🔷 Where You Stand (Important)

You have already worked on:

  1. One-parameter rationally extended harmonic oscillator
  2. Multi-dimensional rational extensions

This means you are already familiar with:

  • Shape invariance beyond textbook forms
  • Exceptional orthogonal polynomials (EOPs)
  • Higher-order SUSY / Darboux transformations
  • Multi-dimensional factorization

So do not go back to:
❌ basic Morse / Coulomb
❌ first-order SUSY examples
❌ standard shape-invariant lists

🔥 HIGH-VALUE RESEARCH DIRECTIONS (Strongly Recommended)

1️⃣ Multi-Step / Higher-Order SUSY in Multi-Dimensions

Natural next step from your work

Problem ideas:

  • Construct two-step and k-step SUSY chains for rationally extended oscillators in (D > 1)
  • Study spectral degeneracy breaking due to higher-order intertwining operators
  • Explore reducible vs irreducible SUSY in higher dimensions

📌 Why important:
Almost no systematic classification exists for higher-order SUSY in multi-D systems.

📌 Possible outcome:
A clean classification paper.

2️⃣ Non-Hermitian / PT-Symmetric SUSY-QM

Hot and under-explored

Concrete problems:

  • Construct PT-symmetric rational extensions of:
    • Harmonic oscillator
    • Scarf / Rosen–Morse potentials
  • Analyze:
    • Reality of spectrum
    • SUSY breaking/restoration
  • Study biorthogonal EOPs

📌 Why publishable:
PT-SUSY is still poorly understood mathematically.

📌 Keywords to use:
PT symmetry, pseudo-Hermiticity, non-Hermitian SUSY

3️⃣ Position-Dependent Mass (PDM) SUSY-QM

Very strong direction

Research problems:

  • Construct SUSY partner Hamiltonians with: [ H = -\frac{1}{2}\frac{d}{dx}\left(\frac{1}{m(x)}\frac{d}{dx}\right) + V(x) ]
  • Find shape-invariant PDM systems
  • Build rational extensions with EOPs

📌 Why valuable:
Relevant to semiconductors, quantum wells, nanostructures.

📌 Almost guaranteed publication if done cleanly.

4️⃣ Matrix SUSY-QM (Coupled Channels)

Advanced but powerful

Problems:

  • Construct 2×2 or 3×3 SUSY partner Hamiltonians
  • Study:
    • Spin-orbit coupling
    • Coupled oscillators
  • Develop matrix shape invariance

📌 Why rare:
Very few researchers work on matrix SUSY systematically.

📌 Ideal if you like algebra.

5️⃣ Conditional and Broken Shape Invariance

Mathematically deep

Possible projects:

  • Systems where shape invariance holds only for specific parameter values
  • Explore spontaneously broken SUSY
  • Study zero-mode disappearance

📌 Connection:
Links SUSY-QM to phase transitions.

🚀 MODERN / EMERGING DIRECTIONS (High Impact)

6️⃣ SUSY-QM on Curved Spaces

Examples:

  • SUSY on:
    • Sphere (S²)
    • Hyperbolic space
  • Rational extensions in curved backgrounds

📌 Why interesting:
Links SUSY-QM with gravity and geometry.

7️⃣ Exceptional Orthogonal Polynomials: Classification Problems

Instead of building potentials:

  • Classify new EOP families
  • Study completeness and orthogonality rigorously
  • Multi-index EOP systems

📌 Highly mathematical, very publishable

8️⃣ Supersymmetry + Integrability

  • Connection between:
    • SUSY-QM
    • Painlevé equations
    • Lax pairs
  • Deformed oscillator algebras

📌 Strong journals like JMP, JPA accept these

📌 SAFE, PUBLISHABLE PROJECT IDEAS (If You Want Fast Output)

Here are three concrete paper-ready problem statements:

🟢 Project A

“Higher-order supersymmetric constructions of rationally extended multi-dimensional harmonic oscillators”

🟢 Project B

“PT-symmetric rational extensions and SUSY breaking in non-Hermitian quantum mechanics”

🟢 Project C

“Shape-invariant position-dependent mass Hamiltonians via supersymmetric quantum mechanics”

🎯 How to Decide What to Work On

Ask yourself:

  • Algebra lover? → Matrix SUSY, higher-order SUSY
  • Physics applications? → PDM SUSY, PT-SUSY
  • Pure math elegance? → EOP classification
  • Fast publication? → PDM or PT-SUSY

🧭 Next Step (Important)

If you want, I can:

  • Help you choose one direction
  • Propose a precise research problem
  • Draft a paper outline (Introduction → Results → Appendix)
  • Suggest journals suited to your topic