<p>We examine the exact a priori majorant <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$M_\gamma =\sup \limits _{q\in A_\gamma }\lambda _0(q)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>M</mi> <mi>γ</mi> </msub> <mo>=</mo> <munder> <mo movablelimits="false">sup</mo> <mrow> <mi>q</mi> <mo>∈</mo> <msub> <mi>A</mi> <mi>γ</mi> </msub> </mrow> </munder> <msub> <mi>λ</mi> <mn>0</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> of the least eigenvalue of the Sturm–Liouville problem <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$-y''+qy=\lambda y$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>-</mo> <msup> <mi>y</mi> <mrow> <mo>′</mo> <mo>′</mo> </mrow> </msup> <mo>+</mo> <mi>q</mi> <mi>y</mi> <mo>=</mo> <mi>λ</mi> <mi>y</mi> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$y(0)=y(1)=0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>y</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>y</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, with a potential <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$q\in C[0,1]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>q</mi> <mo>∈</mo> <mi>C</mi> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation> of the class <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$A_\gamma $</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>A</mi> <mi>γ</mi> </msub> </math></EquationSource> </InlineEquation> determined by the conditions <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$q\le 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>q</mi> <mo>≤</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\int \limits _0^1|q|^\gamma dx=1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <munderover> <mo movablelimits="false">∫</mo> <mn>0</mn> <mn>1</mn> </munderover> <msup> <mrow> <mo stretchy="false">|</mo> <mi>q</mi> <mo stretchy="false">|</mo> </mrow> <mi>γ</mi> </msup> <mi>d</mi> <mi>x</mi> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\gamma \in (0,1/2)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>γ</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">/</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. For this majorant, we prove the strict estimate <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$M_\gamma <\pi ^2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>M</mi> <mi>γ</mi> </msub> <mo><</mo> <msup> <mi>π</mi> <mn>2</mn> </msup> </mrow> </math></EquationSource> </InlineEquation>. The last estimate was known earlier in the case where <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\gamma <1/3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>γ</mi> <mo><</mo> <mn>1</mn> <mo stretchy="false">/</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>.</p>

ON AN A PRIORI MAJORANT OF THE LEAST EIGENVALUES OF THE STURM–LIOUVILLE PROBLEM

期刊论文
发表日期：

2026年2月1日

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A. A. Vladimirov,
E. S. Karulina

摘要

We examine the exact a priori majorant $M_\gamma =\sup \limits _{q\in A_\gamma }\lambda _0(q)$ $M_{γ} = sup_{q \in A_{γ}} λ_{0} (q)$ of the least eigenvalue of the Sturm–Liouville problem $-y''+qy=\lambda y$ $- y^{''} + q y = λ y$ , $y(0)=y(1)=0$ $y (0) = y (1) = 0$ , with a potential $q\in C[0,1]$ $q \in C [0, 1]$ of the class $A_\gamma $ $A_{γ}$ determined by the conditions $q\le 0$ $q \leq 0$ and $\int \limits _0^1|q|^\gamma dx=1$ $\int_{0}^{1} {| q |}^{γ} d x = 1$ , where $\gamma \in (0,1/2)$ $γ \in (0, 1 / 2)$ . For this majorant, we prove the strict estimate $M_\gamma <\pi ^2$ $M_{γ} < π^{2}$ . The last estimate was known earlier in the case where $\gamma <1/3$ $γ < 1 / 3$ .