Threshold Virtual States of a Jacobi operator

Abstract: We prove that the set of parameters for which a virtual level appears at the edge of the continuous spectrum of a Jacobi matrix with a finite-rank diagonal perturbation constitutes an algebraic variety of codimension one. This variety partitions the parameter space into connected components, with their number determined by the size of the perturbation support. We also reveal a hierarchical structure underlying these critical varieties as the rank of the perturbation increases.

• The paper establishes that threshold virtual states arise as zeros of a multivariate Jost function, linking spectral transitions to real affine algebraic varieties.
• It employs a recursive three-term polynomial scheme to map parameter space, showing how boundary crossings lead to discrete eigenvalue creation or annihilation.
• The study reveals a hierarchical, symmetric structure in finite-rank perturbations, offering clear criteria for virtual state formation in lattice Schrödinger models.

Threshold Virtual States and Algebraic Structures in Finite-Rank Jacobi Operator Perturbations

Introduction

This work investigates the emergence and structure of threshold (virtual) states in Jacobi operators subject to finite-rank diagonal perturbations. The context is provided by discrete Schrödinger operators representing, for example, the dynamics of two spinless bosons on a lattice with vanishing center-of-mass momentum. The principal contribution is a rigorous characterization of the parameter sets for which virtual states arise at the edges of the absolutely continuous spectrum and a detailed examination of the underlying algebraic and geometric structure of these sets. The appearance of virtual states is shown to correspond to the vanishing of a multivariate Jost function, translating the spectral problem into the study of real affine algebraic varieties.

Jacobi Operator Perturbations and the Jost Function Framework

The underlying Jacobi matrix is

$$J = \begin{pmatrix} 2 & -\sqrt{2} & 0 & \cdots \ -\sqrt{2} & 2 & -1 & \cdots \ 0 & -1 & 2 & \cdots \ \vdots & \vdots & \vdots & \ddots \end{pmatrix}$$

which is perturbed by a diagonal finite-rank operator $V = \mathrm{diag}(v_1, ..., v_n, 0, ...)$ restricted to the first $n$ basis vectors. The spectral analysis proceeds via the associated Jost function, constructed from solutions asymptotic to Floquet (i.e., Bloch) waves and intimately tied to the perturbation determinant. The Jost function for a finite-rank perturbation is a multivariate polynomial encoded recursively.

The essential technical construct is that threshold virtual states correspond to the zeros of the so-called threshold Jost function $C_n$, with $C_n = Q_n - Q_{n-1}$, where the $Q_k$ are polynomials satisfying a three-term recursion governed by the sequence of potential values.

Algebraic Geometry of Virtual State Manifolds

The work makes explicit that the locus in parameter space $(v_1, ..., v_n)$ for which a virtual level occurs forms a real affine algebraic variety of codimension one. This variety is identified as the zero set of the threshold Jost function. The key results include:

• The zero set $\mathbf{V}(C_n)$ is a smooth, connected algebraic hypersurface partitioning the parameter space $\mathbb{R}^n$ into $n+1$ open, unbounded connected components.
• As one crosses a component boundary in parameter space (i.e., crosses the critical variety $V = \mathrm{diag}(v_1, ..., v_n, 0, ...)$0), the number of bound states below or above the essential spectrum alters by exactly one.
• The number and location of these components are determined inductively via the three-term recursion for the polynomials $V = \mathrm{diag}(v_1, ..., v_n, 0, ...)$1, leading to a well-defined hierarchy of parameter space stratification.
• For each $V = \mathrm{diag}(v_1, ..., v_n, 0, ...)$2, the associated critical variety $V = \mathrm{diag}(v_1, ..., v_n, 0, ...)$3 admits a recursive description in terms of graphs of rational functions constructed from the $V = \mathrm{diag}(v_1, ..., v_n, 0, ...)$4.

A symmetry principle for the Jost function links left and right threshold phenomena by an involutive map on both parameter and spectral coordinates.

Numerical and Spectral Implications

The sharp spectral implications of this structure are as follows:

• For a non-critical operator (no virtual levels at thresholds), the region in parameter space specifies the exact number of negative eigenvalues (below the continuum) and positive eigenvalues above the spectrum. The algebraic structure ensures that every time a parameter path crosses a component boundary defined by $V = \mathrm{diag}(v_1, ..., v_n, 0, ...)$5, a single eigenvalue transitions to or from the continuum threshold.
• In the presence of a threshold virtual state, the system is at a critical point: the number of discrete eigenvalues below/above the continuum remains fixed except that an infinitesimal parameter change can create or annihilate a threshold eigenvalue.
• For large coupling (potential strengths), precise results describe the allocation of $V = \mathrm{diag}(v_1, ..., v_n, 0, ...)$6 negative and $V = \mathrm{diag}(v_1, ..., v_n, 0, ...)$7 positive eigenvalues in terms of the indices of the component of the parameter space.

Hierarchical Structure and Inductive Algorithm

A central technical result is the explicit inductive construction and algorithm for tracking the partitioning of parameter space. The zero set of $V = \mathrm{diag}(v_1, ..., v_n, 0, ...)$8 (the critical variety) is shown to have a laminar, hierarchical structure. The recurrence relations provide a dynamical system on the space of singularities—every increase in perturbation rank ($V = \mathrm{diag}(v_1, ..., v_n, 0, ...)$9) gives rise to new graphs and domains, all linked by the recurrence, rational function structure, and explicit shift invariance.

The work establishes that for finite-rank diagonal perturbations of support size $n$0, the critical varieties stratify $n$1 robustly, with all boundaries and components constructed in closed form, and the spectral transition mechanism is mapped bijectively to the crossing of these boundaries.

Implications and Future Directions

On a practical level, this framework provides a complete and explicit criterion for virtual state formation in finite-rank-perturbed discrete Schrödinger operators. From a theoretical standpoint, the reduction of the spectral transition (bound state creation/annihilation at thresholds) to the real affine algebraic geometry of Jost function zeros links this area with the long-standing problems in real algebraic geometry (cf. aspects of Hilbert's 16th problem referenced in the paper). The rich recursive and symmetric structure suggests avenues for generalization to other symmetry classes and higher-rank perturbations, including more complex interactions and applications to lattice multi-particle systems.

Conclusion

This study offers a comprehensive characterization of threshold virtual states for Jacobi operators with finite-rank diagonal perturbations by fully characterizing the parameter sets supporting such virtual states as codimension-one algebraic varieties. The explicit recursive and hierarchical description of these varieties enables fine-grained control of the spectral transitions at continuum thresholds. This provides both a powerful analytic and geometric perspective on lattice Schrödinger problems and a toolkit for future extensions to more complex regimes in discrete and semi-discrete quantum systems.