We study a two-dimensional charged particle interacting with a magnetic field, in general non-homogeneous, perpendicular to the plane, a confining potential, and a point interaction. If the latter moves adiabatically along a loop, the state corresponding to an isolated eigenvalue acquires a Berry phase. We derive an expression for it and evaluate it in several examples such as a homogeneous field, a magnetic whisker, a particle confined at a ring or in quantum dots, a parabolic and a zero-range one. We also discuss the behavior of the lowest Landau level in this setting obtaining an explicit example of the WilczekZee phase for an infinitely degenerated eigenvalue.

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A method to obtain bounds for the number of solutions with regular behaviour at infinity of selfadjoint ordinary differential equations is given. In this method, which is presented for a polar problem, the sequence of nested ellipsoids in a previous paper of mine is replaced by the use of certain projections. At the same time I take the opportunity to complete the unsatisfactory bibliography of my earlier paper.

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