<p>We study the discrete spectrum of fiber Hamiltonians <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$H_{\mu _1\mu _2}(K)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>H</mi> <mrow> <msub> <mi>μ</mi> <mn>1</mn> </msub> <msub> <mi>μ</mi> <mn>2</mn> </msub> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>K</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> for a two-boson lattice model on <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathbb {Z}^3$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> <mn>3</mn> </msup> </math></EquationSource> </InlineEquation> with on-site and nearest-neighbour interactions. The operators depend on the total quasi-momentum <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$K\in \mathbb {T}^3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>K</mi> <mo>∈</mo> <msup> <mrow> <mi mathvariant="double-struck">T</mi> </mrow> <mn>3</mn> </msup> </mrow> </math></EquationSource> </InlineEquation> and the real coupling constants <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mu _1,\mu _2,$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>μ</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>μ</mi> <mn>2</mn> </msub> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> and act on the Hilbert space <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$L^{2,\textrm{e}}(\mathbb {T}^3)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mrow> <mn>2</mn> <mo>,</mo> <mtext>e</mtext> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">T</mi> </mrow> <mn>3</mn> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> of even square-integrable functions. Each fiber Hamiltonian is a self-adjoint rank-four perturbation of the multiplication operator by the two-particle fiber dispersion relation, so its essential spectrum is explicitly determined. For <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$K=0,$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>K</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> the space <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$L^{2,\textrm{e}}(\mathbb {T}^3)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mrow> <mn>2</mn> <mo>,</mo> <mtext>e</mtext> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">T</mi> </mrow> <mn>3</mn> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> decomposes into three orthogonal invariant subspaces. In one subspace, the <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$(\mu _1,\mu _2)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <msub> <mi>μ</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>μ</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-plane is divided by smooth hyperbolic curves into regions where the number of eigenvalues below the lower edge of the essential spectrum is constant and equal to 0, 1 or 2. On these curves, eigenvalues reach the lower threshold of the essential spectrum and generate threshold resonances. In each of the remaining two subspaces, a unique critical point on the <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\mu _2$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>μ</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation>-axis separates the parameter regimes with 0 and 1 eigenvalue below the essential spectrum; at this point, the lower threshold becomes an eigenvalue. An analogous description holds for the number of eigenvalues above the upper edge of the essential spectrum, together with the corresponding threshold phenomena at the upper spectral edge. In addition, we obtain a complete classification of the discrete spectrum of <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$H_{\mu _1\mu _2}(0)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>H</mi> <mrow> <msub> <mi>μ</mi> <mn>1</mn> </msub> <msub> <mi>μ</mi> <mn>2</mn> </msub> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> describing the exact number of eigenvalues lying below and above the essential spectrum in each region of the parameter plane. Moreover, we establish stability of these eigenvalues under perturbations of the total quasi-momentum <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$K\in \mathbb {T}^3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>K</mi> <mo>∈</mo> <msup> <mrow> <mi mathvariant="double-struck">T</mi> </mrow> <mn>3</mn> </msup> </mrow> </math></EquationSource> </InlineEquation>: for <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$K\ne 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>K</mi> <mo>≠</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, the operator <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$H_{\mu _1\mu _2}(K)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>H</mi> <mrow> <msub> <mi>μ</mi> <mn>1</mn> </msub> <msub> <mi>μ</mi> <mn>2</mn> </msub> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>K</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> has at least as many eigenvalues below and above the essential spectrum as <InlineEquation ID="IEq14"> <EquationSource Format="TEX">$H_{\mu _1\mu _2}(0).$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>H</mi> <mrow> <msub> <mi>μ</mi> <mn>1</mn> </msub> <msub> <mi>μ</mi> <mn>2</mn> </msub> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation> If the lower edge of the essential spectrum is a threshold resonance of <InlineEquation ID="IEq15"> <EquationSource Format="TEX">$H_{\mu _1\mu _2}(0)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>H</mi> <mrow> <msub> <mi>μ</mi> <mn>1</mn> </msub> <msub> <mi>μ</mi> <mn>2</mn> </msub> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, then <InlineEquation ID="IEq16"> <EquationSource Format="TEX">$H_{\mu _1\mu _2}(K)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>H</mi> <mrow> <msub> <mi>μ</mi> <mn>1</mn> </msub> <msub> <mi>μ</mi> <mn>2</mn> </msub> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>K</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> has at least one more eigenvalue below the essential spectrum than <InlineEquation ID="IEq17"> <EquationSource Format="TEX">$H_{\mu _1\mu _2}(0).$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>H</mi> <mrow> <msub> <mi>μ</mi> <mn>1</mn> </msub> <msub> <mi>μ</mi> <mn>2</mn> </msub> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation> If the lower edge is a threshold eigenvalue of multiplicity two then <InlineEquation ID="IEq18"> <EquationSource Format="TEX">$H_{\mu _1\mu _2}(K)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>H</mi> <mrow> <msub> <mi>μ</mi> <mn>1</mn> </msub> <msub> <mi>μ</mi> <mn>2</mn> </msub> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>K</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> has at least two more such eigenvalues. Analogous statement holds for the upper edge of the essential spectrum.</p>

Threshold States and Coupling-Constant Classification of Bound States for a Family of Finite-Rank Lattice Hamiltonians

期刊论文
发表日期：

2026年7月17日

查看全文

Saidakhmat N. Lakaev,
Saidakbar S. Abduvayitov,
Shukhrat S. Lakaev

摘要

We study the discrete spectrum of fiber Hamiltonians $H_{\mu _1\mu _2}(K)$ $H_{μ_{1} μ_{2}} (K)$ for a two-boson lattice model on $\mathbb {Z}^3$ $Z^{3}$ with on-site and nearest-neighbour interactions. The operators depend on the total quasi-momentum $K\in \mathbb {T}^3$ $K \in T^{3}$ and the real coupling constants $\mu _1,\mu _2,$ $μ_{1}, μ_{2},$ and act on the Hilbert space $L^{2,\textrm{e}}(\mathbb {T}^3)$ $L^{2, e} (T^{3})$ of even square-integrable functions. Each fiber Hamiltonian is a self-adjoint rank-four perturbation of the multiplication operator by the two-particle fiber dispersion relation, so its essential spectrum is explicitly determined. For $K=0,$ $K = 0,$ the space $L^{2,\textrm{e}}(\mathbb {T}^3)$ $L^{2, e} (T^{3})$ decomposes into three orthogonal invariant subspaces. In one subspace, the $(\mu _1,\mu _2)$ $(μ_{1}, μ_{2})$ -plane is divided by smooth hyperbolic curves into regions where the number of eigenvalues below the lower edge of the essential spectrum is constant and equal to 0, 1 or 2. On these curves, eigenvalues reach the lower threshold of the essential spectrum and generate threshold resonances. In each of the remaining two subspaces, a unique critical point on the $\mu _2$ $μ_{2}$ -axis separates the parameter regimes with 0 and 1 eigenvalue below the essential spectrum; at this point, the lower threshold becomes an eigenvalue. An analogous description holds for the number of eigenvalues above the upper edge of the essential spectrum, together with the corresponding threshold phenomena at the upper spectral edge. In addition, we obtain a complete classification of the discrete spectrum of $H_{\mu _1\mu _2}(0)$ $H_{μ_{1} μ_{2}} (0)$ describing the exact number of eigenvalues lying below and above the essential spectrum in each region of the parameter plane. Moreover, we establish stability of these eigenvalues under perturbations of the total quasi-momentum $K\in \mathbb {T}^3$ $K \in T^{3}$ : for $K\ne 0$ $K \neq 0$ , the operator $H_{\mu _1\mu _2}(K)$ $H_{μ_{1} μ_{2}} (K)$ has at least as many eigenvalues below and above the essential spectrum as $H_{\mu _1\mu _2}(0).$ $H_{μ_{1} μ_{2}} (0) .$ If the lower edge of the essential spectrum is a threshold resonance of $H_{\mu _1\mu _2}(0)$ $H_{μ_{1} μ_{2}} (0)$ , then $H_{\mu _1\mu _2}(K)$ $H_{μ_{1} μ_{2}} (K)$ has at least one more eigenvalue below the essential spectrum than $H_{\mu _1\mu _2}(0).$ $H_{μ_{1} μ_{2}} (0) .$ If the lower edge is a threshold eigenvalue of multiplicity two then $H_{\mu _1\mu _2}(K)$ $H_{μ_{1} μ_{2}} (K)$ has at least two more such eigenvalues. Analogous statement holds for the upper edge of the essential spectrum.