Semiclassical tunneling for some 1D Schrödinger operators with complex-valued potentials

This paper establishes a semiclassical tunneling result for a non-selfadjoint Schrödinger operator with a complex-valued potential, demonstrating that its low-lying spectrum consists of exponentially close eigenvalue pairs whose gaps exhibit a distinct rotational behavior as the semiclassical parameter vanishes when the potential's phase is non-zero.

The Gist

Imagine you are a tiny quantum particle, like an electron, trapped in a landscape. In the classic version of this story (the "self-adjoint" case), the landscape is a valley with two deep pits (wells) separated by a high hill.

Normally, if you are in the left pit, you stay there. But because you are a quantum particle, you have a spooky ability called tunneling. You can occasionally "dig" through the hill and pop up in the right pit.

This paper explores what happens when we change the rules of the game. Instead of a standard, flat landscape, we introduce a complex-valued potential. In physics terms, this means the "hill" isn't just a barrier; it has a twist, a phase, or a rotation to it (represented by the angle $\alpha$).

Here is the breakdown of what the authors discovered, using simple analogies:

1. The Setup: Two Pits and a Twisted Hill

Think of the two pits as Lefty and Righty.

• The Old Way (Real Potential): If you put a particle in Lefty, it eventually tunnels to Righty. The time it takes depends on how high the hill is. The energy levels of the particle in Lefty and Righty are almost identical, but not quite. They form a "pair" of energy levels that are extremely close together.
• The New Way (Complex Potential): The authors added a "twist" to the hill. Imagine the hill isn't just a wall; it's a wall that spins or rotates as you try to climb it. This makes the math much harder because the system is no longer "symmetric" in the usual sense (it's non-self-adjoint).

2. The Big Surprise: The Tunneling Doesn't Break

When you introduce a twist to a system, you often expect things to break or cancel out.

• The Fear: The researchers wondered: Will the twist cause the particle's wave to interfere with itself destructively? Will the tunneling stop? Will the energy gap between the two states disappear?
• The Result: No! The tunneling still happens. In fact, the "gap" (the tiny difference in energy between the two states) actually grows as you increase the twist. The particle still tunnels, but it does so with a new, exotic rhythm.

3. The "Dancing" Eigenvalues

In the old world, the two energy levels were like two twins standing side-by-side.
In this new twisted world, the two energy levels are like dancers spinning around each other.

• As the "quantum scale" ($h$, which is like the size of the particle) gets smaller, these two energy levels don't just get closer; they rotate around one another in the complex plane.
• The authors calculated exactly how fast they spin and how far apart they are. It turns out the "distance" between them involves a complex number, meaning the gap has both a size and a direction (an angle).

4. The "Agmon Distance": The New Map

To calculate how hard it is to tunnel, physicists use a concept called the Agmon distance.

• Analogy: Think of this as the "effective cost" of walking through the hill. In the old world, the cost was just the height of the hill.
• The Twist: In this paper, the "cost" of walking through the hill is now a complex number. It's like the hill has a "temperature" and a "direction" to it. The authors found a formula for this new, twisted distance. It turns out that even though the math is complex, the tunneling probability is determined by the real part of this new distance.

5. Why This Matters

You might ask, "Why do we care about a twisted hill?"

• Real-World Physics: Complex potentials often appear in physics when dealing with absorption (like light being absorbed by a material) or gain (like in lasers). They are also used to model systems with magnetic fields or PT-symmetry (a special kind of symmetry where time and space are flipped).
• The Takeaway: This paper proves that even when the rules get weird (complex numbers, non-symmetric forces), the fundamental phenomenon of quantum tunneling is robust. It doesn't vanish; it just changes its dance steps.

Summary in a Nutshell

Imagine two rooms (wells) separated by a door (barrier).

• Normal Physics: You can sneak through the door. The time it takes is predictable.
• This Paper: The door is now a spinning, magical portal.
• The Discovery: You can still sneak through! The portal doesn't block you; it just makes you spin as you pass. The authors figured out exactly how fast you spin and how the "cost" of passing through changes. They showed that the "magic" of quantum tunneling is strong enough to survive even the weirdest twists in the laws of physics.

The Bottom Line: Quantum particles are like acrobats. Even if you put a spinning, twisting obstacle in their path, they don't fall off the tightrope; they just learn a new, more complex way to balance.

Technical Summary

1. Problem Statement

The paper investigates the semiclassical spectral properties of a non-selfadjoint Schrödinger operator in one dimension with a complex-valued potential. The operator is defined as:
$$L(h) = -h^2 \partial_x^2 + e^{i\alpha}V(x)$$
where:

• $h > 0$ is the semiclassical parameter ($h \to 0$).
• $\alpha \in (-\pi, \pi)$ is a fixed phase parameter.
• $V: \mathbb{R} \to \mathbb{R}^+$ is a smooth, even potential with exactly two non-degenerate global minima at $x_\ell$ and $x_r = -x_\ell$.
• $V$ vanishes only at these two points and grows at infinity.

The Core Challenge:
In the selfadjoint case ($\alpha = 0$), the low-lying spectrum consists of pairs of exponentially close eigenvalues due to quantum tunneling between the two wells. The gap between the first two eigenvalues is governed by the Agmon distance.
When $\alpha \neq 0$, the operator $L(h)$ is non-selfadjoint. This introduces significant difficulties:

1. The spectrum is no longer necessarily real.
2. Standard spectral theorems (e.g., for selfadjoint operators) do not apply.
3. It is unclear if the tunneling phenomenon persists, how the eigenvalue gap behaves, and whether destructive interference (common in non-selfadjoint settings) might cause the gap to collapse or behave qualitatively differently.

2. Methodology

The authors employ a rigorous semiclassical analysis combining several advanced techniques adapted for non-selfadjoint operators:

A. Decoupling and Single-Well Analysis

The strategy involves "sealing" the potential wells to define single-well operators $L_\ell(h)$ and $L_r(h)$.

• Exponential Localization: Using Agmon-type estimates, the authors prove that eigenfunctions of the single-well operators are exponentially localized near their respective minima. This relies on constructing a specific weight function $\Phi$ and proving an elliptic estimate for the conjugated operator $e^{\Phi/h} L(h) e^{-\Phi/h}$.
• Resolvent Bounds: Since the spectral theorem is unavailable, the authors derive precise resolvent bounds for the single-well operator. They utilize the fact that near the minima, the operator behaves like a complex harmonic oscillator ($H(h) = (hD_x)^2 + e^{i\alpha}x^2$). By relating the resolvent of $L_\ell(h)$ to that of the complex harmonic oscillator, they establish the existence and algebraic simplicity of eigenvalues.

B. WKB Approximations

The authors construct WKB (Wentzel-Kramers-Brillouin) quasimodes for the single-well operators.

• They derive formal asymptotic expansions for eigenvalues and eigenfunctions.
• Crucially, they prove the optimality of these approximations: the WKB ansatz describes the true spectrum of the single-well operator with exponential accuracy.
• The phase function $\phi_\ell(x)$ is complex-valued: $\phi_\ell(x) = e^{i\alpha/2} \int_{x_\ell}^x \sqrt{V(s)} ds$.

C. Interaction and Spectral Gap Calculation

To analyze the double-well operator $L(h)$, the authors:

1. Quasi-Orthogonality: Prove that the eigenfunctions of the left and right wells are almost orthogonal, with an overlap decaying as $O(e^{-\text{Re}(S(\alpha))/h})$.
2. Riesz Projectors: Approximate the Riesz projector of the double-well operator by the sum of the Riesz projectors of the single-well operators. This allows them to reduce the spectral problem to a $2 \times 2$ matrix problem in the subspace spanned by the localized states.
3. Wronskian Calculation: The eigenvalue gap is computed via the interaction term, which is expressed using the Wronskian of the WKB solutions at the midpoint between the wells.

3. Key Contributions and Results

Main Theorem (Theorem 1.1)

The paper establishes that for small $h$, the spectrum of $L(h)$ near the origin consists of pairs of algebraically simple eigenvalues. Specifically, the two eigenvalues with the smallest modulus, $\mu_1(h)$ and $\mu_2(h)$, satisfy:
$$\mu_2(h) - \mu_1(h) = \left( A + o_{h\to 0}(1) \right) h^{1/2} e^{-S(\alpha)/h}$$
where:

• Complex Agmon Distance: $S(\alpha) = e^{i\alpha/2} \int_{x_\ell}^{x_r} \sqrt{V(s)} ds$.
• Prefactor: $A$ is an explicit constant depending on $\alpha$ and the curvature of $V$ at the minima.
• Gap Behavior: The magnitude of the gap is $|\mu_2 - \mu_1| \sim |A| \sqrt{h} e^{-\text{Re}(S(\alpha))/h}$.

Key Findings

1. Persistence of Tunneling: Despite the non-selfadjoint nature of the operator, the tunneling phenomenon persists. The eigenvalues still come in exponentially close pairs.
2. No Destructive Interference Collapse: Contrary to expectations in some non-selfadjoint contexts where interference might cancel the tunneling amplitude, the gap here does not collapse. The magnitude of the gap actually increases as $\alpha$ increases (due to the factor $e^{-\text{Re}(S(\alpha))/h} = e^{-\cos(\alpha/2)S/h}$).
3. Rotational Dynamics: When $\alpha \neq 0$, the eigenvalues are complex. The difference $\mu_2 - \mu_1$ has a phase that scales as $1/h$. As $h \to 0$, the two eigenvalues rotate rapidly around each other in the complex plane.
4. Generalized Agmon Distance: The tunneling rate is governed by the real part of the complex Agmon distance $S(\alpha)$. The imaginary part of $S(\alpha)$ dictates the rotation of the eigenvalues.
5. Dynamical Interpretation: The paper provides a dynamical interpretation of the gap. For $\alpha \neq 0$, a state initially localized in one well evolves towards a state with equal mass in both wells (due to the imaginary part of the eigenvalues dominating the decay/growth), whereas for $\alpha=0$, it exhibits oscillatory tunneling.

4. Significance and Impact

• Mathematical Physics: This work extends the foundational Helffer-Sjöstrand theory of semiclassical tunneling to the non-selfadjoint regime. It resolves the question of whether complex potentials destroy the tunneling mechanism, showing instead that the mechanism is robust but modified by complex phase factors.
• Non-Selfadjoint Spectral Theory: The paper provides a blueprint for analyzing non-selfadjoint operators where the spectral theorem fails. The use of complex harmonic oscillators as a proxy for resolvent bounds and the careful construction of subsolutions for elliptic estimates are methodological advances.
• Physical Applications: Complex potentials are used to model absorption, gain/loss systems, and open quantum systems (e.g., in optics or superconductivity). Understanding the spectral gap in these systems is crucial for predicting stability and transport properties. The result suggests that in PT-symmetric or similar systems, tunneling is not suppressed but rather modified in a predictable way.
• Higher Dimensions: The authors note that 1D results are often the building blocks for higher-dimensional problems via dimensional reduction (Feshbach/Grushin methods). This work lays the groundwork for analyzing tunneling in double-well magnetic fields or other complex potential landscapes in higher dimensions.

In summary, the paper demonstrates that semiclassical tunneling in 1D with complex potentials is a well-defined phenomenon characterized by a complex-valued Agmon distance, leading to a spectral gap that rotates rapidly in the complex plane but maintains a non-vanishing magnitude determined by the real part of the distance.