$\delta'(x)$ is a scale invariant barrier, where the S-matrix and phase shifts do not depend of the momentum.

A recent divulgative article is Point interactions: boundary conditions or potentials with the Dirac delta function (De Vincenzo - Sánchez) Canadian Journal of Physics 10/2010; 88(11):809-815. DOI: 10.1139/P10-060 Another interesting reference can be http://arxiv.org/abs/quant-ph/0406158 where it is argued that the parametrisation has some gauge freedom.

But if you want an interpretation, so I argue to look at it as some scale-invariant object. Already the point of being supported in a single point implies some amusing property under scaling, as it must be mapped towards another interaction having support in a point, so you can guess all the families will do for nice fixed points and renormalisation lines in the space of potentials with compact support. Besides, the $\delta'$ potential -alone- can be argued to have dimensions of inverse length squared, the same that the kinetic term, and so some invariance of the whole hamiltonian under scaling $x \to \lambda x$ can be expected.

Indeed if you apply the formulae of the first reference to $V(x) = g_2 \ \delta'(x)$ you get conditions $ u(0^+) =\mu \ u(0^-)$ and $ \mu \ u'(0^+) = u'(0^-)$ that allow to solve for the $S$-matrix, or if you prefer the Transmision and Reflection coefficient. Now for instance for the left wave we will have in $0^-$ the sum of incident and reflected: $$u_k(0^-)= e^{ikx}+ e^{-ikx} R^l = (1 + R^l) $$ and its derivative $$u'_k(0^-)= ik (e^{ikx} - e^{-ikx} R^l) = ik (1 - R^l)$$ and similarly in $0^+$ the transmited wave $$ u_k(0^+)= e^{ikt} T^l = T^l, \ u'_k(0^+)= ik e^{ikt} T^l = ik T^l$$

so you see the magic of this particular boundary condition: the $ik$ factors can cancel and the transmission and reflection coefficients do not depend of $k$

$$ \mu T^l = (1-R^l), T^l = \mu (1+R^l)$$

A modern reference relating delta derivatives to scattering is http://iopscience.iop.org/0305-4470/36/27/311 "On the existence of resonances in the transmission probability for interactions arising from derivatives of Dirac's delta function"

$\delta'(x)$ is a scale invariant barrier, where the S-matrix and phase shifts do not depend of the momentum.

A recent divulgative article is Point interactions: boundary conditions or potentials with the Dirac delta function (De Vincenzo - Sánchez) Canadian Journal of Physics 10/2010; 88(11):809-815. DOI: 10.1139/P10-060 Another interesting reference can be http://arxiv.org/abs/quant-ph/0406158 where it is argued that the parametrisation has some gauge freedom.

But if you want an interpretation, so I argue to look at it as some scale-invariant object. Already the point of being supported in a single point implies some amusing property under scaling, as it must be mapped towards another interaction having support in a point, so you can guess all the families will do for nice fixed points and renormalisation lines in the space of potentials with compact support. Besides, the $\delta'$ potential -alone- can be argued to have dimensions of inverse length squared, the same that the kinetic term, and so some invariance of the whole hamiltonian under scaling $x \to \lambda x$ can be expected.

Indeed if you apply the formulae of the first reference to $V(x) = g_2 \ \delta'(x)$ you get conditions

$$ u(0^+) =\mu \ u(0^-), \ \ \mu\, u'(0^+) = u'(0^-)$$

that allow to solve for the $S$-matrix, or if you prefer the Transmision and Reflection coefficient. Now for instance for the left wave we will have in $0^-$ the sum of incident and reflected: $$u_k(0^-)= e^{ikx}+ e^{-ikx} R^l = (1 + R^l) $$ and its derivative $$u'_k(0^-)= ik (e^{ikx} - e^{-ikx} R^l) = ik (1 - R^l)$$ and similarly in $0^+$ the transmited wave $$ u_k(0^+)= e^{ikt} T^l = T^l, \ u'_k(0^+)= ik e^{ikt} T^l = ik T^l$$

so you see the magic of this particular boundary condition: the $ik$ factors can cancel and the transmission and reflection coefficients do not depend of $k$

$$ \mu T^l = (1-R^l), T^l = \mu (1+R^l)$$

A modern reference relating delta derivatives to scattering is http://iopscience.iop.org/0305-4470/36/27/311 "On the existence of resonances in the transmission probability for interactions arising from derivatives of Dirac's delta function"

tl;dr: $\delta'(x)$ is a scale invariant barrier, where the S-matrix and phase shifts do not depend of the momentum.

A recent divulgative article is Point interactions: boundary conditions or potentials with the Dirac delta function (De Vincenzo - Sánchez) Canadian Journal of Physics 10/2010; 88(11):809-815. DOI: 10.1139/P10-060 Another interesting reference can be http://arxiv.org/abs/quant-ph/0406158 where it is argued that the parametrisation has some gauge freedom.

But if you want an interpretation, so I argue to look at it as some scale-invariant object. Already the point of being supported in a single point implies some amusing property under scaling, as it must be mapped towards another interaction having support in a point, so you can guess all the families will do for nice fixed points and renormalisation lines in the space of potentials with compact support. Besides, the $\delta'$ potential -alone- can be argued to have dimensions of inverse length squared, the same that the kinetic term, and so some invariance of the whole hamiltonian under scaling $x \to \lambda x$ can be expected.

Indeed if you apply the formulae of the first reference to $V(x) = g_2 \ \delta'(x)$ you get conditions $ u(0^+) =\mu \ u(0^-)$ and $ \mu \ u'(0^+) = u'(0^-)$ that allow to solve for the $S$-matrix, or if you prefer the Transmision and Reflection coefficient. Now for instance for the left wave we will have in $0^-$ the sum of incident and reflected: $$u_k(0^-)= e^{ikx}+ e^{-ikx} R^l = (1 + R^l) $$ and its derivative $$u'_k(0^-)= ik (e^{ikx} - e^{-ikx} R^l) = ik (1 - R^l)$$ and similarly in $0^+$ the transmited wave $$ u_k(0^+)= e^{ikt} T^l = T^l, \ u'_k(0^+)= ik e^{ikt} T^l = ik T^l$$

so you see the magic of this particular boundary condition: the $ik$ factors can cancel and the transmission and reflection coefficients do not depend of $k$

$$ \mu T^l = (1-R^l), T^l = \mu (1+R^l)$$

A modern reference relating delta derivatives to scattering is http://iopscience.iop.org/0305-4470/36/27/311 "On the existence of resonances in the transmission probability for interactions arising from derivatives of Dirac's delta function"

$\delta'(x)$ is a scale invariant barrier, where the S-matrix and phase shifts do not depend of the momentum.

A recent divulgative article is Point interactions: boundary conditions or potentials with the Dirac delta function (De Vincenzo - Sánchez) Canadian Journal of Physics 10/2010; 88(11):809-815. DOI: 10.1139/P10-060 Another interesting reference can be http://arxiv.org/abs/quant-ph/0406158 where it is argued that the parametrisation has some gauge freedom.

But if you want an interpretation, so I argue to look at it as some scale-invariant object. Already the point of being supported in a single point implies some amusing property under scaling, as it must be mapped towards another interaction having support in a point, so you can guess all the families will do for nice fixed points and renormalisation lines in the space of potentials with compact support. Besides, the $\delta'$ potential -alone- can be argued to have dimensions of inverse length squared, the same that the kinetic term, and so some invariance of the whole hamiltonian under scaling $x \to \lambda x$ can be expected.

Indeed if you apply the formulae of the first reference to $V(x) = g_2 \ \delta'(x)$ you get conditions $ u(0^+) =\mu \ u(0^-)$ and $ \mu \ u'(0^+) = u'(0^-)$ that allow to solve for the $S$-matrix, or if you prefer the Transmision and Reflection coefficient. Now for instance for the left wave we will have in $0^-$ the sum of incident and reflected: $$u_k(0^-)= e^{ikx}+ e^{-ikx} R^l = (1 + R^l) $$ and its derivative $$u'_k(0^-)= ik (e^{ikx} - e^{-ikx} R^l) = ik (1 - R^l)$$ and similarly in $0^+$ the transmited wave $$ u_k(0^+)= e^{ikt} T^l = T^l, \ u'_k(0^+)= ik e^{ikt} T^l = ik T^l$$

so you see the magic of this particular boundary condition: the $ik$ factors can cancel and the transmission and reflection coefficients do not depend of $k$

$$ \mu T^l = (1-R^l), T^l = \mu (1+R^l)$$

A modern reference relating delta derivatives to scattering is http://iopscience.iop.org/0305-4470/36/27/311 "On the existence of resonances in the transmission probability for interactions arising from derivatives of Dirac's delta function"