In Evans–Lewis [5] and Evans–Lewis–Saitō [6], [7], [8], [9] we have been discussing conditions for the finiteness and for the infiniteness of bound states of Schrödinger-type operators using geometric methods. Here the ideas and results obtained so far are summarized and presented in an expository manner. These bound states correspond to eigenvalues below the essential spectrum of the operator. After basic results are presented, Schrödinger operators of atomic type will be discussed to show how these basic results can be applied to various types of $N$-body Schrödinger operators.

Let $r_d(\nu)$ denote the remainder term of the stationary phase method over $R^d$. Then an estimate of $\nu^{d/2+1}r_d(\nu)$, as $d \to \infty$, is given under certain assumptions, which are tolerable for application to Feynman path integrals.

In this survey, we want to analyze the effect of the presence of a magnetic potential on the spectrum of the Schrödinger operator with magnetic field. We consider three connected problems:

• study of the bottom of the spectrum

• study of the bottom of the essential spectrum

• study of the decay of the eigenfunctions.

We consider the following nonlinear elliptic equations with real parameter $\lambda$: \[ \tag{$P_\lambda$} \Delta u+f(u,\lambda)=0, \quad u>0 \text{ in } \Omega; \quad u=0 \quad \text{on } \partial\Omega, \] where $\Omega$ is a smooth bounded domain in $R^n$ ($n \ge 2$) and $f \ge 0$ satisfies an inequality: \[ f(u,\lambda) \le c_1 + c_2 u^p \quad (c_1,c_2 > 0, p > 1\text{ constants}). \]

We suppose the existence of a family of solutions $\{(u_s, \lambda_s)\}_{0<s\le1}$ of $(P_\lambda)$ with the following properties: $(u_s, \lambda_s) \in C (\bar{\Omega}) \times R$ is continuous in $s$, $\lambda_s$ is bounded, and $\max u_s \to \infty$ ($s \downarrow 0$).

We investigate the asymptotic behavior of solutions near blowing-up points.

Sufficient conditions are given for the boundedness of $f(H)$, $H = -\triangle + V$, in $L^p({\mathbf{R}}^d)$, $1 \le p \le \infty$. Optimal results with respect to the decay of $f$ are obtained for $L^p$-boundedness of $e^{-itH} f(H)$ and the nearly-optimal norm-estimate $\|e^{-itH}f(H)\|_{\mathcal{B}(L^p)} \le C(1 + |t|)^\gamma$, $t \in \mathbf{R}$, $\gamma > d|1/2 - 1/p|$ is proved. Results are also obtained on the mapping properties of $e^{-itH}$ between certain Besov spaces.

We prove a new backward estimate and a new strong unique continuation property for solutions $u \in \mathcal{C} = C^o ((0, T); H^2 ({\mathbf{R}}^n; e^{-\alpha|x|^2} dx)) \cap C^1 ((0, T); L^2 ({\mathbf{R}}^n; e^{-\alpha|x|^2} dx))$ of parabolic differential equations $\frac{\partial u}{\partial t} = \Delta u + V(x, t)u$ under certain conditions on $V$, where $\alpha > 0$ is a fixed number.

The asymptotics of the ground state energy of large atoms as $Z \to \infty$ is given exactly by Thomas–Fermi theory. The introduction of a large magnetic field, $B$, changes the situation. If we set $B = cZ^p$ then, as $Z \to \infty$, there are 5 regions: $p < 4/3$, $p = 4/3$, $4/3 < p < 3$, $p = 3$, $p > 3$. The first three are described exactly by a modified TF theory. The fifth is describable exactly by a one-dimensional Hartree like theory. The fourth is describable exactly by a novel density matrix theory. A surprising conclusion is that although the magnetic field has a profound effect on the atomic energy in regions 2, 3, 4 and 5, the atom remains spherical (to leading order) in regions 2 and 3.