Eigenvalue problems for one-dimensional discrete Schrödinger operators with symmetric boundary conditions (Q2784362)

Consider the interval \(I=[-a,a]\), \(a>0\), \(0<b<a\), \(v>0\), the function NEWLINE\[NEWLINEV(x)=\begin{cases} -v, &\quad x\in[-b,b]\\ 0, &\quad x\notin [-b,b]\end{cases},NEWLINE\]NEWLINE a piecewise constant function \(m\) given by NEWLINE\[NEWLINEm(x)=\begin{cases} m', &\quad x\in[-b,b]\\ m, &\quad x\notin [-b,b]\end{cases}.NEWLINE\]NEWLINE Consider the eigenvalue problem of the Schrödinger operator NEWLINE\[NEWLINE-\left(\frac{1}{m(x)} u'\right)'+V(x)u=\lambda u\tag{1}NEWLINE\]NEWLINE on \(I\), the function \(V\) denoting the quantum well potential. The authors study a discrete version of the problem (1) dividing \(I\) into equal parts of length \(h\) and assuming \(a\) and \(b\) are integer multiples of \(h\), i.e. \(a=(M+N+1)h\), \(b=Nh\). The node points of the discrete equation are \(x_i=hi\), \(-M-N-1\leq i\leq M+N+1\).NEWLINENEWLINENEWLINEUsing a standard central-differencing technique the authors obtain the discrete version of (1) consisting of six equations. The authors impose the following general boundary conditions: NEWLINE\[NEWLINEu_{N+M+1}=u_{-(N+M)}+u_{N+M},\quad u_{-(N+M+1)}=u_{N+M}+u_{-(N+M)}.NEWLINE\]NEWLINE In particular, \(\beta=0\) and \(\gamma=1\) (resp. \(\gamma=0\) and \(\beta=1\)) correspond to Neumann (resp. periodic) boundary conditions.NEWLINENEWLINENEWLINEAn eigenvector \((u_i)_i\), \(-(M+N+1)\leq i\leq M+N+1\), of the discrete version of (1) is said to be symmetric (resp. antisymmetric) if \(u_i=u_{i-1}\) (resp. \(u_i=-u_{i-1}).\) If \(\lambda\) is an eigenvalue whose corresponding eigenvector is symmetric (resp. antisymmetric) then \(\lambda\) is said to be symmetric (resp. antisymmetric).NEWLINENEWLINENEWLINEThe authors prove that each eigenvector of the discrete version of (1) is either symmetric or antisymmetric. The authors define the symmetric and antisymmetric characteristic equations of the discrete version of (1) and observe that the roots of symmetric (resp. antisymmetric) characteristic equations are symmetric (resp. antisymmetric) eigenvalues of the discrete version of (1). The results of the authors concern the number of energy states lying in the wells.