Bethe-Sommerfeld conjecture

2020 Mathematics Subject Classification: Primary: 35P20 [MSN][ZBL]

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 [Ea]) 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 [SoBe], 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 [Sk2].

This conjecture has been proved in dimensions $2$ and $3$ by M.M. Skriganov [Sk], [Sk3] (see also [DaTr]) in 1979, respectively 1984, and in dimension $4$ by B. Helffer and A. Mohamed [HeMo] in 1996.

A complete proof of the general case was only published ten years later (see below) although there are results under particular assumptions on the lattice [Sk2].

One way to prove this conjecture (see [DaTr], [HeMo]) is to analyze the density of states [Sh], 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 $2\le\nu\le4$.

A second approach consists in using a (singular) perturbation theory as presented in [Ve] and in the book [Ka] which is mainly devoted to the case $n\le 3$ in the case of second-order operators.

Finally a perturbative proof was announced by O.A. Veliev in the middle of the eighties but this is only around 2007 that two complete proofs of the Bethe-Sommerfeld conjecture along these lines were finalized with the publication in refereed journals of two papers by O.A. Veliev [Ve2] (2007) and L. Parnovski [Pa] (2008).

Similar questions occur for other operators with periodic coefficients, like the Schrödinger operator with magnetic field [Mo], the Dirac operator and more general elliptic operators ([Ka], [Ku]).

Since 2008, many generalizations and improvments have been obtained by S. Morozov, L. Parnovski, A. Sobolev, R. Schterenberg and O.A. Veliev. We refer to [PaSo], [Ve3] and references therein.

References

[DaTr]	J. Dahlberg, E. Trubowitz, "A remark on two dimensional periodic potentials", Comment. Math. Helvetici, 57 (1982), pp. 130–134 MR0672849
[Ea]	M.S.P. Eastham, "The spectral theory of periodic differential equations", Scottish Acad. Press (1973), MR3075381 Zbl 0287.34016
[HeMo]	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 MR1609321
[Ka]	Y.E. Karpeshina, "Perturbation theory for the Schrödinger operator with a periodic potential", Lecture Notes Math., 1663, Springer (1977) MR1472485 Zbl 0883.35002 Zbl 0731.35074 Zbl 0731.35071
[Ku]	P. Kuchment, "Floquet theory for partial differential equations", Oper. Th. Adv. Appl., 60, Birkhäuser (1993) MR1232660 Zbl 0789.35002
[Mo]	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 MR1459642
[Pa]	L. Parnovski, "Bethe-Sommerfeld conjecture", Ann. Henri Poincaré 9(3) (2008), pp. 457-508 MR2419769
[PaSo]	L. Parnovski, A. Sobolev, "Bethe-Sommerfeld conjecture for periodic operators with strong perturbations", Invent. Math., 181 (3) (2010), pp. 467-540 MR2660451
[Sh]	M. Shubin, "The spectral theory and the index of almost periodic coefficients" Russian Math. Surveys, 34 : 2 (1979), pp. 109–157 MR0535710 Zbl 0448.47032
[Sk]	M.M. Skriganov, "Proof of the Bethe–Sommerfeld conjecture in dimension two" Soviet Math. Dokl., 20 : 5 (1979), pp. 956–959 Zbl 0435.35028
[Sk2]	M.M. Skriganov, "Geometric and arithmetic methods in the spectral theory of multidimensional periodic operators" Proc. Steklov Inst. Math. : 2 (1987) MR0905202 Zbl 0615.47004
[Sk3]	M.M. Skriganov, "The spectrum band structure of the three dimensional Schrödinger operator with periodic potential" Invent. Math., 80 (1985), pp. 107–121 MR0784531 Zbl 0578.47003 Zbl 0567.47004
[SoBe]	A. Sommerfeld, H. Bethe, "Electronentheorie der Metalle", Handbuch Physik, Springer (1933) (Edition: Second)
[Ve]	O.A. Veliev, "Asymptotic formulae for eigenvalues of Schrödinger operators and Bethe-Sommerfeld conjecture." Functional Analysis and Appl., Vol.21 (1987), pp. 87-100
[Ve2]	O.A. Veliev, "Perturbation theory for the periodic multidimensional Schrödinger operator and the Bethe-Sommerfeld conjecture", Int. J. Contemp. Math. Sci., 2 no. 1-4, (2007), pp 19-87 MR2292470
[Ve3]	O.A. Veliev, "Multidimensional Periodic Schrödinger Operator. Perturbation theory and applications",Springer Tracts in Modern Physics, 263. Springer (2015). MR3328527