eLecture:Quantum Graphs

Taksu Cheon

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Quantum Graphs : Physics of Quantum Singularity (4-1)

Two Examples 1

To make rather abstract arguments of the previous section more accessible, we look at three concrete examples with specific choices of parameters. First two will be of invariant torus,and the third one of equispectral sphare. We start from estabishing a useful equality

(4.1)

First, we consider the parameter space wityh fixed values of \mu = 0 and \nu = \nu_0. With the condition \mu = 0, it is easy seen from the definition of Euler angles, that the specific choice of the value of \nu_0 doesn't matter as long as it is a constant. Under these conditions, collections of U spanned by the parameters (\theta_+, \theta_-) form a torus whose any element is mapped onto itself by the action of P_3 (or \sigma_3). This "P_3 invariant torus", represented by

(4.2)

gives, as the connection condition for wave function, equations

A
(4.3)

Each equation contains only wave function at one side of thesingular point x=0, signifying the situation where left and right sides form separate systems. Namely it represents a insurmountable councing barrier at the origin, and \theta_+ and \theta_- separately give the boundary condition at the right and left side of the barrier. For bothy sides, \thata=0 represents Neumann barrier and \theta = \pi is the Dirichlet barrier.

As the second example, we set \mu = \pi/2 and \nu = 0, and consider the whole set of point interaction specified by the parameter space manifold spanned by (\theta_+, \theta_-). This is again a torus. Every point in this torus is transformed into some point in this torus itself by the action pof P_1, thus the manifold can be called "P_1 invariant torus". Since P_1 is the parity transformation in usual sense, this torus represents the collection of all possible point interaction with left-right symmetry. The matrix U is given in the form

(4.4)

and the connection condition by

A
(4.5)

Here, \phi_+ and \phi_- are symmetric and antisymmetrix element of wave function. Because of the parity invariance of the system for this case, \phi_+ and \phi_- separate and their respective properties are specified by \theta_+ and \theta_-, independently. If we set (\theta_+, \theta_) = (\theta, 0), this has no effect on antisymmetric wav functions and produces jump in the derivative of symmetric waave function. This is nothing but the delta potential. If, on the otherhand, we set (\theta_+, \theta_) = (0, \theta), this has no effect on symmetric wav functions and produces jump in the antisymmetric wave function, which is nothing but the delta potential. What we have is clearly an epsilon potential. These facts could be also confired by explicit calculation to obtain their respective transfer matrices.

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