<p>In this paper, we study the eigenvalue problem for the third-order differential operator with a positive delta point interaction under the periodic boundary condition. For a constant <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\alpha >0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, we consider the operator <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$H=i\frac{d^3}{dx^3}+\alpha \delta (x)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>H</mi> <mo>=</mo> <mi>i</mi> <mfrac> <msup> <mi>d</mi> <mn>3</mn> </msup> <mrow> <mi>d</mi> <msup> <mi>x</mi> <mn>3</mn> </msup> </mrow> </mfrac> <mo>+</mo> <mi>α</mi> <mi>δ</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> in <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$L^2(-1/2,1/2)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mo>-</mo> <mn>1</mn> <mo stretchy="false">/</mo> <mn>2</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">/</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\delta (\cdot )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>δ</mi> <mo stretchy="false">(</mo> <mo>·</mo> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is the Dirac’s delta function supported at the origin. We impose a function <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$y\in \textrm{dom}(H)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>y</mi> <mo>∈</mo> <mtext>dom</mtext> <mo stretchy="false">(</mo> <mi>H</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> the periodic boundary conditions <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$y^{(j)}(1/2)=y^{(j)}(-1/2), j=0,1,2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>y</mi> <mrow> <mo stretchy="false">(</mo> <mi>j</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">/</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msup> <mi>y</mi> <mrow> <mo stretchy="false">(</mo> <mi>j</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mo>-</mo> <mn>1</mn> <mo stretchy="false">/</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <mi>j</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>. Then, its spectrum <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\sigma (H)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>σ</mi> <mo stretchy="false">(</mo> <mi>H</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is discrete. We show that there exists exactly one eigenvalue <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\lambda _n$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>λ</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation> in the interval <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$((2n\pi )^3,(2(n+1)\pi )^3)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mi>π</mi> <mo stretchy="false">)</mo> </mrow> <mn>3</mn> </msup> <mo>,</mo> <msup> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mi>π</mi> <mo stretchy="false">)</mo> </mrow> <mn>3</mn> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> for each <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$n\in {{\mathbb {Z}}}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>∈</mo> <mi mathvariant="double-struck">Z</mi> </mrow> </math></EquationSource> </InlineEquation>.</p>

Periodic Eigenvalues for a Third-Order Differential Operator Plus a Positive Delta Interaction

期刊论文
发表日期：

2026年5月26日

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Hiroaki Niikuni

摘要

In this paper, we study the eigenvalue problem for the third-order differential operator with a positive delta point interaction under the periodic boundary condition. For a constant $\alpha >0$ $α > 0$ , we consider the operator $H=i\frac{d^3}{dx^3}+\alpha \delta (x)$ $H = i \frac{d^{3}}{d x^{3}} + α δ (x)$ in $L^2(-1/2,1/2)$ $L^{2} (- 1 / 2, 1 / 2)$ , where $\delta (\cdot )$ $δ (\cdot)$ is the Dirac’s delta function supported at the origin. We impose a function $y\in \textrm{dom}(H)$ $y \in dom (H)$ the periodic boundary conditions $y^{(j)}(1/2)=y^{(j)}(-1/2), j=0,1,2$ $y^{(j)} (1 / 2) = y^{(j)} (- 1 / 2), j = 0, 1, 2$ . Then, its spectrum $\sigma (H)$ $σ (H)$ is discrete. We show that there exists exactly one eigenvalue $\lambda _n$ $λ_{n}$ in the interval $((2n\pi )^3,(2(n+1)\pi )^3)$ $({(2 n π)}^{3}, {(2 (n + 1) π)}^{3})$ for each $n\in {{\mathbb {Z}}}$ $n \in Z$ .