Post Closed as "off-topic" by Daniel Lichtblau, MarcoB, Alex Trounev, José Antonio Díaz Navas, bbgodfrey
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$$[(a+k_i)^2+(b+k_j)^2]X_{i,j}-\sum_{m,n}V_{m,n}X_{i-m,j-n}=\mu X_{i,j}$$. where $$-N<=i,j<=N$$$$-N\le i,j\le N$$ Here we can set $$N=10,a =1, b=1$$ and $$V_{m,n}$$ is the matrix element of $$V$$. Once I write the coefficient matrix of $$X$$, I can solve the eigenvalue $$\mu$$.

$$[(a+k_i)^2+(b+k_j)^2]X_{i,j}-\sum_{m,n}V_{m,n}X_{i-m,j-n}=\mu X_{i,j}$$. where $$-N<=i,j<=N$$ Here we can set $$N=10,a =1, b=1$$ and $$V_{m,n}$$ is the matrix element of $$V$$. Once I write the coefficient matrix of $$X$$, I can solve the eigenvalue $$\mu$$.

$$[(a+k_i)^2+(b+k_j)^2]X_{i,j}-\sum_{m,n}V_{m,n}X_{i-m,j-n}=\mu X_{i,j}$$. where $$-N\le i,j\le N$$ Here we can set $$N=10,a =1, b=1$$ and $$V_{m,n}$$ is the matrix element of $$V$$. Once I write the coefficient matrix of $$X$$, I can solve the eigenvalue $$\mu$$.

2 added 18 characters in body

$$[(a+k_i)^2+(b+k_j)^2]X_{i,j}-\sum_{m,n}V_{m,n}X_{i-m,j-n}=\mu X_{i,j}$$.

Here where $$-N<=i,j<=N$$ Here we can set $$N=10,a =1, b=1$$ and $$V_{m,n}$$ is the matrix element of $$V$$. Once I write the coefficient matrix of $$X$$, I can solve the eigenvalue $$\mu$$.

$$[(a+k_i)^2+(b+k_j)^2]X_{i,j}-\sum_{m,n}V_{m,n}X_{i-m,j-n}=\mu X_{i,j}$$.

Here we can set $$N=10,a =1, b=1$$ and $$V_{m,n}$$ is the matrix element of $$V$$. Once I write the coefficient matrix of $$X$$, I can solve the eigenvalue $$\mu$$.

$$[(a+k_i)^2+(b+k_j)^2]X_{i,j}-\sum_{m,n}V_{m,n}X_{i-m,j-n}=\mu X_{i,j}$$. where $$-N<=i,j<=N$$ Here we can set $$N=10,a =1, b=1$$ and $$V_{m,n}$$ is the matrix element of $$V$$. Once I write the coefficient matrix of $$X$$, I can solve the eigenvalue $$\mu$$.

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# How to write the equation into matrix form

$$[(a+k_i)^2+(b+k_j)^2]X_{i,j}-\sum_{m,n}V_{m,n}X_{i-m,j-n}=\mu X_{i,j}$$.

Here we can set $$N=10,a =1, b=1$$ and $$V_{m,n}$$ is the matrix element of $$V$$. Once I write the coefficient matrix of $$X$$, I can solve the eigenvalue $$\mu$$.