Is there a way to slow down radioactive decay? (with mathematical proof)

It stated that relativistic effect(obviously) and chemical component can slow down the radio activity decay.

My question were:

1. Can you list all the methods to slow down radioactive decay?(Especially through quantum mechanics.)

2. Why by using chemical component to combine the outer shell electrons can slow down the radio active decay? (which happened in the center of nuclear) Can you provide some quantum mechanics calculations or proofs?

3. Is there a way to "actively" change the rate of radio active decay like shining a beam of laser towards the compounds?

Well, OP is asking a rather demanding question. I can't answer all of it, but I can address the observed-physics part of it.

Nuclear decay can be influenced by the atomic electron shell in the case of electron capture. At the nucleon level the reaction is $\mathrm{p^+}+\mathrm{e}^- \longrightarrow \mathrm{n^0} + \nu_\mathrm{e}$. As you might expect, the decay rate is proportional to the product of the nucleon (ie proton) density and the electron density. The nucleus is typically 1000 times smaller in diameter than the electron shell. For electrons in the K-shell (s-orbital), the density is maximal at r=0 (the locus of the nucleus), and it decreases away from the nucleus. For orbitals with higher momentum, the electron density at the nucleus is essentially zero, being proportional to $r^n$, where $n=1,2,3,...$ for p,d,f, and higher orbitals.

So, electron capture in the nucleus can be influenced by a chemical bonding (compared to the lone atom) if that chemical bonding changes the s-orbital. In practice this occurs only in the lightest elements. In the heavier elements the s-orbitals are shielded by too many layers of other electrons.

There are a few unusual cases. $^{163}_{66}\mathrm{Dy}$ is a stable atom. But when fully ionized, the bare nucleus can $\beta$-decay into a bound state where the electron appears in the 1s orbital (K-shell).

Similarly for $^{187}_{75}\mathrm{Re}$. As an atom it has a decay half life of $42\cdot 10^9$ years. When fully ionized, its half life shortens dramatically to 47 days due to bound-state $\beta^−$ decay into the K and L shells.

Finally, $^7_4Be$ decays by electron capture only. If you use a laser to excite its outer electrons, the s-orbital will be affected a little ($<1\%$) and the half life will be changed accordingly, depending on what fraction of the atoms is excited or ionized.

Sources: Beta decay, Changing decay rates

True decay rates (as opposed to electron capture) can be influenced by degeneracy. If a nucleus is surrounded by a degenerate fermion gas, then any decay that produces that fermion will be heavily suppressed if the maximum decay energy of the produced fermion is lower than the Fermi energy of the surrounding gas.

Obviously, the densities required to create degenerate fermion gases with MeV energies are extremely high and generally only found in white dwarfs or neutron stars. A notable example would be that neutron stars are made mostly of neutrons because their beta decay is blocked by a very dense gas of degenerate electrons, with Fermi energies of $>10$ MeV. Without this occuring, then neutron stars would turn into ionised hydrogen stars within half an hour.