Transforming ODEs to Schrödinger-like equations

Many spectral problems in mathematical physics naturally arrive in a Schrödinger-like form

$$- \phi''(\rho) + V(\rho)\,\phi(\rho) = \omega^2 \phi(\rho),$$

where $\omega^2$ is the eigenvalue and $V(\rho)$ an effective potential. Often, however, one starts from more general second-order ODEs, possibly with first-derivative terms, variable weights on the right-hand side, and even matrix-valued couplings.

This note summarizes:

1. The final transformation formulas for a single scalar ODE.
2. The structure of coupled systems, with emphasis on the conditions under which they can be reduced to a set of decoupled Schrödinger equations.

The derivations are standard (Liouville / Sturm–Liouville theory); the goal here is to record the usable end results and the key conditions.

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1. Single equation: transformation summary

Consider the most general scalar second-order linear eigenvalue problem of the form

$$- \psi''(x) + P(x)\,\psi'(x) + Q(x)\,\psi(x) = \omega^2 F(x)^2\,\psi(x). \tag{1}$$

Here $P,Q,F$ are given real (or complex) functions, and $\omega^2$ is the spectral parameter. The following steps take (1) to standard Schrödinger form.

Step 1: remove the first derivative

Define

$$\psi(x) = e^{g(x)}\,u(x),\qquad g'(x) = \frac{P(x)}{2}.$$

Then $u(x)$ satisfies

$$- u''(x) + U(x)\,u(x) = \omega^2 F(x)^2\,u(x), \tag{2}$$

with

$$U(x) = Q(x) - \frac{1}{2}P'(x) - \frac{1}{4}P(x)^2.$$

So every equation of the form (1) is equivalent to a Sturm–Liouville-type equation (2) with no first-derivative term.

Step 2: flatten the weight $F(x)^2$

Introduce a new coordinate $\rho$ via

$$\frac{d\rho}{dx} = F(x),$$

and set

$$u(x) = F(\rho)^{-1/2}\,\phi(\rho),$$

where $F(\rho)$ means $F$ expressed as a function of $\rho$. Then $\phi$ obeys a Schrödinger equation

$$- \phi''(\rho) + V(\rho)\,\phi(\rho) = \omega^2 \phi(\rho), \tag{3}$$

with potential

$$V(\rho) = \frac{U(\rho)}{F(\rho)^2} + \frac{1}{2}\frac{F''(\rho)}{F(\rho)} - \frac{1}{4}\left(\frac{F'(\rho)}{F(\rho)}\right)^2.$$

Combining the steps:

Single-ODE transformation (summary).
Equation (1) is equivalent to the Schrödinger-like eigenvalue problem (3) by the change of variable $x\mapsto\rho$ and rescaling

$$\psi(x) = \exp\!\left(\frac{1}{2}\int^x P(s)\,ds\right) \,F(\rho)^{-1/2}\,\phi(\rho),\qquad \frac{d\rho}{dx} = F(x).$$

The transformation is invertible on any interval where $F(x)\neq 0$.

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2. Coupled ODEs: structure and conditions

We now turn to coupled systems. The key questions are:

1. Can we still remove all first-derivative terms?
2. Can the system be cast into a Schrödinger-like vector equation?
3. Under what conditions can we further decouple it into independent scalar Schrödinger equations?

We address these in increasing order of restrictiveness.

2.1 General setup

Let $\mathbf y(x)\in\mathbb C^n$ be a column vector of $n$ functions, and consider

$$- \mathbf y''(x) + P(x)\,\mathbf y'(x) + Q(x)\,\mathbf y(x) = \omega^2 F(x)^2\,\mathbf y(x). \tag{4}$$

Here

• $P(x),Q(x),F(x)^2$ are $n\times n$ matrix-valued coefficient functions,
• $\omega^2$ is the spectral parameter, and
• primes denote $d/dx$.

Important special cases:

• $P(x)$ diagonal (component-wise first-derivative terms),
• $F(x)^2$ diagonal, e.g. $F(x)^2 = \mathrm{diag}(f_1(x)^2,\dots,f_n(x)^2)$.

We now describe what can and cannot be done in general.

2.2 Removing first-derivative terms (always possible)

Set

$$\mathbf y(x) = S(x)\,\boldsymbol\phi(x),$$

with $S(x)\in GL(n,\mathbb C)$ invertible. A short computation gives an equation for $\boldsymbol\phi$ of the form

$$- \boldsymbol\phi''(x) + A(x)\,\boldsymbol\phi'(x) + B(x)\,\boldsymbol\phi(x) = \omega^2\,W(x)\,\boldsymbol\phi(x),$$

where

$$A(x) = 2S^{-1}S' + S^{-1}P S,\qquad W(x) = S^{-1}F^2 S.$$

To remove first derivatives, impose

$$2S'(x) + P(x)\,S(x) = 0. \tag{5}$$

This is a linear matrix ODE. For any smooth $P(x)$ and any nonsingular initial condition $S(x_0)$, it has a unique local solution with $S(x)\in GL(n,\mathbb C)$. With this choice, $A(x)\equiv 0$, and we obtain

$$- \boldsymbol\phi''(x) + U(x)\,\boldsymbol\phi(x) = \omega^2\,W(x)\,\boldsymbol\phi(x), \tag{6}$$

where

$$U(x) = S^{-1}\bigl(S'' + P S' + Q S\bigr), \qquad W(x) = S^{-1}F(x)^2 S(x).$$

So:

Fact 1. For any coupled system of the form (4), one can always choose $S(x)$ satisfying (5) so that the transformed variables $\boldsymbol\phi(x)=S(x)^{-1}\mathbf y(x)$ satisfy (6) with no first-derivative term.

This is the exact analogue of the scalar case, just with $S(x)$ a matrix.

2.3 Schrödinger-like vector equation: condition on the weight

Equation (6) is a matrix Sturm–Liouville problem: it has a matrix potential $U(x)$ and a matrix weight $W(x)$ multiplying the eigenvalue.

We now ask whether we can reduce it to a vector Schrödinger equation

$$- \boldsymbol\Phi''(\rho) + V(\rho)\,\boldsymbol\Phi(\rho) = \omega^2 \boldsymbol\Phi(\rho), \tag{7}$$

where the eigenvalue multiplies the identity matrix. We allow:

• a scalar change of variable $\rho=\rho(x)$, and
• a further matrix rescaling $\boldsymbol\phi(x)=T(\rho)\,\boldsymbol\Phi(\rho)$.

Under such transformations, the weight matrix $W(x)$ in (6) becomes

$$\widetilde W(\rho) = \frac{1}{(\rho'(x))^2}\,T(\rho)^{-1}W(x)T(\rho).$$

To reach (7), we must have $\widetilde W(\rho) = I$ for all $\rho$, i.e.

$$T(\rho)^{-1} W(x) T(\rho) = \rho'(x)^2 I.$$

However, conjugation does not change eigenvalues: at each $x$ the matrix on the left has the same eigenvalues as $W(x)$, whereas the matrix on the right has all eigenvalues equal to $\rho'(x)^2$. Therefore this is possible if and only if, at each $x$,

$$W(x) \ \text{is proportional to the identity matrix.}$$

In other words:

Fact 2. (Vector Schrödinger form)
The system (6) can be transformed to the Schrödinger-like vector equation (7) by a scalar reparametrization and matrix rescaling if and only if

$$W(x) = c(x)\,I_n \quad\text{for some scalar function } c(x).$$

In the common specialization where $F^2$ is diagonal,

$$F^2(x) = \mathrm{diag}(f_1(x)^2,\dots,f_n(x)^2),$$

and $S(x)$ is chosen, for example, to be diagonal, this condition reduces to

$$f_1(x)^2 = f_2(x)^2 = \dots = f_n(x)^2 \quad\text{for all }x.$$

When this holds, one may choose $\rho$ such that $d\rho/dx = f(x)$ with $f(x)^2=f_i(x)^2$, and (6) becomes (7) with some matrix potential $V(\rho)$.

If the eigenvalues of $W(x)$ are not all equal, no combination of scalar reparametrizations and similarity transformations can make the weight equal to the identity for all $x$; one is left with a genuine matrix Sturm–Liouville problem with a nontrivial weight.

2.4 Full decoupling into independent scalar Schrödinger equations

Assume now that Fact 2 holds (so we have brought the system to (7)). We ask: when can we find a constant invertible matrix $S_0$ such that

$$\boldsymbol\chi(\rho) = S_0^{-1}\boldsymbol\Phi(\rho)$$

satisfies $n$ independent scalar equations

$$- \chi_i''(\rho) + V_i(\rho)\,\chi_i(\rho) = \omega^2 \chi_i(\rho), \qquad i=1,\dots,n?$$

In terms of $V(\rho)$ this means we want

$$S_0^{-1} V(\rho) S_0 \quad\text{to be diagonal for all }\rho.$$

Equivalently:

There must exist $n$ linearly independent constant vectors $v_1,\dots,v_n$ and scalar functions $V_i(\rho)$ such that

$$V(\rho)\,v_i = V_i(\rho)\,v_i \quad\text{for all }\rho,\ i=1,\dots,n.$$

The $v_i$ are an $\rho$-independent eigenbasis for the family of matrices $\{V(\rho)\}$. Packing them as columns of $S_0$, we have

$$V(\rho) = \sum_{i=1}^n V_i(\rho)\,P_i,\qquad P_i = v_i w_i^{\mathsf T},$$

where $w_i^{\mathsf T}$ are the dual covectors and $P_i$ are constant rank-one projectors satisfying

$$P_iP_j = \delta_{ij}P_i,\qquad \sum_{i=1}^n P_i = I.$$

So the decoupling condition can be phrased invariantly as:

Fact 3. (Decoupling condition)
The Schrödinger vector equation (7) can be reduced by a constant similarity transformation to $n$ independent scalar equations if and only if there exists a set of constant projectors $P_1,\dots,P_n$ with

$$P_iP_j = \delta_{ij}P_i,\qquad \sum_i P_i=I,$$

and scalar functions $V_i(\rho)$ such that

$$V(\rho) = \sum_{i=1}^n V_i(\rho)\,P_i \quad\text{for all }\rho.$$

In more elementary language: all matrices $V(\rho)$ must share the same eigenvectors; only their eigenvalues are allowed to depend on $\rho$.

When $V(\rho)$ is diagonalizable for each $\rho$, a useful sufficient condition is:

• the matrices $\{V(\rho)\}$ commute pairwise, and
• at least one $V(\rho)$ has a simple (nondegenerate) spectrum.

Then the family can be simultaneously diagonalized by a constant matrix $S_0$.

Two-equation case ($n=2$)

For $n=2$ one can make the decoupling condition more explicit. After going through

1. first-derivative removal (Fact 1), and
2. weight flattening (Fact 2, with $f_1^2=f_2^2$),

the system can be written as

$$- \begin{pmatrix}\phi_1\\\phi_2\end{pmatrix}'' + \begin{pmatrix}a(\rho) & b(\rho)\\ c(\rho) & d(\rho)\end{pmatrix} \begin{pmatrix}\phi_1\\\phi_2\end{pmatrix} = \omega^2 \begin{pmatrix}\phi_1\\\phi_2\end{pmatrix}.$$

Decoupling asks for a constant change of basis $\boldsymbol\chi = S_0^{-1}\boldsymbol\phi$ such that the potential matrix is diagonal in that basis for all $\rho$. Equivalently, one seeks constant coefficients $\kappa$ such that combinations

$$u(\rho) = \phi_1(\rho) + \kappa\,\phi_2(\rho)$$

are eigenvectors of the matrix potential for all $\rho$. This leads to the quadratic condition

$$c(\rho)\,\kappa^2 + (a(\rho)-d(\rho))\,\kappa - b(\rho) = 0.$$

Requiring two constant roots $\kappa_{1,2}$ for all $\rho$ forces the coefficients of this quadratic to be proportional as functions of $\rho$. Assuming $c(\rho)\not\equiv 0$, this is equivalent to

$$\frac{b(\rho)}{c(\rho)} = \text{const},\qquad \frac{a(\rho)-d(\rho)}{c(\rho)} = \text{const},$$

i.e. two independent functional constraints among $a,b,c,d$ for the two-component system.

This is the $n=2$ version of Fact 3 in components.

2.5 Summary of conditions for coupled systems

Putting everything together, a coupled $n$-component system of the form (4) admits a reduction to decoupled scalar Schrödinger equations

$$- \chi_i''(\rho) + V_i(\rho)\,\chi_i(\rho) = \omega^2 \chi_i(\rho)$$

if and only if the following hold, at least locally:

1. First-derivative removal (always possible):
   There exists $S(x)\in GL(n)$ solving

   $$2S'(x) + P(x)S(x) = 0,$$

   giving (6).

2. Scalar weight condition (Schrödinger vector form):
   The resulting weight matrix $W(x)=S^{-1}F^2S$ is proportional to the identity,

   $$W(x) = c(x)I_n.$$

   In particular, if $F^2$ is diagonal with entries $f_i(x)^2$, this requires $f_1^2=\dots=f_n^2$. One then chooses a new coordinate $\rho$ with $d\rho/dx=\sqrt{c(x)}$ to obtain (7).

3. Common eigenbasis of the potential (decoupling):
   The matrix potential $V(\rho)$ in (7) has an $\rho$-independent eigenbasis. Equivalently, there exist constant projectors $P_i$ and scalar potentials $V_i(\rho)$ such that

   $$V(\rho) = \sum_{i=1}^n V_i(\rho)\,P_i.$$

For generic data, these conditions impose $n^2-n$ independent functional constraints among the entries of the matrix potential (after weight flattening), as can be seen by counting degrees of freedom: a general $n\times n$ matrix has $n^2$ independent entries, whereas a diagonalizable matrix with fixed eigenvectors but $\rho$-dependent eigenvalues has only $n$ functional degrees of freedom.

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3. Closing remarks

• For a single ODE, transforming to Schrödinger form is always possible under mild regularity assumptions; the formulas are explicit and widely used.
• For coupled systems, one can always eliminate first derivatives, but turning the result into a scalar-weight Schrödinger-type vector equation already imposes a nontrivial condition on the weight matrix.
• Full decoupling into independent scalar Schrödinger equations requires a further strong condition: the family of matrix potentials must share a common, $x$-independent eigenbasis.

These conditions often appear implicitly when one searches for normal modes that separate cleanly in multi-component systems (e.g. coupled perturbations in curved spacetime or multi-field models), and make precise when such decoupling is mathematically possible.