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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.

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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