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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
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