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We give a simplified account of the properties of the transfer matrix for a complex one-dimensional potential, paying special attention to the particular instance of unidirectional invisibility. In appropriate variables, invisible potentials appear as performing null rotations, which lead to the helicity-gauge symmetry of massless particles. In hyperbolic geometry, this can be interpreted, via Möbius transformations, as parallel displacements, a geometric action that has no Euclidean analogy.

The work of Bender and coworkers [

Quite recently, the prospect of realizing

In all these matters, the time-honored transfer-matrix method is particularly germane [

To remedy this flaw, we have been capitalizing on a number of geometrical concepts to gain further insights into the behavior of one-dimensional scattering [

We stress that our formulation does not offer any inherent advantage in terms of efficiency in solving practical problems; rather, it furnishes a general and unifying setting to analyze the transfer matrix for complex potentials, which, in our opinion, is more than a curiosity.

To be as self-contained as possible, we first briefly review some basic facts on the quantum scattering of a particle of mass

The problem at hand is governed by the time-independent Schrödinger equation
^{2} and ^{2}, _{±}_{x}_{→±∞}

Since

Here, ^{2} = _{±}_{±}

The problem requires to work out the exact solution of _{±}_{±}

M is the transfer matrix, which depends in a complicated way on the potential _{+}_{−} = 0)] and to a left-moving wave [(_{+}_{−} = 1)], both of unit amplitude. The result can be displayed as
^{ℓ,r}^{ℓ,r}

With this in mind, ^{ℓ}^{ℓ}

Because of the Wronskian of the solutions

We thus arrive at the important conclusion that, irrespective of the potential, the transmission coefficient is always independent of the input direction.

Taking this constraint into account, we go back to the system

A straightforward check shows that det M = +1, so M ∊ SL(2, ℂ); a result that can be drawn from a number of alternative and more elaborate arguments [

One could also relate outgoing amplitudes to the incoming ones (as they are often the magnitudes one can externally control): this is precisely the scattering matrix, which can be concisely formulated as

Finally, we stress that transfer matrices are very convenient mathematical objects. Suppose that _{1} and _{2} are potentials with finite support, vanishing outside a pair of adjacent intervals _{1} and _{2}. If M_{1} and M_{2} are the corresponding transfer matrices, the total system (with support _{1} U _{2}) is described by

This property is rather helpful: we can connect simple scatterers to create an intricate potential landscape and determine its transfer matrix by simple multiplication. This is a common instance in optics, where one routinely has to treat multilayer stacks. However, this important property does not seem to carry over into the scattering matrix in any simple way [

The scattering solutions

Indeed, the two independent solutions of

The kernels _{±}_{+} = {

Let us look at the Wronskian of the Jost functions
_{+}. A spectral singularity is a point _{+} of the continuous spectrum of the Hamiltonian

The asymptotic behavior of

Using
_{22}(_{12}(_{21}(

One could also consider the more general case that the Hamiltonian _{+}, a possibly complex discrete spectrum. The latter corresponds to the square-integrable solutions of that represent bound states. They are also zeros of _{22}(

The eigenvalues of _{+}_{11}(

As heralded in the Introduction, unidirectional invisibility has been lately predicted in

The potential ^{ℓ}(^{r}^{r}^{ℓ}(

The potential is called invisible from the left (right), if it is reflectionless from left (right) and in addition

Next, we scrutinize the role of

On the other hand, the time reversal inverts the sense of time evolution, so that

Consequently, under a combined

Let us apply this to a general complex scattering potential. The transfer matrix of the
^{(
)}, fulfils

Comparing with

When the system is invariant under this transformation [M^{(
)} = M], it must hold

This can be equivalently restated in the form
_{ℓ,r} =_{ℓ,r}^{ℓ}, R^{r}^{ℓ}^{r}

A direct consequence of

In optics, beam propagation is governed by the paraxial wave equation, which is equivalent to a Schrödinger-like equation, with the role of the potential played here by the refractive index. Therefore, a necessary condition for a complex refractive index to be

To move ahead, let us construct the Hermitian matrices
_{±}_{±}_{±};

One can verify that M acts on

The matrix _{±}_{±}.

Let us consider the set ^{μ}^{μ}^{μ}_{μ}

The congruence ^{μ}

Having set the general scenario, let us have a closer look at the transfer matrix corresponding to right invisibility (the left invisibility can be dealt with in an analogous way); namely,
^{r}^{μ}^{μ}^{μ}^{μ}

If we write _{i}_{i}

Let us take, for the time being, Re _{2} + _{1} as the differential operator

As we can appreciate, the combinations
^{2}, the flow lines of the Killing vector ^{0} − ^{3} = _{2} with a hyperboloid (^{0})^{2} − (^{1})^{2} − (^{3})^{2} = _{3}. The case _{3} = 0 has the hyperboloid degenerate to a light cone with the orbits becoming parabolas lying in corresponding null planes.

Although the relativistic hyperboloid in Minkowski space constitute by itself a model of hyperbolic geometry (understood in a broad sense, as the study of spaces with constant negative curvature), it is not the best suited to display some features.

Let us consider the customary tridimensional hyperbolic space ℍ^{3}, defined in terms of the upper half-space {(^{3}|

The geodesics are the semicircles in ℍ^{3} orthogonal to the plane

We can think of the plane ^{3} as the complex plane ℂ with the natural identification (^{3}.

Every matrix M in SL(2, ℂ) induces a natural mapping in ℂ via Möbius (or bilinear) transformations [

Observe that we can break down the action ^{3} as follows:
_{3} as an inversion in the hemisphere spanned by the circle and composing appropriate pairs of inversions gives us these formulas.

In fact, one can show that PSL(2, ℂ) preserves the metric on ℍ_{3}. Moreover every isometry of ℍ_{3} can be seen to be the extension of a conformal map of ℂ̂ to itself, since it must send hemispheres orthogonal to ℂ̂ to hemispheres orthogonal to ℂ̂, hence circles in ℂ̂ to circles in ℂ̂. Thus all orientation-preserving isometries of ℍ_{3} are given by elements of PSL(2, ℂ) acting as above.

In the classification of these isometries the notion of fixed points is of utmost importance. These points are defined by the condition ^{2} is lesser than, greater than, or equal to 4, respectively. The canonical representatives of those matrices are [_{3}, and every hyperplane in ℍ_{3} that contains the geodesic joining the two fixed points in ℂ̂ is invariant); and a parallel displacement of magnitude λ, respectively. We emphasize that this later action is the only one without Euclidean analogy Indeed, in view of

We have studied unidirectional invisibility by a complex scattering potential, which is characterized by a set of

We have shown how to cast this phenomenon in term of space-time variables, having in this way a relativistic presentation of invisibility as the set of null rotations. By resorting to elementary notions of hyperbolic geometry, we have interpreted in a natural way the action of the transfer matrix in this case as a parallel displacement.

We think that our results are yet another example of the advantages of these geometrical methods: we have devised a geometrical tool to analyze invisibility in quite a concise way that, in addition, can be closely related to other fields of physics.

We acknowledge illuminating discussions with Antonio F. Costa, José F. Carineña and José María Montesinos. Financial support from the Spanish Research Agency (Grant FIS2011-26786) is gratefully acknowledged.

The authors declare no conflict of interest.