Biorthonormal Basis Expansions

• Biorthonormal basis expansions are spectral representations using paired dual bases that satisfy mutual biorthonormality, offering flexibility for non-Hermitian and non-self-adjoint systems.
• They are applied in quantum chemistry, computational physics, and galactic dynamics to address eigenvalue problems, potential expansions, and density reconstructions with precision.
• Algorithmic frameworks like iBTMRG and BMPS enable iterative updating of dual tensors, ensuring high-precision, scalable computations in both finite and infinite-dimensional spaces.

A biorthonormal basis expansion is a class of spectral representation in which the target object (typically a vector, function, density, or operator) is expressed as a sum or integral over pairs of dual bases. Each left and right basis is not, in general, orthonormal in itself, but the pair is mutually biorthonormal. Such expansions are fundamental in a range of applications including computational physics, quantum chemistry, Banach space theory, and galactic dynamics, especially where non-Hermitian, non-self-adjoint, or generalized structures require the flexibility of duality rather than strict orthogonality.

1. Definitions and Core Properties

Let $V$ be a finite- or infinite-dimensional vector (or function) space, over $\mathbb{R}$ or $\mathbb{C}$. A pair of sequences $\{|r_i\rangle\}_{i=1}^m$ (right basis) and $\{|l_i\rangle\}_{i=1}^m$ (left basis) is called biorthonormal if

$\langle l_i | r_j \rangle = \delta_{ij}.$

The corresponding completeness relation is

$$\sum_{i=1}^m |r_i\rangle\langle l_i| = \mathbb{I}_{m}.$$

A biorthonormal expansion for a vector $|v\rangle \in V$ takes the form

$$|v\rangle = \sum_{i=1}^m \langle l_i | v \rangle |r_i\rangle,$$

with analogous expressions for operators and functionals. In infinite-dimensional or function spaces, the basis sets may be countable or continuous, and the definition applies in the weak topology if necessary.

In Banach spaces, a biorthogonal system $\{(x_n,x_n^*)\}$ satisfies $x_m^*(x_n) = \delta_{mn}$, with the totality and minimality properties leading to the Markushevich basis (M-basis) structure (Gill, 2013).

2. Biorthonormal Basis Expansions in Non-Hermitian Problems

Biorthonormality is essential when the underlying operator or inner product is not self-adjoint. For a non-Hermitian transfer matrix $T$, with largest-modulus eigenvalue $\lambda_{\max}$ and corresponding right/left (Perron) eigenvectors $|\Psi_R\rangle$ and $\langle\Psi_L|$, the biorthonormal reduced bases efficiently encode these eigenstates: $$T\,|\Psi_R\rangle = \lambda_{\max}|\Psi_R\rangle, \qquad \langle\Psi_L|\,T = \lambda_{\max}\langle\Psi_L|.$$ Representations involve expansions over dual bases as

$$|\Psi_R\rangle = \sum_{i_1,\ldots,i_L} (\Psi_R)^{i_1,\ldots,i_L} |r_{i_1}\rangle\otimes\dots\otimes|r_{i_L}\rangle, \quad \langle\Psi_L| = \sum_{i_1,\ldots,i_L} (\Psi_L)^{i_1,\ldots,i_L} \langle l_{i_1}| \otimes\cdots\otimes\langle l_{i_L}|$$

with the biorthogonality and completeness constraints (Huang, 2011). This framework underpins non-Hermitian extensions of transfer-matrix renormalization group (TMRG) and matrix-product state (MPS) analyses.

3. Biorthonormal Matrix-Product States and Algorithmic Construction

In the thermodynamic limit, Perron eigenstates of non-Hermitian transfer matrices can be represented as infinite biorthonormal matrix-product states (BMPS), yielding a translationally invariant structure. For a two-site unit cell: $$|\Psi_R\rangle = \cdots [A^{s_{n-1}}\Lambda B^{s_n}\Lambda A^{s_{n+1}}]\cdots$$

$$\langle \Psi_L| = \cdots [\widetilde{B}^{s_{n-1}}\Lambda \widetilde{A}^{s_n}\Lambda \widetilde{B}^{s_{n+1}}]\cdots$$

Here, $A^s$ and $B^s$ are right MPS tensors, $\widetilde{A}^s$, $\widetilde{B}^s$ the duals, and $\Lambda$ the diagonal singular-value matrix (Huang, 2011). The normalization and mixed-pivot canonical forms are enforced via: $$\sum_s (A^s \Lambda) (\widetilde{B}^s)^T = \mathbb{I}_m, \quad \sum_s (B^s \Lambda) (\widetilde{A}^s)^T = \mathbb{I}_m.$$

Two principal algorithms—iBTMRG A (biorthonormal SVD bases) and iBTMRG B (orthonormal SVD, then projection to biorthonormal bases)—iteratively update these dual tensors, restoring biorthonormality at each step via SVD and non-unitary transformations. At convergence, all observables can be computed at cost independent of the system size; bulk quantities are given by contractions within a single two-site unit cell (Huang, 2011).

4. Biorthonormal Expansions in Function Spaces and Galactic Dynamics

Biorthonormal basis-function expansions provide a spectral decomposition of densities and potentials, particularly in galactic dynamics. Consider two paired sets of basis functions—$\{\psi_{n\ell}(r)\}$ (potential) and $\{\rho_{n\ell}(r)\}$ (density)—combined with spherical harmonics: $$\Phi_{n\ell m}(r, \theta, \phi) = \psi_{n\ell}(r) Y_{\ell m}(\theta, \phi), \qquad \rho_{n\ell m}(r, \theta, \phi) = \rho_{n\ell}(r) Y_{\ell m}(\theta, \phi)$$ with Poisson’s equation $\nabla^2\Phi_{n\ell m} = 4\pi G \rho_{n\ell m}$ and

$$\int_0^\infty \psi_{n\ell}(r)\,\rho_{n'\ell}(r)\,r^2 dr = \delta_{n n'}.$$

The overall expansion for a target density $\rho$ is

$$\rho(r,\theta,\phi) = \sum_{n\ell m} C_{n\ell m}\, \rho_{n\ell}(r)\, Y_{\ell m}(\theta, \phi)$$

with direct recovery of $C_{n\ell m}$ via projection onto the potential basis (Sanders et al., 2020). The Hernquist-Ostriker pair is commonly used, but general double-power-law biorthonormal expansions (e.g., Zhao, NFW, Jaffe) are fully characterized via recurrence relations and closed-form normalization (Lilley et al., 2018).

The selection of basis (inner slope, outer slope, scale parameters) is dictated by the physical system; for many realistic profiles, biorthonormal expansions with $n_{\mathrm{max}} \gtrsim 15$, $\ell_{\mathrm{max}} \gtrsim 6$ provide near machine-precision accuracy for orbit calculations and halo potential reconstructions (Sanders et al., 2020).

5. Biorthonormal Expansions in Quantum Chemistry and Nonadiabatic Dynamics

In coupled-cluster theories, electronic states are natively biorthonormal: left and right CC states are not Hermitian conjugates, but satisfy $$\langle \tilde{\psi}_m | \psi_n \rangle = \delta_{mn}$$. The total molecular wavefunction is expanded as

$$\Psi(r,R; t) = \sum_n \chi_n(R, t) \, \psi_n(r; R), \qquad \widetilde{\Psi}(r, R; t) = \sum_n \tilde{\chi}_n(R, t) \, \tilde{\psi}_n(r; R),$$

yielding paired left and right nuclear Schrödinger equations. The corresponding nonadiabatic coupling elements are expressed uniquely in the biorthonormal basis, circumventing the need for expensive full-CI normalization and preserving invariance under invertible basis transformations (Kjønstad et al., 2020). All observables are extracted as

$$\langle \hat{O} \rangle (t) = \langle \widetilde{\Psi}(t) |\hat{O}| \Psi(t)\rangle,$$

with the cost scaling like $O(N^6)$ for CCSD-based couplings.

6. Existence, Uniqueness, and Generalizations in Banach Spaces

In Banach space theory, Markushevich bases (M-bases) provide biorthogonal expansions in general separable Banach spaces. For every separable Banach space $B$, there exists a biorthogonal system $\{(x_n, x^*_n)\}$ such that the closed span of $\{x_n\}$ is $B$, the functionals $\{x^*_n\}$ are total, and $x^*_m(x_n) = \delta_{mn}$ with $\| x_n \| \| x^*_n \| = 1$ (Gill, 2013). Unlike Schauder bases, such biorthogonal systems need not yield norm-convergent expansions—only weak convergence is guaranteed in general.

Construction proceeds by embedding $B$ in a separable Hilbert space, selecting a fundamental minimal system $\{x_n\}$, and extending evaluation functionals via Hahn-Banach, normalizing to $\|x_n\|_B = 1$, $\|x^*_n\|_{B^*}=1$, ensuring the pair is biorthonormal under the duality bracket. For classical $\ell_p$ and $C([0,1])$, canonical normalized (even unconditional) Schauder bases are biorthonormal; in arbitrary separable Banach spaces, biorthonormal bases always exist but may lack completeness in norm (Gill, 2013).

7. Applications, Computational and Analytical Significance

Biorthonormal basis expansions are deployed in:

• Spectral solutions to the Poisson equation in spherical and axisymmetric potentials (double-power-law bases for galactic halos, Hernquist–Ostriker expansions, and generalizations to NFW or Jaffe models) (Lilley et al., 2018, Sanders et al., 2020).
• Non-Hermitian quantum lattice systems, enabling efficient calculation of bulk observables and correlation functions without boundary effects in the thermodynamic limit (iBTMRG and iBMPS frameworks) (Huang, 2011).
• Nonadiabatic quantum molecular dynamics, as in the coupled-cluster biorthonormal expansions optimizing both cost and formal rigor (Kjønstad et al., 2020).
• Generalized Banach space decompositions, especially in contexts lacking true orthonormal bases (Gill, 2013).

In all cases, biorthonormal expansions provide essential algorithmic and analytical tools where non-self-adjointness, weak topology, or physical duality demands more than a standard orthonormal framework.

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Key References:

• Biorthonormal MPS for non-Hermitian TMRG: (Huang, 2011)
• Biorthonormal expansions in coupled-cluster and Born-Huang nonadiabatic dynamics: (Kjønstad et al., 2020)
• M-basis existence and Banach space theory: (Gill, 2013)
• Biorthonormal function expansions for galactic dynamics and halo modeling: (Sanders et al., 2020, Lilley et al., 2018)