Consider the time-independent Schrödinger equation in one dimension for a particle of mass $m$ with potential $V(x)$.

(a) Show that if the potential is an even function then any non-degenerate stationary state has definite parity.

(b) A particle of mass $m$ is subject to the potential $V(x)$ given by

$$V(x)=-\lambda(\delta(x-a)+\delta(x+a))$$

where $\lambda$ and $a$ are real positive constants and $\delta(x)$ is the Dirac delta function.

Derive the conditions satisfied by the wavefunction $\psi(x)$ around the points $x=\pm a$.

Show (using a graphical method or otherwise) that there is a bound state of even parity for any $\lambda>0$, and that there is an odd parity bound state only if $\lambda>\hbar^{2} /(2 m a)$. [Hint: You may assume without proof that the functions $x \tanh x$ and $x \operatorname{coth} x$ are monotonically increasing for $x>0$.]