Hitherto, a finitely thick barrier next to a well or a rigid wall has been considered the potential of simplest shape giving rise to resonances (metastable states) in one dimension x ∈ (−∞, ∞). In such a potential, there are three real turning points at any energy below the barrier. Resonances are Gamow's (time-wise) decaying states with discrete complex energies (En = En − iΓn/2). These are also spatially catastrophic states that manifest as peaks/wiggles in Wigners reflection time-delay at E = n ≈ En. Here we explore potentials with simpler shapes giving rise to resonances − twopiece rising potentials having just one-turning point. We demonstrate our point by using rising exponential profile in various ways.In quantum mechanics, the bound state refers to trapping of a particle with a quantized energy (say, E 0 ) for ever in a potential well, V (x). In this case, there are necessarily two real classical turning points x 1 , x 2 such that resonance is a metastable state in which a particle is trapped temporarily due the very shape ( Fig. 1) of a potential and then it leaks out of it. Automatic (spontaneous) emission of α particle from a nucleus [1] in radioactivity and field ionization [1] of an atom when it is subject to an intense electric field are the well known examples. So far, the minimal shape of a one-dimensional or central potential for these states is required to be a finite barrier next to a wall ( Fig. 1(a)) or a well ( Fig. 1(b)). Consequently, these potentials are such that at energies below the barrier height there are at three real turning points (roots of E = V (x)).See Figs. 1(b,c).The shape-resonances are discrete complex energy states which are determined by imposing [2] an out-going boundary condition of Gamow (1928) at the exit of the potential. If the barrier is on the left, one demands ψ(x) ∼ e −ikx for x ∼ −∞ and on the other side of the well we impose ψ(0) = 0. This prescription of Gamow yields discrete complex at E ≈ E n . To understand τ (E), let us take one resonance situation(2) * Electronic address: 1: zahmed@barc.gov.in, 2:

We utilize the coset construction to derive the effective field theory of magnon-phonon interactions in (anti)-ferromagnetic and ferrimagnetic insulating materials. The action is used to calculate the equations of motion which generalize the Landau-Lifshitz and stress equations to allow for magneto-acoustic couplings to all orders in the fields at lowest order in the derivative expansion. We also include the symmetry breaking effects due to Zeeman, and Dzyaloshinsky-Moriya interactions. This effective theory is a toolbox for the study of magneto-elastic phenomena from first principles. As an example we use this theory to calculate the leading order contribution to the magnon decay width due to its the decay into phonons.

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