Recently a peculiar quantum behaviour of a free particle on a plane (R2) was discussed  , where free particles with 0 angular momentum ( ) appear to be attracted to the origin of the plane (the origin of the coordinate system on the plane). The authors  have considered a Dirac δ-function potential at the origin, which breaks the translational invariance of the plane and introduces a length scale in the problem. It turns out that a Hamiltonian with a Dirac δ-function potential on the plane is equivalent to a free Hamiltonian on a plane with one point (at the origin) taken out  . Then the plane becomes non simply connected and the consequences for quantum mechanics are studied in  : namely the self adjointness of operators. In the present paper we will discuss a non simply connected surface and will study the consequences of this topological property of the surface for the stability of the anti centrifugal states. We will follow our previous work  on the generalised anti centrifugal potential. In this paper  we have analysed the Laplace-Beltrami (LBO) operator on a surface where we have introduced half-geodesic coordinates  . The radial coordinate represents the geodesic and the other closes around the hole at the origin. The line element in these coordinates is given by: . The LBO separates in two parts: kinetic and potential part V for the radial coordinate given by
Here represents the Gaussian curvature of the surface and is the geodesic curvature. For angular momentum (note that the anti centrifugal potential appears only for ) the above potential takes the form:
In addition for a general surface there is a quantum geometric potential (QGP). Usually it is shown that this potential is induced on a surface, using the method of a very high sandwich potential around the surface. This potential has the form 
Here ’s are the principal curvatures of the surface, is the Planck’s constant and is the effective mass. M is the Mean curvature and K is the Gauss curvature of a two-dimensional surface embedded in three-dimensional space.
The combined potential reads:
2. The Normal n of a Surface as a Main Variable
The investigations until now ware concentrated on different geometries and the corresponding QGP. So far the link between the QGP and the topology of the underlying geometry has not been discussed.
In this paper we will use the normal to the surface as the basic variable describing the surface. Indeed the link between the normal of a surface and its Gaussian and Mean curvatures is given by the following expressions  :
Now we are ready to express in terms of the normal only.
Here we have used the following identity:
Now we will consider here a surface that represents a double layer connected by a smooth neck. The normals to this surface at the infinity on the upper layer are directed upwards and on the lower layer are directed downwards. Let us note that the catenoid posses the same topology as the above described double layer. Usually the anti centrifugal potential is considered in cylindrical coordinates because there its presence becomes immediately evident. In this case:
where r is the radial coordinate—in our case this is .
3. The Number of States
Now we would like to estimate the number of radial states (for the effective one dimensional Schrödinger equation in the direction). The momenta in the potential vary between 0 and where
In the inequality we have used the fact that and are always positive. When = constant vector field, therefore (note that for and that the potential represents a smooth function of because we assume that the surface is smooth surface and as a consequence is a smooth function of ) and for , so represents a turning point for the quasi-classical approximation in the framework of which we are estimating the number of localised states. Now we are ready to evaluate the density of states along the axis. The density of states is given by the following expression  :
where is the azymutal angle in the direction. The unit vector that represents the normal to the surface is given by . Note that usually in the context of different spin models represents the energy density and is the one dimensional non linear sigma model. Then the total number of states is:
The integration path represents the equivalence of two disconnected half circles to the left of the origin and to the right of the origin of the coordinate system. The total gives ( ).
It is clear that using topological arguments we cannot determine the exact number of states. But on the other hand the localised states cannot be destroyed by any smooth deformation of the considered surface i.e. they are topologically stable. These topological considerations give the localised anti-centrifugal states additional legitimacy.