# zbMATH — the first resource for mathematics

Semiclassical asymptotics of eigenvalues for Schrödinger operators with magnetic fields. (English) Zbl 0859.35081
The author studies the semi-classical asymptotics of eigenvalues for Schrödinger operators $$H(\lambda)$$ with magnetic fields either on two-dimensional compact manifolds $$M$$ or on $$\mathbb{R}^2$$. Here $$H(\lambda)=1/2\nabla^{A_*}_\lambda\nabla^A_{\lambda}+\lambda^2V+\lambda v$$, $$\lambda>0$$, with $$\nabla^A_\lambda u=du+i\lambda uA$$ $$(i=\sqrt{-1})$$, $$A$$ a $$C^\infty$$-differential form and $$v$$, $$V$$ smooth real functions. In the compact case it is assumed that $$V$$ is nonnegative and that the set of zeros of $$V$$ is a finite subset $$\{p^a\}^N_{a=1}$$ of $$M$$ and that the Hessian $$\nabla^2V$$ is nondegenerate at each zero. In the case of $$\mathbb{R}^2$$ a growth condition is assumed, namely, $$\liminf V(x)>0$$, $$|x|\to\infty$$.
Let $$k_1(a)^2$$, $$k_2(a)^2$$ be the eigenvalues of the Hessian $$(\nabla^2V)(p^a)$$ and set $$B(a)=|dA(p^a)||dA|$$. Set $$m_1=\sqrt{(k_1+k_2)^2+B^2}$$, $$m_2=\sqrt{(k_1-k_2)^2+B^2}$$ and $$m^a_i=m_i(k_1(a),k_2(a),B(a))$$, $$i=1,2$$. For the harmonic oscillator $$H(k_1,k_2,B)$$ on $$\mathbb{R}^2$$ with uniform magnetic field is unitarily equivalent to the operator $H_1={1\over 2}\sum^2_{j=1} \Biggl(i{\partial\over \partial x_j}-Q_j\Biggr)^2+ {1\over 2}(k^2_1x^2_1+ k^2_2x^2_2),$ with $$Q_1=c_2Bx_2$$, $$Q_2=-c_1Bx_1$$ and $$c_i=k_i/(k_1+k_2)$$, the ideas involved in the van Vleck-Pauli formula may be used to calculate that $\text{Tr}(\exp(-tH))=\text{Tr}(\exp(-tH_1))={1\over 2(\text{cosh}{m_1t\over 2}-\text{cosh}{m_2t\over 2})}.$ On the other hand the Malliavin calculus may be used to establish a generalization of a result of Watanabe, namely, $\lim_{t\to\infty}\text{ Tr}\Biggl(\exp\Biggl(-{tH(\lambda)\over \lambda}\Biggr)\Biggr)= \lim_{\lambda\to\infty} \sum^\infty_{j=1} \exp\Biggl(-{t\mu_n(\lambda)\over \lambda}\Biggr)=\sum_a {1\over 2(\text{cosh } m^a_1t/2-\text{cosh }m^a_2t/2)}.$ These two formulae together with the continuity theorem of the Laplace transform establish that $$\mu_n(\lambda)/\lambda$$ converges as $$\lambda\to\infty$$ to an eigenvalue of some harmonic oscillator under a uniform magnetic field. Let $V^a(x)={1\over 2}\sum^2_{j,k=1} V^a_{j,k}x_jx_k,\;V^a_{jk}={\partial^2V(p^a)\over \partial x_j\partial x_k}\text{ and } \overline A^a_k=\sum^2_{j=1} \partial_jA_k(p^a)x_j,\quad k=1,2.$ Set $\overline K^a={1\over 2} \sum^2_{j=1} \Biggl(i{\partial\over \partial x_j}-\overline A^a\Biggr)^2+V^a+v(p^a)$ and $$\{\mu_n\}^\infty_{n=1}$$, $$\mu_1\leq\mu_2\leq\cdots$$ be the eigenvalues of $$\bigoplus^N_{a=1}\overline K^a$$. Suppose that $$\partial_kB(p^a)=0$$, $$a=1,2,\dots,N$$, $$k=1,2$$. Then $$\lim_{\lambda\to\infty} \mu_n(\lambda)/\lambda=\mu_n$$ for every $$n\in \mathbb{N}$$.
The proof of this result is obtained directly by variational methods previously used by B. Simon [Ann. Inst. Henri Poincaré, Sect. A 38, 295-308 (1983; Zbl 0526.35027)], together with the IMS localization formula.

##### MSC:
 35P15 Estimates of eigenvalues in context of PDEs 81Q20 Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory 35J10 Schrödinger operator, Schrödinger equation
##### Keywords:
Malliavin calculus; uniform magnetic field
Full Text: