---
title: Demkov–Osherov Model (DOM) Overview
url: https://www.emergentmind.com/topics/demkov-osherov-model-dom
type: topic
---

# Demkov–Osherov Model (DOM) Overview

The Demkov–Osherov model (DOM), often written as the DO model, is an exactly solvable multilevel Landau–Zener problem in which a single diabatic level with linear time dependence crosses a band of parallel or time-independent diabatic levels, with constant couplings only between the slanted level and each band state. In the standard reduction used in the literature, the Hamiltonian is obtained from a generic multilevel Landau–Zener form \(H_{LZ}(t)=\hat A+\hat B\,t\) when \(\hat B\) has one eigenvalue distinct from an \(n\)-fold degenerate subspace, after subtracting a trivial diagonal term and rescaling time. The model is a paradigmatic exactly solvable system of nonadiabatic transitions, and later work uses it both as a reference point for nonlinear generalizations with curved levels and as an exact local scattering problem in integrable constructions based on Knizhnik–Zamolodchikov equations and multivariable Painlevé-II systems [2112.12866] [1401.0682] [2603.22470].

## 1. Canonical Hamiltonian and reduction from multilevel Landau–Zener dynamics

A standard starting point is the generic multilevel Landau–Zener Hamiltonian
\[
H_{LZ}(t)=\hat A+\hat B\,t.
\]
If \(\hat B\) has one eigenvalue \(b_1\) and \(n\) degenerate eigenvalues \(b_2\), one diagonalizes \(\hat B\), uses the degeneracy of the \(n\)-dimensional subspace to diagonalize the corresponding minor of \(\hat A\), subtracts the diagonal piece \(b_2 t\,\mathbb 1\), and rescales time by \(b_1-b_2\). This yields the Demkov–Osherov Hamiltonian
\[
H_{DO}(t)=
\begin{pmatrix}
t+a_{00} & v_{01} & \cdots & v_{0n}\\
v_{01} & a_{01} & 0 & \cdots\\
\vdots & 0 & \ddots & \vdots\\
v_{0n} & \cdots & \cdots & a_{0n}
\end{pmatrix},
\]
or, in bra–ket form,
\[
H_{DO}(t)=(t+a_{00})\,|0\rangle\langle 0|
+\sum_{i=1}^{n} a_{0i}\,|i\rangle\langle i|
+\sum_{i=1}^{n} v_{0i}\big(|0\rangle\langle i|+|i\rangle\langle 0|\big).
\]
Only the level \(|0\rangle\) has a linearly time-dependent diabatic energy. The levels \(|i\rangle\), \(i=1,\dots,n\), have constant diabatic energies \(a_{0i}\), and there is no direct coupling between \(|i\rangle\) and \(|j\rangle\) for \(i\neq j\). In this basis, the model is precisely one moving level interacting with a band of static levels [2112.12866].

The two-level Landau–Zener problem is recovered for \(n=1\). In that sense, the DOM is the minimal multilevel extension that preserves exact solvability while retaining a highly structured coupling graph: a star topology centered on the sweeping level \(|0\rangle\) [2112.12866].

## 2. Scattering problem, exact structure, and factorization

The Schrödinger equation is considered in the scattering picture,
\[
-i\partial_t\Psi(t)=H_{DO}(t)\Psi(t),
\]
with evolution from \(t\to -\infty\) to \(t\to +\infty\), where the diabatic levels are asymptotically well separated. Transition probabilities are then defined in the diabatic basis \(\{|k\rangle\}\) [2112.12866].

An algebraic characterization of the exact solution uses the eigenstate ansatz
\[
|x_m(t)\rangle=\sum_{k=0}^n\frac{\gamma_k}{x_m(t)-\epsilon_k}\,|k\rangle,
\qquad
E_m^0(t)=\frac{\gamma_0^2}{x_m(t)-\epsilon_0},
\]
with consistency conditions
\[
v_{0i}=\frac{\gamma_0\gamma_i}{\epsilon_0-\epsilon_i},
\qquad
a_{0i}=\frac{\gamma_0^2}{\epsilon_i-\epsilon_0},
\]
and
\[
a_{00}=\sum_{i=1}^n\frac{\gamma_i^2}{\epsilon_i-\epsilon_0},
\qquad
t=\sum_{i=0}^n\frac{\gamma_i^2}{x_m(t)-\epsilon_i}.
\]
These relations show that the parameters of the Hamiltonian admit a nontrivial integrable parametrization in terms of \(\{\gamma_i,\epsilon_i\}\) [2112.12866].

In the standard DOM, transition probabilities factorize in a Landau–Zener-like product form. For the three-level case with one sweeping level and two parallel levels, the exact DOM scattering amplitudes coincide with the result of the independent crossing approximation: one multiplies the individual two-level Landau–Zener scattering matrices in chronological order and inserts adiabatic phase accumulation between crossings. At each isolated two-level crossing with coupling \(g\) and relative slope \(v\), the corresponding Landau–Zener probability is
\[
P_{\text{LZ}}=e^{-2\pi |g|^2/|v|}.
\]
In the DOM setting used for asymptotic analysis of multivariable Painlevé-II, this factorized construction is not merely heuristic; it coincides with the exact DOM solution [2603.22470].

## 3. Curved-level generalization and the DOM limit

A later exactly solvable model replaces the band of parallel levels by a Coulomb band and can be regarded as a generalization of the DOM. In the original time variable \(\tau\), the diabatic Hamiltonian is
\[
H_{00}(\tau)=\beta \tau,\qquad
H_{jj}(\tau)=\frac{k_j}{\tau},\qquad
H_{0j}(\tau)=H_{j0}(\tau)=g_j,
\]
with all other matrix elements zero. Thus the sweeping level remains linear,
\[
E_0(\tau)=\beta \tau,
\]
while the band states become curved,
\[
E_j(\tau)=\frac{k_j}{\tau},\qquad j=1,\ldots,N.
\]
The DOM is recovered in the specific limit
\[
k_i \gg |k_i-k_j| \sim |g_i|^2/\beta,
\]
for \(i,j=1,\ldots,N\). In that limit the Coulomb band behaves locally as a band of nearly parallel levels, and the Demkov–Osherov solution is recovered [1401.0682].

After the change of variables
\[
t=\frac{\tau^2}{2},\qquad a_j(\tau)=\tau\, b_j(t),
\]
the system reduces to
\[
i\frac{d b_0}{dt}=\beta b_0+\sum_{j=1}^N g_j b_j,
\qquad
2it\frac{d b_j}{dt}=(k_j-i)b_j+g_j b_0.
\]
Elimination of the band amplitudes yields an \((N+1)\)-th order ordinary differential equation of Meijer \(G\)-type for \(b_0(t)\). The exact survival probability of the initially populated level \(0\) is
\[
P_{00}=\prod_{j=1}^{N}\left(\frac{e^{-2\pi l_j}+1}{e^{\pi k_j}+1}\right),
\]
where \(l_1,\dots,l_N\) are the real roots of the characteristic polynomial
\[
g(y)=\prod_{j=1}^N\left(y+\frac{k_j}{2}\right)
-\sum_{j=1}^N\frac{g_j^2}{2\beta}
\prod_{\substack{m=1\\m\neq j}}^N\left(y+\frac{k_m}{2}\right).
\]
This formula retains a product structure reminiscent of DOM, but the exponents are renormalized by the full many-level interaction through the roots \(l_j\) [1401.0682].

Two limiting regimes are particularly informative. In the degenerate-band case \(k_i=k\) for all \(i\),
\[
P_{00}=
\frac{\exp\!\left(\pi\left[k-\sum_{i=1}^N g_i^2/\beta\right]\right)+1}
{\exp(\pi k)+1},
\]
so the survival probability remains finite even for arbitrarily strong couplings and many band states. In the well-separated regime \(|k_i-k_j|\gg |g_s^2/\beta|\),
\[
l_j\approx -\frac{k_j}{2}+\frac{g_j^2}{2\beta},
\]
and
\[
P_{00}\approx
\prod_{i=1}^N
\left(
\frac{\exp\!\left(\pi\left[k_i-g_i^2/\beta\right]\right)+1}
{\exp(\pi k_i)+1}
\right),
\]
which reproduces the product of effectively independent two-level Landau–Zener-like contributions familiar from DOM. By contrast, the curved-band model also shows that transition probabilities within the band generally do not saturate asymptotically; only probabilities involving the isolated level \(0\) are well defined at large time [1401.0682].

## 4. Distinction from the two-level Demkov pulse model

The DOM should be distinguished from the two-level Demkov model used in optical Bloch dynamics. The latter is a pulsed two-state problem with constant detuning and exponential envelope,
\[
\Delta=\text{const},\qquad
\Omega(t)=\Omega_0 e^{-|t|/T},\qquad
\Gamma=\text{const},
\]
where \(\Gamma=1/T_2\) is a pure dephasing rate [1402.5648].

| Model | Time dependence | Role |
|---|---|---|
| Standard DOM | One level linear in time; band levels constant or parallel | Multilevel nonadiabatic scattering |
| Two-level Demkov model | \(\Omega(t)=\Omega_0 e^{-|t|/T}\), \(\Delta=\text{const}\) | Exactly solvable pulsed two-state problem |
| Coulomb-band generalization | One linear level and \(E_j(\tau)=k_j/\tau\) | Nonlinear-time generalization of DOM |

In the two-level Demkov model with dephasing, the Bloch equations are
\[
\frac{d}{dt}
\begin{bmatrix}
u(t)\\
v(t)\\
w(t)
\end{bmatrix}
=
\begin{bmatrix}
-\Gamma & \Delta & 0\\
-\Delta & -\Gamma & -\Omega(t)\\
0 & \Omega(t) & 0
\end{bmatrix}
\begin{bmatrix}
u(t)\\
v(t)\\
w(t)
\end{bmatrix},
\]
with
\[
u(t)=2\Re\rho_{12}(t),\qquad
v(t)=2\Im\rho_{12}(t),\qquad
w(t)=\rho_{22}(t)-\rho_{11}(t).
\]
The exact solution is obtained by reducing the problem to a third-order differential equation for \(w(t)\), solving it in terms of generalized hypergeometric functions \({}_1F_2\), and matching across the cusp at \(t=0\). On resonance, \(\Delta=0\), the solution simplifies further and reduces to Bessel functions [1402.5648].

This two-level Demkov problem is not the multilevel Demkov–Osherov model, but it functions as one of the elementary building blocks underlying the general DOM: in the full DOM, transitions are described as a sequence of independent two-level Demkov- or Landau–Zener-type crossings between one slanted level and several parallel levels. With dephasing included, the final population transfer is
\[
P_2(+\infty)=\frac{1+w(+\infty)}{2}=\frac{1+A_+}{2},
\]
and increasing \(\Gamma\) leads to a monotonic suppression of the final transition probability [1402.5648].

## 5. Integrable reformulations through EKZ equations and boundary WZNW theory

The DOM and its close variants admit a reformulation in terms of extended Knizhnik–Zamolodchikov equations and boundary Wess–Zumino–Novikov–Witten theory. In that construction, the boundary WZNW action is
\[
S_{BWZNW}(g)=S_{WZNW}(g)+S_{bound}^L(\mathcal C)+S_{bound}^R(\bar{\mathcal C}),
\]
with contour terms
\[
S_{bound}^L(\mathcal C)=\alpha\oint_{\mathcal C}dw\,w\,J^3(w),
\qquad
S_{bound}^R(\bar{\mathcal C})=\alpha\oint_{\bar{\mathcal C}}d\bar w\,\bar w\,\bar J^3(\bar w).
\]
The resulting correlators satisfy extended KZ equations
\[
\big[(\mathsf{k}+2)\partial_{w_l}-\hat H_l^R\big]G(\{w\})=0,
\qquad
\hat H_l^R=\lambda S_l^3+\hat H_l^G,
\]
where \(\hat H_l^G\) are Gaudin Hamiltonians and \([\hat H_l^R,\hat H_{l'}^R]=0\) [2112.12866].

The paper then studies an altered Demkov–Osherov model (ADO), in which two levels have the same slope in time,
\[
H_{ADO}(t)=
\sum_{k=0,1}(t+v_{kk})|k\rangle\langle k|
+\sum_{i=2}^n a_i|i\rangle\langle i|
+v_{01}(|0\rangle\langle 1|+|1\rangle\langle 0|)
+\sum_{k=0,1}\sum_{i=2}^n v_{ki}(|k\rangle\langle i|+|i\rangle\langle k|).
\]
After Fourier transformation in time and elimination of the static levels, one obtains an EKZ-type equation
\[
i\partial_\omega\Phi(\omega,\{a_k\})=H_1(\omega,\{a_k\})\,\Phi(\omega,\{a_k\}),
\]
together with companion equations in the parameters \(a_i\). The integrability condition is the rank-one coupling constraint
\[
v_{ij}=\gamma_i\gamma_j,
\]
which implies that the corresponding classical vectors \(b_k^\mu\) are parallel and ensures the zero-curvature condition \([H_i,H_j]=0\) [2112.12866].

Under this constraint the ADO system is solved exactly. For the simplest nontrivial case \(n=2\), the transition probability takes the explicit Landau–Zener form
\[
P=\exp\big[-2\pi(\gamma_0^2+\gamma_1^2)\gamma_2^2\big].
\]
This reformulation places DOM-type Hamiltonians at the interface of non-equilibrium quantum dynamics, Gaudin-type integrable systems, and boundary conformal field theory [2112.12866].

## 6. DOM as local scattering data in multivariable Painlevé-II theory

A later application uses the exact DOM solution as the central linear ingredient in an asymptotically exact WKB analysis of a multivariable Painlevé-II system. For the \(n\)-component nonlinear equations
\[
u_k''(x)=x u_k(x)-2u_k(x)\sum_{j=1}^n u_j^2(x)+e_k u_k(x),
\qquad k=1,\dots,n,
\]
the consistency condition of a two-time Schrödinger problem defines a Lax pair with Hamiltonians \(H(t,x)\) and \(H_1(t,x)\). In the worked-out \(n=2\) case, one studies the \(t\)-evolution at fixed large \(|x|\) in a three-dimensional diabatic basis \(|0\rangle,|1\rangle,|2\rangle\) [2603.22470].

For large negative \(x\), after the rescaling
\[
t=\tau\sqrt{|x|},
\]
and expansion near \(\tau=\pm \tfrac12\), the local Hamiltonians become exact three-level DOM Hamiltonians. Near \(\tau=-\tfrac12\),
\[
H_{-1/2}(\tau_m)=
\begin{pmatrix}
-4|x|^{3/2}\tau_m & g_1^- & g_2^-\\
(g_1^-)^* & 4|x|^{3/2}\tau_m & 0\\
(g_2^-)^* & 0 & 4|x|^{3/2}\tau_m+2\sqrt{|x|}e
\end{pmatrix},
\]
and near \(\tau=+\tfrac12\),
\[
H_{1/2}(\tau_p)=
\begin{pmatrix}
4|x|^{3/2}\tau_p & g_1^+ & g_2^+\\
(g_1^+)^* & -4|x|^{3/2}\tau_p & 0\\
(g_2^+)^* & 0 & -4|x|^{3/2}\tau_p+2\sqrt{|x|}e
\end{pmatrix}.
\]
Each local Hamiltonian has one linearly sweeping level, two parallel levels, and time-independent couplings within the local variable. In this framework the exact DOM S-matrix replaces conventional complex-plane Stokes data: path-independence of the flat two-time connection allows the evolution operator to be computed in different asymptotic regions, and the DOM solution supplies the nonadiabatic matching data [2603.22470].

The resulting connection formulas between \(x\to -\infty\) and \(x\to +\infty\) are written in terms of
\[
p_1=e^{-\pi\alpha_1^2},\qquad p_2=e^{-\pi\alpha_2^2},
\]
and phase combinations \(\Phi_1,\Phi_2\). Their structure contains the characteristic DOM ingredients: Landau–Zener exponentials \(p_j\), coherent interference of multiple semiclassical paths through the three-level crossing, and phase factors involving \(\arg\Gamma(\cdot)\) [2603.22470].

In the physical application to unstable vacuum decay through a second-order phase transition, the Painlevé-II connection formulas determine the scaling of produced excitations. In the near-vacuum regime
\[
{\cal J}_1={\cal J}_2\equiv{\cal J}\ll 1,
\]
the phase-averaged quantities satisfy
\[
\overline{I_2+2I_1}\approx \frac{1}{2\pi}\ln\!\left(\frac{1}{2\pi{\cal J}}\right),
\]
\[
I_1\approx \frac{1}{4\pi}\left[\ln\!\left(\frac{1}{2\pi{\cal J}}\right)-c_1\right],
\qquad
c_1\approx 1.166,
\]
and
\[
I_2\approx \frac{c_2}{2\pi},
\qquad
c_2=c_1\approx 1.166.
\]
Semiclassically, \(I_1/\hbar\) is the number of Higgs-like excitations and \(I_2/\hbar\) the number of Goldstone-like excitations. In this sense, DOM is not merely a solvable crossing model but an exact local scattering mechanism that determines asymptotic data for an integrable nonlinear system [2603.22470].

Source: https://www.emergentmind.com/topics/demkov-osherov-model-dom