# Non real eigenvalues for elliptic equations

I am looking for an example of a pure second order uniformly elliptic operator $$L=\sum_{i,j=1}^da_{ij}(x)D_{ij}$$ in a bounded domain $$\Omega$$ (with Dirichlet boundary conditions, for example) having a non-real eigenvalue in $$L^2(\Omega)$$. Under the above assumptions, the operator $$L$$ has a discrete spectrum and the principal eigenvalue is real; however it is non-symmetric. Note that if $$a_{ij}=a$$ (in particular if $$d=1$$), then every eigenvalue is real since the measure $$d\mu=a^{-1}(x)\, dx$$ symmetrizes $$L$$. Does anybody know such an example?

• Small anecdote: I remember Wolfgang Arendt mentioning the same question a few years ago during coffee. He said that he already had discussed this with several people who know quite a deal about elliptic operators, but none of them knew the answer. Jan 12, 2020 at 23:34
• Thank you for the comment. Let me point out that the spectrum is real also for operators like $\Delta+\nabla \phi \cdot \nabla$, since it is symmetric with respect to the the measure $d\mu=e^{\phi}\, dx$. In particular the spectrum is always real in 1d and this perhaps explains why it is not easy to produce counteraxamples. Jan 13, 2020 at 19:19

Here is a construction. It elaborates from perturbation analysis of eigenvalues. However it starts from the situation of a non-simple eigenvalue.

So, let me start with the standard self-adjoint $$L_0=-\Delta$$. I assume that the Dirichlet problem admits an eigenvalue $$\lambda_0$$ of multiplicity $$2$$ exactly. I denote $$(u,v)$$ an orthonormal basis of the eigenspace. A well-known example is $$\lambda_0=5$$ for the domain $$K=(0,\pi)\times(0,\pi)$$ and $$u(x)=\frac2\pi\,\sin x_1\sin2x_2,\quad v=\frac2\pi\,\sin2x_1\sin x_2.$$

Let $$M=\sum a_{ij}D_{ij}$$ be given with bounded (smooth) functions $$a_{ij}$$, and form $$L_\epsilon=L+\epsilon M$$. Perturbation analysis tells us that $$L_\epsilon$$ admits a stable plane $$\Pi_\epsilon$$, which depends smoothly upon $$|\epsilon|<\!<1$$, and $$P_0={\rm vec}(u,v)$$. In addition, the restriction of $$L_\epsilon$$ over $$\Pi_\epsilon$$ is similar to a matrix $$C_\epsilon$$ such that on the one hand $$C_0=\lambda_0I_2$$, and on the other hand $$\left.\frac{d}{d\epsilon}\right|_{\epsilon=0}C_\epsilon=\begin{pmatrix} \langle Mu,u\rangle & \langle Mu,v\rangle \\ \langle Mv,u\rangle & \langle Mv,v\rangle \end{pmatrix}=:X_0.$$ Suppose now that $$M$$ has been chosen so that $$(\dagger)\qquad ( \langle Mu,u\rangle - \langle Mv,v\rangle )^2+4 \langle Mu,v\rangle \langle Mv,u\rangle <0.$$ Then its eigenvalues are complex conjugated. Therefore, for $$\epsilon\ne0$$ small enough, $$C_\epsilon$$ will have non-real eigenvalues. Since these are eigenvalues of the Dirichlet problem for $$L_\epsilon$$, this provides an example.

There remains to find coefficients $$a_{ij}$$ satisfying ($$\dagger$$). Let me just choose in my example (on the square $$K$$) $$M=aD_{12}$$. The functions $$uD_{12}u-vD_{12}v$$, $$uD_{12}v$$ and $$vD_{12}u$$ are linearly independent. Thus there exists a function $$a$$ such that $$\int_Ka(uD_{12}u-vD_{12}v)\,dx=0,\qquad\int_KauD_{12}v\,dx=1,\qquad\int_KavD_{12}u\,dx=-1.$$ This ends the construction.

Edit. It is known that one-dimensional elliptic second-order operators have a real spectrum. Actually the eigenvalues are simple, therefore the construction described above cannot be implemented.

Redit. In the construction above, the eigenvalues depend smoothly upon $$\epsilon$$. There is no branching phenomenon. This happens because the unperturbed operator $$L_0$$ is self-adjoint, thus diagonalisable.

• Thank you very much. Nov 9, 2020 at 18:25