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Eigenfunctions of deformed Schrödinger equations

Matijn François, Alba Grassi, Tommaso Pedroni

TL;DR

This work introduces a solvable deformed Schrödinger-type problem given by the finite-difference operator $H_N = 2\cosh(p) + V_N(x)$, where $V_N$ is a degree-$N$ polynomial, arising from Seiberg–Witten quantization of SU($N$) gauge theories. Using the open TS/ST correspondence, the authors construct exact analytic off-shell eigenfunctions that are entire in $x$ and become $L^2$-normalizable only at discrete energies, with even/odd $N$ producing bound and resonant states, respectively. The generalized Matone relations and quantum mirror maps relate the moduli $h_k$ to Coulomb-branch parameters, and explicit eigenfunctions are provided for both confining ($N$ even) and unbounded ($N$ odd) potentials, including illustrative cubic and quartic examples. In the four-dimensional limit, these open-string constructions reproduce the deformed quantum Seiberg–Witten eigenfunctions, connecting gauge theory, topological strings, and quantum integrable systems and offering a rare exactly solvable quantum problem with explicit bound and resonant states.

Abstract

We study the spectral problems associated with the finite-difference operators $H_N = 2 \cosh(p) + V_N(x)$, where $V_N(x)$ is an arbitrary polynomial potential of degree $N$. These systems can be regarded as a solvable deformation of the standard Schrödinger operators $p^2 + V_N(x)$, and they arise naturally from the quantization of the Seiberg-Witten curve of four-dimensional, $\mathcal{N} = 2$, SU($N$) supersymmetric Yang-Mills theory. Using the open topological string/spectral theory correspondence, we construct exact, analytic eigenfunctions of $H_N$, valid for arbitrary polynomial potentials and describing both bound and resonant states. Our solutions are entire in $x$ for generic values of the energy, and become $L^2$-normalizable only at a discrete set of energies. An interesting feature of these Hamiltonians is the existence of special loci in the parameter space of the potential, the so-called Toda points. The eigenfunctions exhibit enhanced decay at these points, leading to spectral degeneracies for confining potentials and to a real energy spectrum for unbounded ones. Our results provide a rare example of a quantum-mechanical spectral problem that is exactly solvable, admitting explicit, analytic eigenfunctions for both bound and resonant states.

Eigenfunctions of deformed Schrödinger equations

TL;DR

This work introduces a solvable deformed Schrödinger-type problem given by the finite-difference operator , where is a degree- polynomial, arising from Seiberg–Witten quantization of SU() gauge theories. Using the open TS/ST correspondence, the authors construct exact analytic off-shell eigenfunctions that are entire in and become -normalizable only at discrete energies, with even/odd producing bound and resonant states, respectively. The generalized Matone relations and quantum mirror maps relate the moduli to Coulomb-branch parameters, and explicit eigenfunctions are provided for both confining ( even) and unbounded ( odd) potentials, including illustrative cubic and quartic examples. In the four-dimensional limit, these open-string constructions reproduce the deformed quantum Seiberg–Witten eigenfunctions, connecting gauge theory, topological strings, and quantum integrable systems and offering a rare exactly solvable quantum problem with explicit bound and resonant states.

Abstract

We study the spectral problems associated with the finite-difference operators , where is an arbitrary polynomial potential of degree . These systems can be regarded as a solvable deformation of the standard Schrödinger operators , and they arise naturally from the quantization of the Seiberg-Witten curve of four-dimensional, , SU() supersymmetric Yang-Mills theory. Using the open topological string/spectral theory correspondence, we construct exact, analytic eigenfunctions of , valid for arbitrary polynomial potentials and describing both bound and resonant states. Our solutions are entire in for generic values of the energy, and become -normalizable only at a discrete set of energies. An interesting feature of these Hamiltonians is the existence of special loci in the parameter space of the potential, the so-called Toda points. The eigenfunctions exhibit enhanced decay at these points, leading to spectral degeneracies for confining potentials and to a real energy spectrum for unbounded ones. Our results provide a rare example of a quantum-mechanical spectral problem that is exactly solvable, admitting explicit, analytic eigenfunctions for both bound and resonant states.

Paper Structure

This paper contains 27 sections, 132 equations, 10 figures.

Table of Contents

  1. Introduction
  2. The spectral problem and its solution
  3. The spectral problem
  4. N even: confining potential.
  5. N odd: unbounded potential.
  6. The generalized Matone relations
  7. The eigenfunctions
  8. N even: confining potential.
  9. N odd: unbounded potential.
  10. Examples
  11. Cubic potentials
  12. Quartic potentials
  13. Comments on the Toda points
  14. Derivation from the TS/ST correspondence
  15. Open TS/ST for the Y(N, 0) geometries
  16. ...and 12 more sections

Figures (10)

  • Figure 1: Numerical ground state eigenfunction ($E_0\approx 0.54151 + {\rm i} \, 0.25905$) of \ref{['eq:hn']} with $V_3(x) = x^3$, obtained via complex dilatation. Left: rotated eigenfunction $\psi_0^{\theta}(x,\boldsymbol{h})$. Right: true eigenfunction $\psi_0(x,\boldsymbol{h}) = \psi_0^{\theta}({\rm e}^{-{\rm i} \theta}x,\boldsymbol{h})$. Here we set $\Lambda=1/2$, $\hbar=1$ and $\theta=-1/10$. The eigenfunction is normalized so that $\psi_0(0,\boldsymbol{h})=1$. Solid lines denote the real part; dashed lines denote the imaginary part.
  • Figure 2: Left: third excited state of the $V_3(x)$ potential with $h_2=0$, $\hbar=1$, and $\Lambda=\tfrac{1}{2}$. Dashed lines denote the imaginary part of the eigenfunction, while solid lines denote the real part. Right: difference between the numerical eigenfunction and the analytic expression from \ref{['eq:eigenfeven']}. The coloured curves show the effect of including an increasing number of terms in the $\Lambda$-expansion of the eigenfunction: red (0 terms), green (1 term), blue (2 terms).
  • Figure 3: Off-shell eigenfunction of the potential $V_3(x)$ with $h_2=-4$, $h_3= -2.5$, $\hbar=1$ and $\Lambda=2/5$. From left to right: first saddle, second saddle and the full eigenfunction given by \ref{['eq:eigenfodd']}. Dashed lines denote the imaginary part, while solid lines denote the real part.
  • Figure 4: The ground state of the $V_4(x)$ potential with $h_2=-3$, $h_3=-0.1$, $\Lambda=(1/2)^{1/2}$ and $\hbar=1$; the corresponding energy is $E_0 \approx -0.527521$. Dashed lines represent the imaginary part of the eigenfunction, while solid lines represent the real part. Left: illustration of the failure of the oscillation theorem. Centre: eigenfunction obtained from \ref{['eq:eigenfeven']}. Right: difference between the numerical eigenfunction and the analytic expression from \ref{['eq:eigenfeven']}. The coloured curves show the effect of including an increasing number of terms in the $\Lambda$-expansion of the eigenfunction: red (0 terms), green (1 term), blue (2 terms).
  • Figure 5: Off-shell eigenfunction of the potential $V_4(x)$ with $h_2 = -1.6$, $h_3 = 0.45$, $h_4 = 0.06$, $\hbar = 1$ and $\Lambda = 13/23$. From left to right: first saddle, second saddle and the full eigenfunction given by \ref{['eq:eigenfeven']}. Dashed lines represent the imaginary part, while solid lines the real part.
  • ...and 5 more figures