In [3], Chen et al. considered the operator for high dimension n.
If we consider the operator (4) for (xin{mathbb{Z}}) only mathcal{D}: u(x_{d})longmapstosum _{-infty}^{+infty }a_{k}u(x_{d}+ beta_{k}), quad x_{d}in{mathbb{Z}}, (5) then its symbol can be defined by the discrete Fourier transform [7, 8] sigma_{d}(xi =sum_{-infty}^{+infty}a_{k}e^{ibeta_{k}xi =sum_{-infty[-pi,pi].
Consider the operator such that for all (3.5).
Consider the operator (L_{a}), then for (ngeq N) the component-function (y_{1} x,lambda_{n,a})) of the eigenfunction (y x,lambda_{n,a})) and the component-function (y_{1} x,lambda_{-n,a})) of the eigenfunction (y x,lambda_{-n,a})) for the operator (L_{a}) both have precisely n zeros in the interval ((a,1)), corresponding to the eigenvalues (lambda_{n,a}>0) and (lambda _{-n,a}<0), respectively.
To calculate (kappa _2), we can consider the operator with constant potential W, and for this operator, we calculate (-frac{1}{2}int |x-y|^{-1}{text |x-y|^{-1}{text |x-y|^{-1}{texty,nu )bigr ), dy) obtaining (-mathsf {tr}}bigl+nu )^{2} h^{-4}), the^dagug in (W=W(x)) and integrate over x.
For every we consider the operator.
Consider the operator (5.51).
Consider the operator equation.
Proof Consider the operator (4).
Similarly, consider the operator, (3.25).
