Suppose I want to solve $u_{xx}=u_t$ on the interval $[0,2\pi]$ with initial condition $u_0(x),x\in[0,2\pi]$ at $t=0$. I am interested in an informal discussion first (i.e. writing dwon a solution etc.) for the boundary conditions: $$ u(0,t)=-u(2\pi,t),\quad u_x(0,t)=u_x(2\pi,t). $$ Obviously if the minus sign wasn't present then this can be studied via Fourier series but can a similar method be applied to this 'antiperiodic' case?
Has this been studied in the literature?
Indeed, there is not enough information to reduce this to a discrete eigenvalue problem. Separating variables $u(x,t) = X(x)T(t)$ gives a linear ODE for $X$ of the form $X''/X = -k^2$ (the other one being $T'/T=-k^2$). This has solutions of the form $A\cos{(k(x-\pi)-a)}$, and we find that all we need to satisfy both boundary conditions is $\cos{a}=0$, so any function of the form $A\cos{(k(x-\pi)-\pi/2)}$ will work, for any value of $k$, real or complex (the $k=0$ solution may be found by dividing by $k$ and taking the limit as $k \to 0$, which gives $A(x-\pi)$, unsurprisingly).
Another way of looking at it is to consider a solution of the form $A\cos{kx}+B\sin{kx}$. We find that the boundary conditions reduce to one equation, $A\cos{k\pi}+B\sin{k\pi}=0$, so any $A,B,k$ satisfying this will work.