There is an exact solution in terms of hypergeometric functions for the so-called Eckart potential. For some information see: http://demonstrations.wolfram.com/ScatteringByASymmetricalEckartPotential/

A more detailed derivation can be found in the book "Quantum Electrochemistry" by Bockris and Kahn, starting on p. 252. The original work is by C.  Eckart: The Penetration of a Potential Barrier by Electrons,  Phys. Rev. 35 (1930), 1303. If you don't have access I can probably get you a copy. In R.P. Bell's paper "Quantum Mechanical Effects in Reactions involving Hydrogen" , Proc. Roy. Soc. A, 148, (1935), 241, this potential is also discussed and compared to others, like the truncated parabola.

There is also a solution for a rectangular barrier, and maybe for some other geometric shapes, But these are not as interesting as the Eckart potential if you want to understand the asymptotic expressions better. Which Eckart himself does. In the abstract  he states: "The approximate formula, [...] is shown to agree very well with the exact formula when the width of the barrier is great compared to the de Broglie wave-length of the incident electron, and W

I checked the references. My task is of other type: at time t=0 all wave function is inside barrier. Then it partly goes outside of barrier. I want to know probability to be inside barrier on time t. It must be sorta exponent.

Well, in principle it should be possible to construct a localized wave packet from Eckart's solutions u, but I don't think this would lead to an analytic expression for the transmission rate, or the probability of finding the particle inside the barrier. I am not aware of any attempts in this direction. There is at least some numerical work, however, from the Bohm (particle trajectory) perspective: Lopreore and Wyatt, who calculate the scattering of a Gaussian wave packet on an Eckart barrier. "Quantum Wave Packet Dynamics with Trajectories", Phys. Rev. Lett. 82, (1999), 5190. And more recently C. Chou: "Computation of transmission probabilities for thin potential barriers with transmitted quantum trajectories", Phys. Lett. A, 379, (2-015), 2174. Maybe the references in those papers can help you further.

It is "quasi-stationary" state. The total probability is not conserved because of probability flow at infinity.

Total probability is integral from 0 to  infinity. If it's = 1 in initial time t=0, then it will sonserved in the future t>0 as followed from Shrodinger equation.

Gert Van der Zwan,

As I understand, scattering and decay are different processes with different parameters. In classical textbook on quantum mechanics I find out decay formula, derived from pass metohd of complex integration, but this is approximation.

I agree, scattering and decay may be different. I did not mean to imply that they answer your question, but that their method may be adapted to do so. What I understand (and remember, it's been a while) is that in Bohm theory you take a number of initial values from the original wave packet, construct the trajectories from those, and for each trajectory calculate the property you need. In the end you sum over the initial points with the appropriate weights  pertaining to the initial wave function, a Gaussian for instance.

Actually I was  still thinking of  trying to answer your original question whether there are analytical solutions, and trying to indicate that the above mentioned approach would show the unlikelihood of such a solution. Even if you have analytical expressions for the trajectories (and I can imagine that for the Eckart barrier you could construct those as an integral over hypergeometric functions), the integration over the initial states will (very) probably not lead to an analytical result.  So, maybe I should not have mentioned this numerical approach. You will also note if you read Chou's paper that he is referring to an exact solution, but he means another numerical approach based on grid methods.  However,  if you go to numerical methods there are many other approaches.

This time dependent problem was solved explicitly for the delta barrier: R.G. Winter, Phys. Rev. 123 (4) 1503 (1961). The probability to remain in the trap was found to decay as t^{-3/2}, not exponentially, as one might have expected.

You can read our papers on the problem of quantum decay: There we found an analytical solution of the time evolution of quantum decay of an arbitrary initial state located within a finite range arbitrary potential. We showed that the decay is exponential for a long period of time, but there is a crossover to an asymptotically nonexponential power-law decay.  

1) G. García-Calderon, J. L. Mateos, and M. Moshinsky. "Resonant Spetra and the Time Evolution of the Survival and Nonescape Probabilities".

Phys. Rev. Lett. 74 (3) (1995) 337.

2) G. García-Calderon, J. L. Mateos, and M. Moshinsky. Survival and Nonescape Probabilities for Resonant and Nonresonant Decay".

Annals of Physics 249 (1996) 430.

Also, G. García-Calderon, J. L. Mateos, and M. Moshinsky. Phys Rev. Lett. 80 (1998) 4354; Phys. Rev. Lett. 90 (2003) 028902. 

Best regards. 

Thank you for references. Now I understand scale of a problem!