Biorthogonal Formalism: Theory & Applications

• Biorthogonal formalism is a framework where paired dual families replace standard orthonormal bases via bilinear or sesquilinear pairings to provide a complete resolution of identity.
• It is applied in quantum mechanics, operator theory, and signal processing to reformulate observables, spectral expansions, and reconstruction methods.
• The approach addresses challenges in non-Hermitian dynamics, metric ambiguities, and computational representations while ensuring robust and accurate analyses.

Searching arXiv for recent and foundational papers on biorthogonal formalism across quantum mechanics, operator theory, and signal processing. Biorthogonal formalism denotes a family of constructions in which the role of an orthonormal basis is replaced by a pair of dual families linked by a bilinear or sesquilinear pairing. In quantum mechanics, this replacement is used when a non-Hermitian but diagonalizable operator has real or complex eigenvalues and complete right and left eigenvectors; orthogonality in the original inner product is then replaced by biorthogonality, and observables, probabilities, density operators, and dynamics are reformulated accordingly (Brody, 2013). In operator theory, the same idea appears through biorthogonal families, $\mathcal D$-quasi bases, $\mathcal G$-quasi bases, generalized Riesz systems, and reproducing-kernel systems (Bagarello et al., 2018). In signal analysis, it appears as an overview-analysis exchange between a localized nonorthogonal family and its biorthogonal dual, yielding exact yet sparse representations for band-limited finite signals (Shimshovitz et al., 2012). Across these settings, the common structure is a dual pairing, a resolution of identity or weak reconstruction formula, and a transfer of spectral or expansion data between a space and an appropriate dual system.

1. Finite-dimensional quantum formulation

In finite-dimensional biorthogonal quantum mechanics, one works on an $N$-dimensional complex Hilbert space $H$ with dual $H^*$ and assumes that all operators under consideration are diagonalizable with $N$ distinct eigenvalues, so that no Jordan blocks arise (Brody, 2013). If $A$ is a linear operator on $H$ with nondegenerate real spectrum $\{a_n\}$, its right eigenvectors $\phi_n\in H$ satisfy

$\mathcal G$0

while the eigenvectors $\mathcal G$1 of the ordinary Hermitian adjoint $\mathcal G$2 satisfy

$\mathcal G$3

The defining relation is the biorthonormality condition

$\mathcal G$4

together with completeness,

$\mathcal G$5

This structure immediately yields an operator expansion. Any linear operator $\mathcal G$6 admits

$\mathcal G$7

and in particular

$\mathcal G$8

The formalism therefore replaces the ordinary spectral resolution by a right-left decomposition on the biorthogonal basis.

A metric operator is then introduced as

$\mathcal G$9

With the $N$0-inner product

$N$1

the right eigenvectors become orthonormal:

$N$2

In this setting, a normalized pure state $N$3 satisfies $N$4, equivalently $N$5, and the probability of outcome $N$6 is

$N$7

For an observable $N$8 whose matrix in the $N$9-$H$0 basis satisfies $H$1, the expectation value is

$H$2

The same framework extends to projectors and mixed states. The projector onto the $H$3-ray is

$H$4

with $H$5 and $H$6. A density operator has the expansion

$H$7

with $H$8, eigenvalues $H$9, and $H^*$0; its expectation of $H^*$1 is $H^*$2 (Brody, 2013).

A concrete two-level realization is given by

$H^*$3

with right eigenvectors $H^*$4, $H^*$5 and left eigenvectors $H^*$6, $H^*$7. The associated metric is

$H^*$8

and in the $H^*$9-inner product the $N$0’s become orthonormal (Brody, 2013).

The ordinary Hermitian theory is recovered when $N$1 and $N$2 share the same eigenvectors, so that $N$3, $N$4, and the biorthogonal basis coincides with an orthonormal one (Brody, 2013). This identifies the biorthogonal formalism not as a distinct kinematics unrelated to standard quantum mechanics, but as a relaxation of Hermiticity under the assumptions of diagonalizability, completeness, and suitable spectral reality.

2. Infinite-dimensional operator-theoretic extensions

In infinite-dimensional Hilbert spaces, biorthogonal formalism is typically developed through sequences $N$5 and $N$6 satisfying

$N$7

or, with alternate inner-product conventions, $N$8 (Bagarello et al., 2018). Here the principal issue is not merely biorthogonality but the strength of reconstruction, the domains on which expansions hold, and whether the systems are bases, quasi bases, or generalized Riesz systems.

A central notion is that of $N$9-quasi bases. For a dense subspace $A$0 such that

$A$1

the two families are called $A$2-quasi bases if for all $A$3,

$A$4

This reproducing formula implies that both linear spans are dense in $A$5 at least when the series converge in the norm of $A$6, and it ensures weak reconstruction on $A$7 (Bagarello et al., 2018).

The associated sesquilinear forms are

$A$8

Each is a densely defined, closed, positive sesquilinear form, and by the representation theorem there are unique positive self-adjoint operators $A$9 and $H$0 with

$H$1

This transfers biorthogonal data into operator-theoretic objects that resemble metric operators but are defined by closed forms rather than finite sums (Bagarello et al., 2018).

Another important notion is the generalized Riesz system. A sequence $H$2 is a generalized Riesz system if there exist an orthonormal basis $H$3 and a closed, densely defined operator $H$4 with densely defined inverse such that

$H$5

Under suitable domain assumptions, a biorthogonal pair that is also a $H$6-quasi basis is equivalent to such a generalized Riesz realization (Bagarello et al., 2018). This provides a similarity transform from orthonormal spectral data to nonorthogonal physical data.

A related weakening is the $H$7-quasi basis. If $H$8 is dense, two biorthogonal sets are $H$9-quasi bases if, for all $\{a_n\}$0,

$\{a_n\}$1

When $\{a_n\}$2 this is often called a weak resolution of the identity (Bagarello et al., 2017). The formulation is weaker than basishood on the whole Hilbert space, but it is sufficient for defining non-self-adjoint Hamiltonians with purely point real spectra through the expansion

$\{a_n\}$3

The notion of regularity can be relaxed further. A pair is regular if both $\{a_n\}$4 and $\{a_n\}$5 are dense; it is semi-regular if only one of the relevant density-and-domain conditions is imposed (Inoue, 2016). In that case one still obtains generalized Riesz bases for one side of the pair. This suggests that biorthogonal formalism is robust under weakening of basis assumptions, but only at the cost of asymmetric reconstruction and more delicate domain control.

These developments clarify a common misconception: biorthogonality alone is not equivalent to basis completeness, norm convergence of expansions, or bounded metric operators. The operator-theoretic literature distinguishes carefully between biorthogonal bases, regular biorthogonal pairs, semi-regular pairs, $\{a_n\}$6-quasi bases, $\{a_n\}$7-quasi bases, and generalized Riesz systems precisely because these properties do not coincide (Bagarello et al., 2017).

3. Metrics, pseudo-Hermiticity, and normalization issues

A recurrent theme in biorthogonal formalism is the construction of positive operators that intertwine non-self-adjoint operators with self-adjoint or orthonormal models. In finite dimension, the metric operator is

$\{a_n\}$8

and the $\{a_n\}$9-inner product makes the right eigenvectors orthonormal (Brody, 2013). In infinite-dimensional settings, analogous operators appear as

$\phi_n\in H$0

at least on a dense span of eigenvectors, with formal intertwining relations

$\phi_n\in H$1

This yields the pseudo-Hermitian relation

$\phi_n\in H$2

and the self-adjoint Hamiltonian

$\phi_n\in H$3

satisfies $\phi_n\in H$4 (Bagarello et al., 2017).

For invariant lattice constructions, one seeks a commuting-unitary family $\phi_n\in H$5 and a dual family $\phi_n\in H$6, with $\phi_n\in H$7 for an intertwiner

$\phi_n\in H$8

The Fourier-symbol equation

$\phi_n\in H$9

determines the coefficients via $\mathcal G$00 when $\mathcal G$01 almost everywhere and $\mathcal G$02 (Bagarello et al., 2014). Under these hypotheses, $\mathcal G$03 is bounded and invertible, the two families are Riesz bases of the closed span, and the weak resolution of the identity becomes

$\mathcal G$04

In the associated pseudo-Hermitian Hamiltonian construction, $\mathcal G$05 serves as the metric, and the “physical” inner product $\mathcal G$06 makes the non-Hermitian Hamiltonian self-adjoint (Bagarello et al., 2014).

A more recent controversy concerns the scaling, or gauge, ambiguity of the conventional biorthogonal inner product. If right and left eigenvectors are rescaled as

$\mathcal G$07

the condition $\mathcal G$08 is preserved, but transition probabilities and expectation values defined with the conventional associated-state inner product change under the $\mathcal G$09 (Edvardsson et al., 2022). To remove this ambiguity, a gauge-invariant inner product is introduced:

$\mathcal G$10

with

$\mathcal G$11

The operator $\mathcal G$12 is positive and Hermitian, the right eigenvectors become orthogonal under $\mathcal G$13, and expectation values

$\mathcal G$14

are independent of the original scaling choice (Edvardsson et al., 2022).

This debate reveals two distinct uses of “metric” in the literature. In some works the metric is a positive operator restoring orthonormality in a fixed biorthogonal basis; in others the emphasis is on eliminating normalization dependence between different admissible biorthogonal bases. A plausible implication is that “biorthogonal formalism” is not a single canonical inner-product prescription but a family of related prescriptions whose equivalence depends on the physical question being asked.

4. Non-Hermitian dynamics, topology, and quench theory

For non-Hermitian dynamics, the right-left spectral decomposition is combined with an associated-state construction. If

$\mathcal G$15

with

$\mathcal G$16

then an arbitrary state $\mathcal G$17 has associated state

$\mathcal G$18

The biorthogonal inner product becomes

$\mathcal G$19

and the transition probability between $\mathcal G$20 and $\mathcal G$21 is

$\mathcal G$22

(Jing et al., 2023).

On this basis one defines an automatically normalized biorthogonal Loschmidt amplitude for the evolution $\mathcal G$23:

$\mathcal G$24

The intensive rate function is

$\mathcal G$25

and zeros of momentum-resolved factors produce non-analytic cusps in $\mathcal G$26, signaling biorthogonal dynamical quantum phase transitions (Jing et al., 2023). The practical recipe is explicit: diagonalize $\mathcal G$27 and $\mathcal G$28, expand the initial state in the right basis, form associated states using the same coefficients in the left basis, evolve the state, compute the normalized $\mathcal G$29, and identify critical times where $\mathcal G$30 or the rate function becomes non-analytic (Jing et al., 2023).

In the non-Hermitian Su-Schrieffer-Heeger model, this formalism produces a half-integer jump in the dynamical topological order parameter

$\mathcal G$31

a feature that does not appear in self-normalized treatments. The periodicity of biorthogonal dynamical quantum phase transitions depends on whether the critical two-level subsystem oscillates or asymptotically reaches a steady state (Jing et al., 2023).

A closely related development in topological transport concerns charge pumping. For a one-dimensional lattice with Bloch Hamiltonian $\mathcal G$32 and left/right Bloch eigenvectors $\mathcal G$33, $\mathcal G$34, the biorthogonal Berry curvature is

$\mathcal G$35

The pumped charge equals the shift of the average position computed with left and right states,

$\mathcal G$36

where $\mathcal G$37 is the biorthogonal Chern number (Zhang et al., 2024). When the non-Hermitian skin effect is present, one replaces the Bloch momentum by the generalized Brillouin zone variable $\mathcal G$38, defines the non-Bloch curvature $\mathcal G$39, and obtains

$\mathcal G$40

The key message is explicit: quantized transport in non-Hermitian Thouless pumping arises only when position or current is measured in a biorthogonal sense (Zhang et al., 2024).

Quench dynamics in PT-symmetric systems adds entanglement and quantum geometry to this picture. With right and left eigenstates, the natural density matrix is

$\mathcal G$41

which is generally non-Hermitian, though $\mathcal G$42 (Lu et al., 27 Jul 2025). The biorthogonal quantum geometric tensor is

$\mathcal G$43

whose real part defines the metric and whose imaginary part encodes the Berry curvature (Lu et al., 27 Jul 2025). In a PT-broken post-quench regime, observables grow as $\mathcal G$44 and the TTC entropy behaves differently in interacting and free-fermion systems: it grows exponentially in generic interacting systems but shows linear decay in the free-fermion special case,

$\mathcal G$45

(Lu et al., 27 Jul 2025).

These results collectively indicate that in non-Hermitian dynamics the biorthogonal formalism is not merely a notational variant. It changes the definitions of state normalization, transition probabilities, Loschmidt echoes, geometric tensors, pumped charge, and entanglement diagnostics, and in several models it shifts the critical times or topological signatures relative to self-normal prescriptions (Jing et al., 2023).

5. Signal processing and computational representations

Outside quantum mechanics, biorthogonal formalism appears in an exact and computationally effective form in periodic Gabor analysis. Starting from the classical Gabor family

$\mathcal G$46

one truncates to a finite grid of $\mathcal G$47 functions. The truncated family is incomplete under periodic boundary conditions, a difficulty tied to the Balian-Low theorem (Shimshovitz et al., 2012).

The remedy is to periodize the Gabor basis via the Dirichlet kernel

$\mathcal G$48

whose translates form an exact basis for band-limited, $\mathcal G$49-periodic functions. The periodic Gabor functions are then

$\mathcal G$50

By construction, the $\mathcal G$51 span exactly the same $\mathcal G$52-dimensional space as the sample-recovery Dirichlet basis, are stable, and bypass the Balian-Low obstruction (Shimshovitz et al., 2012).

Since the $\mathcal G$53 are nonorthogonal, there exist dual functions $\mathcal G$54 such that

$\mathcal G$55

If the overlap matrix is

$\mathcal G$56

then the duals are given by

$\mathcal G$57

which implies

$\mathcal G$58

The key step is the biorthogonal exchange. A signal can be written in the pg basis,

$\mathcal G$59

or in the dual basis,

$\mathcal G$60

Because the pg functions are localized in time-frequency, many overlaps $\mathcal G$61 vanish or are very small for signals occupying only a subset of the time-frequency plane, whereas multiplication by the dense matrix $\mathcal G$62 destroys locality. The formalism therefore exchanges roles: use the localized pg functions only to compute the overlaps, and use the delocalized duals as the synthesis basis. Dropping the small coefficients yields a large compression factor without sacrificing exactness for band-limited, finite-support signals (Shimshovitz et al., 2012).

A 64-point discrete rectangular pulse on an $\mathcal G$63 pg grid illustrates the gain. Keeping only the 25 largest overlaps gives reconstruction errors

• DGE error $\mathcal G$64,
• pgb error $\mathcal G$65,

which is reported as almost an order-of-magnitude improvement (Shimshovitz et al., 2012).

This setting shows that biorthogonality is not tied intrinsically to non-Hermiticity or quantum observables. It can also be understood as an analysis-synthesis asymmetry: localized, nonorthogonal functions are optimal for coefficient extraction, while their duals are optimal for exact reconstruction.

6. Biorthogonality beyond quantum mechanics: kernels, random matrices, and special functions

In reproducing-kernel Hilbert spaces, an exact system $\mathcal G$66 has a unique biorthogonal system $\mathcal G$67 satisfying

$\mathcal G$68

However, biorthogonality does not guarantee completeness of the dual system in a general Hilbert space (Baranov et al., 2010). The reproducing-kernel setting is exceptional because extra analytic structure constrains this defect. For a class of spaces with a Riesz basis of reproducing kernels, Baranov and Belov identify regimes where the biorthogonal to every exact system is complete and regimes where incomplete biorthogonals exist (Baranov et al., 2010). In the notation of their construction, slowly decaying weights force completeness of the biorthogonal system, while rapidly decaying weights permit incomplete biorthogonals.

Biorthogonal ensembles in probability theory are built from families $\mathcal G$69 and $\mathcal G$70 satisfying

$\mathcal G$71

The associated projection kernel

$\mathcal G$72

is idempotent,

$\mathcal G$73

and generates a determinantal point process through the density

$\mathcal G$74

(Cafasso et al., 2024). Cafasso and Claeys develop a double-contour construction of such kernels and use it to represent the partition functions of the Log Gamma polymer, the O'Connell-Yor polymer, and the mixed polymer in terms of explicit biorthogonal measures (Cafasso et al., 2024). They also show that these measures converge to random matrix eigenvalue distributions in small temperature limits.

A different branch of the subject concerns basic hypergeometric biorthogonal functions. Rosengren studies two families built from Rahman’s $\mathcal G$75 functions and obtains explicit continuous and discrete biorthogonality measures (Rosengren, 2016). In that framework the inner product is realized either as a contour integral with a specified weight or, under stronger interlacing conditions, as a discrete sum over nodes. The same objects arise in superconformal indices for three-dimensional quantum field theories and in solvable lattice models (Rosengren, 2016).

These examples suggest that the most stable abstract core of biorthogonal formalism is the existence of dual families with a reproducing pairing and a corresponding kernel or resolution operator. What changes from field to field is the ambient structure: a Hilbert-space metric in pseudo-Hermitian quantum mechanics, an overlap matrix in signal compression, a determinantal kernel in random matrices, or a contour/discrete measure in special-function theory.

7. Conceptual scope and recurring issues

Several recurring themes organize the literature.

First, biorthogonality is weaker than orthonormality but stronger than arbitrary duality. It typically supplies a reconstruction formula, a spectral expansion, or a determinantal kernel, yet it does not by itself imply completeness, boundedness of the metric, or norm-convergent expansions (Baranov et al., 2010). This is why the literature develops refinements such as regularity, semi-regularity, $\mathcal G$76-quasi bases, $\mathcal G$77-quasi bases, and generalized Riesz systems (Inoue, 2016).

Second, the relation between right and left objects is not uniform across applications. In finite-dimensional non-Hermitian quantum mechanics, the left system is usually the eigenbasis of $\mathcal G$78 and the metric restores a positive-definite inner product (Brody, 2013). In dynamical non-Hermitian settings, associated states carry the same expansion coefficients into the left basis, which makes probabilities and Loschmidt amplitudes automatically normalized (Jing et al., 2023). In signal processing, by contrast, one intentionally separates localization properties between the analysis family and the synthesis family (Shimshovitz et al., 2012).

Third, metric selection is both a technical and conceptual issue. In pseudo-Hermitian constructions, a positive intertwining operator implements similarity to a self-adjoint Hamiltonian (Bagarello et al., 2017). In renormalized biorthogonal quantum mechanics, a gauge-invariant operator

$\mathcal G$79

is introduced to remove scaling ambiguities of the conventional biorthogonal inner product (Edvardsson et al., 2022). This suggests that physical predictions may depend on which inner-product structure is regarded as primary.

Finally, the formalism has become increasingly important in non-Hermitian topology and nonequilibrium physics. Quantized charge pumping is guaranteed only under a biorthogonal formalism, with the pumped charge identified with a Chern number or, in the presence of the non-Hermitian skin effect, a non-Bloch Chern number on the generalized Brillouin zone (Zhang et al., 2024). Likewise, biorthogonal dynamical quantum phase transitions produce critical times and topological signatures that differ from conventional self-normal approaches (Jing et al., 2023). These developments indicate that, in contemporary usage, “biorthogonal formalism” often marks the point at which a non-Hermitian theory becomes probabilistically and topologically well posed rather than merely algebraically diagonalizable.