Finite square well: bound-state conditions, simplicity and count

Adopt the postulates P1, P2, P4, P5 and the declared line model The declared one-dimensional line model: state space, Hamiltonian, bound states and generalized stationary solutions with mass parameter $m>0$ and spin label $s=0$, and assume the Axiom of Countable Choice and the Axiom of Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), The Axiom of Choice). Fix $a>0$, $U>0$ and let $V(x):=\{\begin{array}{ll}-U,& \mathrm{\mid }x\mathrm{\mid }<a,\\ 0,& \mathrm{\mid }x\mathrm{\mid }>a,\end{array}$ with arbitrary real endpoint values $V(-a)$ and $V(a)$, so that $V$ is a total bounded measurable function. It is read in joules, with $U$ in joules and $a$ in metres; let $H_{\mathrm{w}}$ be the assigned energy operator of the line model with this potential, so that by The one-dimensional free Schr"odinger operator as the weak second-derivative operator $D(H_{\mathrm{w}})=H^2(\mathbb{R};\mathbb{C})$ and $H_{\mathrm{w}}\psi =-\frac{{\mathrm{\hslash }}^2}{2m}{\psi }^{\mathrm{\prime }\mathrm{\prime }}+V\psi$; finally put $K:=\sqrt{2mU}\mathrm{/}\mathrm{\hslash }$. Then:

(a) $(H_{\mathrm{w}},\mathrm{J})$ is an assigned sharp observable; $H_{\mathrm{w}}$ is self-adjoint on $H^2(\mathbb{R};\mathbb{C})$ and bounded below.

(b) Bound-state conditions. For $E<0$ with $-U<E<0$ write $\epsilon :=-E$, $\kappa :=\sqrt{2m\epsilon }\mathrm{/}\mathrm{\hslash }>0$ and $k:=\sqrt{2m(U-\epsilon )}\mathrm{/}\mathrm{\hslash }>0$, so that $k^2+{\kappa }^2=K^2$. Then $E$ is an eigenvalue of $H_{\mathrm{w}}$ if and only if $k\tan (ka)=\kappa \text{(even case)}\text{or}-k\cot (ka)=\kappa \text{(odd case)}\mathrm{.}$ Every such eigenvalue is simple; its eigenfunction is, up to a nonzero scalar, $x\mapsto \cos (kx)$ on $(-a,a)$ continued by $c{e}^{-\kappa \mathrm{\mid }x\mathrm{\mid }}$ outside in the even case, and $x\mapsto \sin (kx)$ on $(-a,a)$ continued by $c\mathrm{sgn}(x)e^{-\kappa \mathrm{\mid }x\mathrm{\mid }}$ outside in the odd case. No $E\le -U$ and no $E\ge 0$ is an eigenvalue, so every eigenvalue of $H_{\mathrm{w}}$ lies in $(-U,0)$; the bound-state energies of the model are exactly these eigenvalues, and each bound state is stationary with energy equal to its eigenvalue.

(c) Count. The number of negative eigenvalues of $H_{\mathrm{w}}$ equals $\lceil 2aK\mathrm{/}\pi \rceil$; more precisely, the equations of (b) have exactly one solution $k$ in each interval $(\frac{n\pi }{2a},\frac{(n+1)\pi }{2a})$, $n\ge 0$, whose left endpoint satisfies $\frac{n\pi }{2a}<K$, and no solution in $(0,K)$ elsewhere, and the solutions are alternatively of even type ($n$ even) and odd type ($n$ odd). In particular $H_{\mathrm{w}}$ has at least one bound state for all $a,U>0$.

(d) Scope. The statement is conditional on P1, P2, P4, P5, the declared well data $a,U$ and the cited mathematics; the energies are read in joules when $a$ is in metres, $U$ in joules and $\mathrm{\hslash }$ in $\mathrm{J}\mathrm{s}$. No material well, impurity or measured spectrum is identified with the model, and no empirical result is asserted.

• Boundedness below. The quadratic-form bound $H_{\mathrm{w}}\ge -UI$ follows from $-U\le V\le 0$ almost everywhere and $⟨H_{\mathrm{w}}\psi ,\psi ⟩=\frac{{\mathrm{\hslash }}^2}{2m}\mathrm{\parallel }D\psi {\mathrm{\parallel }}_2^2+\int V\mathrm{\mid }\psi {\mathrm{\mid }}^2$, or directly from [F1]. Absence of eigenvalues below $-U$ alone would not exclude continuous spectrum there. The lower bound is not claimed to be attained by a spectral value.
• What is not asserted. The theorem does not compute the eigenvalues numerically beyond their defining root conditions, does not compare them with data, and does not claim that any material system realizes the square well. The example page of the pair carries exact finite checks of the root conditions and of the count.