Bethe-Sommerfeld conjecture

Motivated by the study of the electronic spectrum of a crystal in solid state quantum physics, this conjecture becomes in mathematics a problem in spectral theory for a Schrödinger operator (cf. also Schrödinger equation) on $\R^n$ with a real periodic $C^\infty$-potential $V$. More precisely, one considers the unbounded self-adjoint operator $-\Delta+V(x)$ on $L^2(\R^n)$, where $\Delta$ is the Laplace operator, $\Delta=\sum_{j=1}^n \def\pa{\partial}\pa^2/\pa x_j^2$ and $V$ satisfies $V(x+e_j)=V(x)$ for $j=1,\dots,n$. Here, $e_j$ is a basis in $\R^n$ which generates a lattice $\def\G{\Gamma}\G$ by

$$\G=\big\{\sum_{j=1}^n k_je_j : k=(k_1,\dots,k_n)\in\Z^n\big\},$$ and one denotes by $\def\cK{ {\cal K} }\cK$ a fundamental cell

$$\cK = \big\{\sum_{j=1}^n t_je_j : t_j\in [0,1]\big\}.$$ In this case the spectrum coincides with a union of bands on the real axis. This can be seen using Floquet theory, which consists of introducing a family of problems on the torus $T^n=\R^n/\G$, parametrized by $\theta\in\cK^*$, where $\cK^*$ is a fundamental cell of the dual lattice $\Gamma^*$ generated by the dual basis $(e_j^*)$ of the basis $(e_j)$.

For each $\theta$, the operator considered on $T^n$ is the operator

$$P^\theta=\sum_{j=1}^n\big(\frac{1}{i}\frac{\pa}{\pa x_j}+\theta_j\big)^2+V.$$ Its spectrum consists of a discrete increasing sequence of eigenvalues $\def\l{\lambda}\l(\def\th{\theta}\th)$ ($j\in\N$) tending to $+\infty$ and the $j$th band is then described as

$$B_j=\bigcup_{\th\in \cK^*}\l_j(\th).$$ For $n=1$, this spectrum has been analyzed in detail (e.g., see [a2]) and it is possible to show that the bands do not overlap and that generically the number of lacunae in the spectrum is infinite. The typical model is the Mathieu operator $u\mapsto -d^2u/dx^2+(\cos x)u$.

If the dimension is $>1$, it was conjectured in the 1930s by the physicists A. Sommerfeld and H. Bethe [a11], probably on the basis of what is observed for potentials of the form $V(x)=v_1(x_1)+v_2(x_2)+v_3(x_3)$, that the number of lacunae in the spectrum is always finite. This is what is called the Bethe–Sommerfeld conjecture and this has become a challenging problem in spectral theory, with relations to number theory[a9].

This conjecture has been proved in dimensions $2$ and $3$ by M.M. Skriganov [a8], [a10] (see also [a1]) in 1979, respectively 1984, and in dimension $4$ by B. Helffer and A. Mohamed [a3] in 1996.

The general case seems open (1998) although there are results under particular assumptions on the lattice [a9].

One way to prove this conjecture (see [a1], [a3]) is to analyze the density of states [a7], which is defined via Floquet theory and for a given $\mu\in\R$ by

$$N(\mu)=\frac{1}{|\cK^*|}\int_{\cK^*}\big(\sum_{\l_j(\th)<\mu}1\big)d\th,$$ with $|\cK^*|=\int_{\cK^*}d\th$, and to give, under the assumptions $\int_\cK Vdx = 0$ and $n\ge 2$, a precise asymptotic formula, as $\mu\to +\infty$, for $N(\mu)$ in the following form:

$$N(\mu)=a_\mu^{n/2}+\mathfrak{O}_\epsilon(\mu^{(n-3+\epsilon)/2}+\mathfrak{O}_\epsilon(1)$$ for all $\epsilon>0$, with $a_n=(2\pi)^{-n}|S^{n-1}|/n$ (here, $|S^{n-1}|$ denotes the volume of the sphere).

This leads to a proof of this conjecture if $2\le\nu\le4$.

Another approach consists of using a (singular) perturbation theory as presented in [a4] (which is mainly devoted to the case $n\le 3$ in the case of second-order operators). Similar questions occur for other operators with periodic coefficients, like the Schrödinger operator with magnetic field [a6], the Dirac operator and more general elliptic operators ([a4], [a5]).

References

[a1]	J. Dahlberg, E. Trubowitz, "A remark on two dimensional periodic potentials" Comment. Math. Helvetici , 57 (1982) pp. 130–134
[a2]	M.S.P. Eastham, "The spectral theory of periodic differential equations" , Scottish Acad. Press (1973)
[a3]	B. Helffer, A. Mohamed, "Asymptotic of the density of states for the Schrödinger operator with periodic electric potential" Duke Math. J. , 92 : 1 (1998) pp. 1–60
[a4]	Y.E. Karpeshina, "Perturbation theory for the Schrödinger operator with a periodic potential" , Lecture Notes Math. , 1663 , Springer (1977)
[a5]	P. Kuchment, "Floquet theory for partial differential equations" , Oper. Th. Adv. Appl. , 60 , Birkhäuser (1993)
[a6]	A. Mohamed, "Asymptotic of the density of states for Schrödinger operator with periodic electro-magnetic potential" J. Math. Phys. , 38 : 8 (1997) pp. 4023–4051
[a7]	M. Shubin, "The spectral theory and the index of almost periodic coefficients" Russian Math. Surveys , 34 : 2 (1979) pp. 109–157
[a8]	M.M. Skriganov, "Proof of the Bethe–Sommerfeld conjecture in dimension two" Soviet Math. Dokl. , 20 : 5 (1979) pp. 956–959
[a9]	M.M. Skriganov, "Geometric and arithmetic methods in the spectral theory of multidimensional periodic operators" Proc. Steklov Inst. Math. : 2 (1987)
[a10]	M.M. Skriganov, "The spectrum band structure of the three dimensional Schrödinger operator with periodic potential" Invent. Math. , 80 (1985) pp. 107–121
[a11]	A. Sommerfeld, H. Bethe, "Electronentheorie der Metalle" , Handbuch Physik , Springer (1933) (Edition: Second)