Extracting density information from finite Hamiltonian matrices We demonstrate how to extract approximate, yet highly accurate, density-of-state information over a continuous range of energies from a finite Hamiltonian matrix. The approximation schemes which we present make use of the theory of orthogonal polynomials associated with tridiagonal matrices. However, the methods work as well with non-tridiagonal matrices. We demonstrate the merits of the methods by applying them to problems with single, double, and multiple density bands, as well as to a problem with infinite spectrum.

With every Hamiltonian (hermitian matrix), there is an associated positive definite density of states function (in energy space). With every Hamiltonian (hermitian matrix), there is an associated positive definite density of states function (in energy space). Simple arguments could easily be under- stood when the Hamiltonian matrix is tridiagonal. Simple arguments could easily be under- stood when the Hamiltonian matrix is tridiagonal. We exploit the intimate connection and interplay between tridiagonal matrices and the theory of orthogonal polynomials. We exploit the intimate connection and interplay between tridiagonal matrices and the theory of orthogonal polynomials.

Solutions of the three-term recursion relation are orthogonal polynomials. Solutions of the three-term recursion relation are orthogonal polynomials. Regular p n (x) and irregular q n (x) solutions. Regular p n (x) and irregular q n (x) solutions. Homogeneous and inhomogeneous initial relations, respectively. Homogeneous and inhomogeneous initial relations, respectively.

p n (x) is a polynomial of the “first kind” of degree n in x. p n (x) is a polynomial of the “first kind” of degree n in x. q n (x) is a polynomial of the “second kind” of degree (n  1) in x. q n (x) is a polynomial of the “second kind” of degree (n  1) in x. The set of n zeros of p n (x) are the eigenvalues of the finite n  n matrix H. The set of n zeros of p n (x) are the eigenvalues of the finite n  n matrix H. The set of (n  1) zeros of q n (x) are the eigenvalues of the abbreviated version of this matrix obtained by deleting the first raw and first column. The set of (n  1) zeros of q n (x) are the eigenvalues of the abbreviated version of this matrix obtained by deleting the first raw and first column.

They satisfy the Wronskian-like relation: They satisfy the Wronskian-like relation: The density (weight) function associated with these polynomials: The density (weight) function associated with these polynomials: The density function associated with the Hamiltonian H : The density function associated with the Hamiltonian H :

x y  G 00 (x+iy) Discrete spectrum of H Continuous band spectrum of H

Connection:

For a single limit: The density is single-band with no gaps and with the boundary For some large enough integer N

Note the reality limit of the root and its relation to the boundary of the density band

One-band density example

Two-Band Density giving again a quadratic equation for T(z) The boundaries of the two bands are obtained from the reality of T as

Two-band density example Three-band density example Infinite-band density example

Asymptotic limits not known? Analytic continuation method Analytic continuation method Dispersion correction method Dispersion correction method Stieltjes Imaging method Stieltjes Imaging method

Non-tridiagonal Hamiltonian matrices? Solution will be formulated in terms of the matrix eigenvalues instead of the coefficients.

Analytic Continuation 

One-Band Density Two-Band Density Infinite-Band Density

Dispersion Correction Gauss quadrature: Numerical weights:

()() 11 22 33 44 00 0       n

One-Band Density Two-Band Density Infinite-Band Density

Stieltjes Imaging

One-Band Density Two-Band Density Infinite-Band Density

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