Dimensional Reduction of Stationary States

• Dimensional reduction of stationary states is the process by which time-independent high-dimensional solutions are effectively represented by lower-dimensional structures through exploiting symmetries and scaling limits.
• This framework employs rigorous techniques such as Γ-convergence, subspace projection, and block-diagonalization to yield sharp analytical results and computational efficiency.
• Applications span nonlinear Schrödinger systems, quantum many-body problems, stochastic processes, and eigenstate compression, providing unified insights across mathematical physics.

Dimensional reduction of stationary states refers to the mathematical and physical processes by which stationary (time-independent) solutions of high-dimensional systems are rigorously or effectively described by lower-dimensional structures. This reduction exploits symmetries, scaling limits, integrability, or block-diagonalization, providing either sharp limits (e.g., via Γ-convergence, kinetic theory, or spectral decomposition) or algorithmic compression (e.g., via subspace projection in numerical linear algebra) for describing equilibria, invariant measures, or ground states. In quantum, classical, and stochastic contexts, dimensional reduction both advances fundamental understanding of the structure of stationary solutions and enables efficient analytical and computational analysis.

1. Variational and Spectral Reduction in Nonlinear Schrödinger Systems

For nonlinear dispersive PDEs, particularly the focusing nonlinear Schrödinger equation (NLS) on two-dimensional domains with geometric thinness or a network-like structure, dimensional reduction arises in the thin-domain limit. Consider the "open-book" geometry, defined as Ωₑ = G × [0,ε], where G is a finite or periodic metric graph. The stationary states are mass-constrained minimizers of the NLS action:

$$S_ε(ψ) = \int_{Ω_ε} \frac{1}{2} |\nablaψ|^2 - \frac{1}{p+1} |ψ|^{p+1} \, dx\, dy, \quad \|\psi\|_{L^2(Ω_ε)}^2 = M.$$

A major result is the rigorous Γ-convergence of this variational problem, as ε→0, to the corresponding problem on the metric graph:

$$S_G(φ) = \sum_{e\in E} \int_e \left[ \frac{1}{2} |\phi'_e(x)|^2 - \frac{1}{p+1} |\phi_e(x)|^{p+1} \right] dx, \quad \sum_e \int_e |\phi_e|^2 dx = M,$$

with Neumann boundary conditions imposed transversely and Kirchhoff continuity at graph vertices (Coz et al., 29 Dec 2025). The sharp transition in stationary-state structure occurs at the critical transverse width

$ε^* = \frac{π}{\sqrt{λ_G}},$

where λ_G is the Lagrange multiplier (chemical potential) for the ground-state solution on G. The regimes are as follows:

• For ε < ε*, stationary states are fully reduced and take the product form ψₑ(x,y) = φ_G(x) w₀(y/ε), constant in the thin direction.
• For ε > ε*, genuinely 2D non-factorizing ground states appear due to the bifurcation induced by the first excited transverse mode.

This analytical framework extends to "fractured strips" with delta conditions: as the strip width shrinks, ground states of the 2D NLS converge in H¹ to the unique ground state of the corresponding 1D δ-line NLS problem. This dimension reduction is established via variational compactness, explicit Pohozaev identities, and sharp Γ-limit techniques. For small enough transverse width, minimizers become strictly 1D, while for larger widths, truly multidimensional minimizers exist (Coz et al., 2024).

2. Kinetic Equations and Soliton Gas Reductions

In integrable PDEs, such as the Kadomtsev–Petviashvili II (KP-II) equation, time-independent dimensional reduction provides rigorous closure between (2+1)D and (1+1)D kinetic descriptions. For stationary states,

$$\bigl(u_t - u_x + 6 u u_x + u_{xxx}\bigr)_x + u_{yy} = 0$$

is reduced via a stationary ansatz $u(x, y, t) = U(x, y)$, leading to the stationary Boussinesq equation:

$U_{yy} - U_{xx} + 3 (U^2)_{xx} + U_{xxxx} = 0.$

Generalized hydrodynamics yields a kinetic description for a 2D stationary line-soliton gas, in which the 2D field is characterized by the spectral measure of soliton amplitudes and local conservation laws (Bonnemain et al., 2024). Analytical formulas predict how composite soliton gases interact (refraction, interference), and all predictions are confirmed through large-N exact soliton solutions. The reduction allows the direct application of 1D soliton-gas kinetic theory to stationary patterns in the 2D plane.

3. Stationary State Reduction in Quantum and Stochastic Systems

In finite-state Markov processes, the probability normalization constraint ($\sum_i p_i(t) = 1$) allows exact elimination of one degree of freedom in the stationary master equation:

$\frac{d}{dt} p(t) = Q p(t).$

By projecting onto a reduced space and enforcing normalization explicitly, the dimension is reduced by one, and the resulting $(M-1)$-dimensional system is strictly stable (all eigenvalues have negative real part) except for the removed stationary mode. The stationary distribution is then found explicitly via inversion of the reduced generator $\widetilde Q = J Q H$:

$$p_\infty = \mathbf{e}_M - H \widetilde Q^{-1} J Q \mathbf{e}_M.$$

This reduction carries over, mutatis mutandis, for the stochastic differential equation analog, yielding nondegenerate diffusion on the simplex (Soudry et al., 2012).

For discrete-time quantum channels (completely positive trace-preserving maps), a block-decomposition procedure (irreducible-decomposition) allows reduction of the Hilbert space to a direct sum of minimal enclosures, each supporting a unique extremal invariant state. Any global invariant (stationary) state is then a convex combination (plus possible off-diagonal terms in families of isomorphic minimal blocks) of these finite or lower-dimensional stationary states (Carbone et al., 2015).

4. Dimensional Reduction Algorithms for High-Dimensional Quantum Systems

Algorithmic dimensional reduction is critical for computing stationary states in very large Hilbert spaces. The Jacobi-Davidson algorithm, when implemented in a matrix-free fashion, efficiently finds a low-dimensional optimal subspace capturing the few relevant stationary eigenstates of large Hermitian or generalized-Hermitian operators:

$H ψ = E ψ\ \ (or) \ \ H ψ = S ψ E.$

The search subspace is iteratively expanded and orthogonalized; a small-dimensional projected eigenproblem is solved at each step. The process converges superlinearly, and when operator application routines are $\mathcal{O}(n)$, the total computational cost for $k$ stationary states is $\mathcal{O}(k n)$, achieving dramatic compression of the stationary-state sector (Cook, 2024).

5. Block-Diagonal Reductions in Quantum Walks

In discrete-time quantum walks, the stationary states (1-eigenvectors) of the walk operator $U$ can be exactly characterized in terms of uniform and "flip" subspaces determined by graph structure and marked vertices. The existence and dimension of the stationary subspace, parameterized by the number and structure of marked and unmarked connected components, allows for block-diagonalization:

$$U = \begin{pmatrix} I_{k_u + k_m} & 0 \ 0 & U_{\rm dyn} \end{pmatrix}$$

in a basis adapted to the stationary/dynamic split. Thus, the quantum walk dynamics reduce to analysis of a possibly far smaller block $U_{\rm dyn}$, providing computational and interpretive efficiency (Prūsis et al., 2016).

6. Dimensional Reduction in Quantum Many-Body Systems: Free Fermion Example

For quadratic free-fermion Hamiltonians on 2D lattices with translation invariance, dimensional reduction via partial Fourier transform converts the 2D problem into a set of decoupled 1D chains indexed by transverse momentum. The entropy and stationary properties of subsystems are then computable as superpositions over the reduced 1D sectors. The exact criterion for convergence to a stationary reduced density matrix after a quantum quench is that initial state occupations are diagonal in the transverse mode basis; if not, the system may saturate entanglement entropy without genuine equilibration—necessitating modification of the 1D quasiparticle picture (Yamashika et al., 2023).

7. Universality, Limitations, and Physical Relevance

Dimensional reduction for stationary states is universally leveraged across disciplines whenever symmetries, geometric thinness, spectral projections, or conservation constraints exist. It provides both rigorous analytical reduction and practical computational tractability. However, full reduction is limited to symmetry-protected, thin, or decoupled regimes; above sharp thresholds (e.g., in NLS open books, strip widths, or broken symmetry cases), genuinely high-dimensional stationary states emerge, and reduction may fail or require nontrivial generalization. The systematic study of these reductions unifies topics from quantum field theory, statistical mechanics, integrable systems, kinetic theory, and numerical algorithms, directly impacting the analysis of ground states, invariant distributions, and equilibrium measures.