n$. In particular, the number of bound states can exceed any fixed integer for $N$ large enough.

By the minimax principle it is sufficient to check the claim for $H_2(\beta)$. We choose the coordinate system in such a way that the two “arms” correspond to $\arg\theta = \pm\beta/2$ and employ trial functions of the form $\Phi(x,y) = f(x)g(y)$ supported in the strip $L\le x \le 2L$, with $f\in C^2$ satisfying $f(L)=f(2L)=0$, and $$g(y) = \left\{ \begin{array}{cll} 1 & \quad \dots \quad &
|y|\le 2d \\ {\mathrm{e}}^{-\alpha(|y|-2d)} & \quad \dots \quad & |y|\ge 2d
\end{array} \right.$$ with $d:= L \tan(\beta/2)$. Evaluating the quadratic form of $H_2(\beta)$ and using minimax principle again we get the result, see [@EN1] for details.

\[r:sharp\] The above variational estimate also shows that the bound state number for a sharply broken line is roughly proportional to the inverse angle, $$n \gtrsim {3^{3/2}\over 8\pi \sqrt{5}}\: \beta^{-1}$$ as $\beta\to 0$. This can be regarded as an expected result, since the number is given by the length of the effective potential well which exists in the region where the two lines are so close that they roughly double the depth of the transverse well.

Let us see how the BS 