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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Jost function</span></span>
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</style><p>In <a href="Scattering_theory" class="mw-redirect" title="Scattering theory">scattering theory</a>, the <b>Jost function</b> is the <a href="Wronskian" title="Wronskian">Wronskian</a> of the regular solution and the (irregular) Jost solution to the <a href="Differential_equation" title="Differential equation">differential equation</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\psi ''+V\psi =k^{2}\psi }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo>−<!-- − --></mo>
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<mi>V</mi>
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<mi>k</mi>
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<mi>ψ<!-- ψ --></mi>
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<annotation encoding="application/x-tex">{\displaystyle -\psi ''+V\psi =k^{2}\psi }</annotation>
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</math></span><img src="./205d675d012684732f3a2d59d19b52051f0f7fe6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.476ex; height:3.009ex;" alt="{\displaystyle -\psi ''+V\psi =k^{2}\psi }" loading="lazy"></span>.
</p><p>It was introduced by <a href="Res_Jost" title="Res Jost">Res Jost</a>.
</p>
<meta property="mw:PageProp/toc">
<div class="mw-heading mw-heading2"><h2 id="Background">Background</h2></div>
<p>We are looking for solutions <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi (k,r)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>ψ<!-- ψ --></mi>
<mo stretchy="false">(</mo>
<mi>k</mi>
<mo>,</mo>
<mi>r</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \psi (k,r)}</annotation>
</semantics>
</math></span><img src="./3cacd142ff53c7f7e646db4d674020f6274f2334.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.616ex; height:2.843ex;" alt="{\displaystyle \psi (k,r)}" loading="lazy"></span> to the radial <a href="Schr%C3%B6dinger_equation" title="Schrödinger equation">Schrödinger equation</a> in the case <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell =0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
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<mi>ℓ<!-- ℓ --></mi>
<mo>=</mo>
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<annotation encoding="application/x-tex">{\displaystyle \ell =0}</annotation>
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</math></span><img src="./73607cac64f11029ccb86b9403c0ec2bd1629ded.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.231ex; height:2.176ex;" alt="{\displaystyle \ell =0}" loading="lazy"></span>,
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\psi ''+V\psi =k^{2}\psi .}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo>−<!-- − --></mo>
<msup>
<mi>ψ<!-- ψ --></mi>
<mo>″</mo>
</msup>
<mo>+</mo>
<mi>V</mi>
<mi>ψ<!-- ψ --></mi>
<mo>=</mo>
<msup>
<mi>k</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
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<mi>ψ<!-- ψ --></mi>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle -\psi ''+V\psi =k^{2}\psi .}</annotation>
</semantics>
</math></span><img src="./db2428346e4b7e0add602137bd659731487a26d2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.123ex; height:3.009ex;" alt="{\displaystyle -\psi ''+V\psi =k^{2}\psi .}" loading="lazy"></span></dd></dl>
<div class="mw-heading mw-heading2"><h2 id="Regular_and_irregular_solutions">Regular and irregular solutions</h2></div>
<p>A <i>regular solution</i> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi (k,r)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>φ<!-- φ --></mi>
<mo stretchy="false">(</mo>
<mi>k</mi>
<mo>,</mo>
<mi>r</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \varphi (k,r)}</annotation>
</semantics>
</math></span><img src="./2eba38a286b82d09b6d88c6820930de90d65fa9e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.623ex; height:2.843ex;" alt="{\displaystyle \varphi (k,r)}" loading="lazy"></span> is one that satisfies the boundary conditions,
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\varphi (k,0)&=0\\\varphi _{r}'(k,0)&=1.\end{aligned}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true">
<mtr>
<mtd>
<mi>φ<!-- φ --></mi>
<mo stretchy="false">(</mo>
<mi>k</mi>
<mo>,</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<mn>0</mn>
</mtd>
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<mtr>
<mtd>
<msubsup>
<mi>φ<!-- φ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>r</mi>
</mrow>
<mo>′</mo>
</msubsup>
<mo stretchy="false">(</mo>
<mi>k</mi>
<mo>,</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<mn>1.</mn>
</mtd>
</mtr>
</mtable>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\varphi (k,0)&=0\\\varphi _{r}'(k,0)&=1.\end{aligned}}}</annotation>
</semantics>
</math></span><img src="./f59ba2499a1a2788d4904a8e4e6ce7c094a72c95.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:13.37ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}\varphi (k,0)&=0\\\varphi _{r}'(k,0)&=1.\end{aligned}}}" loading="lazy"></span></dd></dl>
<p>If <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{0}^{\infty }r|V(r)|<\infty }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msubsup>
<mo>∫<!-- ∫ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
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<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">∞<!-- ∞ --></mi>
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</msubsup>
<mi>r</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
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<mi>V</mi>
<mo stretchy="false">(</mo>
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<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mo><</mo>
<mi mathvariant="normal">∞<!-- ∞ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \int _{0}^{\infty }r|V(r)|<\infty }</annotation>
</semantics>
</math></span><img src="./fd08f11e058d60c9b4fa4f604b9be3393d5fc346.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:17.136ex; height:5.843ex;" alt="{\displaystyle \int _{0}^{\infty }r|V(r)|<\infty }" loading="lazy"></span>, the solution is given as a <a href="Volterra_integral_equation" title="Volterra integral equation">Volterra integral equation</a>,
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi (k,r)=k^{-1}\sin(kr)+k^{-1}\int _{0}^{r}dr'\sin(k(r-r'))V(r')\varphi (k,r').}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>φ<!-- φ --></mi>
<mo stretchy="false">(</mo>
<mi>k</mi>
<mo>,</mo>
<mi>r</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<msup>
<mi>k</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msup>
<mi>sin</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<mi>k</mi>
<mi>r</mi>
<mo stretchy="false">)</mo>
<mo>+</mo>
<msup>
<mi>k</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msup>
<msubsup>
<mo>∫<!-- ∫ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
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<mrow class="MJX-TeXAtom-ORD">
<mi>r</mi>
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</msubsup>
<mi>d</mi>
<msup>
<mi>r</mi>
<mo>′</mo>
</msup>
<mi>sin</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<mi>k</mi>
<mo stretchy="false">(</mo>
<mi>r</mi>
<mo>−<!-- − --></mo>
<msup>
<mi>r</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">)</mo>
<mo stretchy="false">)</mo>
<mi>V</mi>
<mo stretchy="false">(</mo>
<msup>
<mi>r</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">)</mo>
<mi>φ<!-- φ --></mi>
<mo stretchy="false">(</mo>
<mi>k</mi>
<mo>,</mo>
<msup>
<mi>r</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">)</mo>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \varphi (k,r)=k^{-1}\sin(kr)+k^{-1}\int _{0}^{r}dr'\sin(k(r-r'))V(r')\varphi (k,r').}</annotation>
</semantics>
</math></span><img src="./bb2567b0f66bc463c25e45c4c568c9b954195118.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:61.103ex; height:5.843ex;" alt="{\displaystyle \varphi (k,r)=k^{-1}\sin(kr)+k^{-1}\int _{0}^{r}dr'\sin(k(r-r'))V(r')\varphi (k,r').}" loading="lazy"></span></dd></dl>
<p>There are two <i>irregular solutions</i> (sometimes called Jost solutions) <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{\pm }}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>±<!-- ± --></mo>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f_{\pm }}</annotation>
</semantics>
</math></span><img src="./820ee0fab0fb010fbfbfeda510698a614d435b72.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.65ex; height:2.509ex;" alt="{\displaystyle f_{\pm }}" loading="lazy"></span> with asymptotic behavior <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{\pm }=e^{\pm ikr}+o(1)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>±<!-- ± --></mo>
</mrow>
</msub>
<mo>=</mo>
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>±<!-- ± --></mo>
<mi>i</mi>
<mi>k</mi>
<mi>r</mi>
</mrow>
</msup>
<mo>+</mo>
<mi>o</mi>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f_{\pm }=e^{\pm ikr}+o(1)}</annotation>
</semantics>
</math></span><img src="./695f83a00bfc833d6a2db6734dffd1a0c5a11d35.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.448ex; height:3.176ex;" alt="{\displaystyle f_{\pm }=e^{\pm ikr}+o(1)}" loading="lazy"></span> as <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r\to \infty }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>r</mi>
<mo stretchy="false">→<!-- → --></mo>
<mi mathvariant="normal">∞<!-- ∞ --></mi>
</mstyle>
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<annotation encoding="application/x-tex">{\displaystyle r\to \infty }</annotation>
</semantics>
</math></span><img src="./dcd3a85ea2e3d6b4027434e502cace4177d7a3e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.986ex; height:1.843ex;" alt="{\displaystyle r\to \infty }" loading="lazy"></span>. They are given by the <a href="Volterra_integral_equation" title="Volterra integral equation">Volterra integral equation</a>,
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{\pm }(k,r)=e^{\pm ikr}-k^{-1}\int _{r}^{\infty }dr'\sin(k(r-r'))V(r')f_{\pm }(k,r').}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
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<msub>
<mi>f</mi>
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<mo stretchy="false">(</mo>
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<mo>,</mo>
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<mo stretchy="false">)</mo>
<mo>=</mo>
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>±<!-- ± --></mo>
<mi>i</mi>
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<mo>−<!-- − --></mo>
<msup>
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<mrow class="MJX-TeXAtom-ORD">
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<msubsup>
<mo>∫<!-- ∫ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>r</mi>
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<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">∞<!-- ∞ --></mi>
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<mi>d</mi>
<msup>
<mi>r</mi>
<mo>′</mo>
</msup>
<mi>sin</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<mi>k</mi>
<mo stretchy="false">(</mo>
<mi>r</mi>
<mo>−<!-- − --></mo>
<msup>
<mi>r</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">)</mo>
<mo stretchy="false">)</mo>
<mi>V</mi>
<mo stretchy="false">(</mo>
<msup>
<mi>r</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">)</mo>
<msub>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>±<!-- ± --></mo>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>k</mi>
<mo>,</mo>
<msup>
<mi>r</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">)</mo>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f_{\pm }(k,r)=e^{\pm ikr}-k^{-1}\int _{r}^{\infty }dr'\sin(k(r-r'))V(r')f_{\pm }(k,r').}</annotation>
</semantics>
</math></span><img src="./89e4e0bbce4c4f20da75f7d5beac2b184caa0ad3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:58.168ex; height:5.843ex;" alt="{\displaystyle f_{\pm }(k,r)=e^{\pm ikr}-k^{-1}\int _{r}^{\infty }dr'\sin(k(r-r'))V(r')f_{\pm }(k,r').}" loading="lazy"></span></dd></dl>
<p>If <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\neq 0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>k</mi>
<mo>≠<!-- ≠ --></mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle k\neq 0}</annotation>
</semantics>
</math></span><img src="./3e4367cc52bf0bebde550f396dde7bf07fa67bdd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.472ex; height:2.676ex;" alt="{\displaystyle k\neq 0}" loading="lazy"></span>, then <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{+},f_{-}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>+</mo>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f_{+},f_{-}}</annotation>
</semantics>
</math></span><img src="./d6125dbbc8c12f92c9077399e2c74fd71b4bf411.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.334ex; height:2.509ex;" alt="{\displaystyle f_{+},f_{-}}" loading="lazy"></span> are linearly independent. Since they are solutions to a second order differential equation, every solution (in particular <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>φ<!-- φ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation>
</semantics>
</math></span><img src="./33ee699558d09cf9d653f6351f9fda0b2f4aaa3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }" loading="lazy"></span>) can be written as a linear combination of them.
</p>
<div class="mw-heading mw-heading2"><h2 id="Jost_function_definition">Jost function definition</h2></div>
<p>The <i>Jost function</i> is
</p><p><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega (k):=W(f_{+},\varphi )\equiv \varphi _{r}'(k,r)f_{+}(k,r)-\varphi (k,r)f_{+,r}'(k,r)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>ω<!-- ω --></mi>
<mo stretchy="false">(</mo>
<mi>k</mi>
<mo stretchy="false">)</mo>
<mo>:=</mo>
<mi>W</mi>
<mo stretchy="false">(</mo>
<msub>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>+</mo>
</mrow>
</msub>
<mo>,</mo>
<mi>φ<!-- φ --></mi>
<mo stretchy="false">)</mo>
<mo>≡<!-- ≡ --></mo>
<msubsup>
<mi>φ<!-- φ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>r</mi>
</mrow>
<mo>′</mo>
</msubsup>
<mo stretchy="false">(</mo>
<mi>k</mi>
<mo>,</mo>
<mi>r</mi>
<mo stretchy="false">)</mo>
<msub>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>+</mo>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>k</mi>
<mo>,</mo>
<mi>r</mi>
<mo stretchy="false">)</mo>
<mo>−<!-- − --></mo>
<mi>φ<!-- φ --></mi>
<mo stretchy="false">(</mo>
<mi>k</mi>
<mo>,</mo>
<mi>r</mi>
<mo stretchy="false">)</mo>
<msubsup>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>+</mo>
<mo>,</mo>
<mi>r</mi>
</mrow>
<mo>′</mo>
</msubsup>
<mo stretchy="false">(</mo>
<mi>k</mi>
<mo>,</mo>
<mi>r</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \omega (k):=W(f_{+},\varphi )\equiv \varphi _{r}'(k,r)f_{+}(k,r)-\varphi (k,r)f_{+,r}'(k,r)}</annotation>
</semantics>
</math></span><img src="./98aa89a9a1fbae177ebf54cd60cf49868d3e797b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:54.525ex; height:3.176ex;" alt="{\displaystyle \omega (k):=W(f_{+},\varphi )\equiv \varphi _{r}'(k,r)f_{+}(k,r)-\varphi (k,r)f_{+,r}'(k,r)}" loading="lazy"></span>,
</p><p>where W is the <a href="Wronskian" title="Wronskian">Wronskian</a>. Since <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{+},\varphi }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>+</mo>
</mrow>
</msub>
<mo>,</mo>
<mi>φ<!-- φ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f_{+},\varphi }</annotation>
</semantics>
</math></span><img src="./4d0a0fa99626251d6fa02df7b0f1603fbbd97244.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.204ex; height:2.676ex;" alt="{\displaystyle f_{+},\varphi }" loading="lazy"></span> are both solutions to the same differential equation, the Wronskian is independent of r. So evaluating at <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r=0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>r</mi>
<mo>=</mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle r=0}</annotation>
</semantics>
</math></span><img src="./894a83e863728b4ee2e12f3a999a09f5f2bf1c89.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.31ex; height:2.176ex;" alt="{\displaystyle r=0}" loading="lazy"></span> and using the boundary conditions on <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>φ<!-- φ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation>
</semantics>
</math></span><img src="./33ee699558d09cf9d653f6351f9fda0b2f4aaa3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }" loading="lazy"></span> yields <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega (k)=f_{+}(k,0)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>ω<!-- ω --></mi>
<mo stretchy="false">(</mo>
<mi>k</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<msub>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>+</mo>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>k</mi>
<mo>,</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \omega (k)=f_{+}(k,0)}</annotation>
</semantics>
</math></span><img src="./7e17278514618c1cbe422c247a32506d0f149315.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.432ex; height:2.843ex;" alt="{\displaystyle \omega (k)=f_{+}(k,0)}" loading="lazy"></span>.
</p>
<div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2></div>
<p>The Jost function can be used to construct <a href="Green's_functions" class="mw-redirect" title="Green's functions">Green's functions</a> for
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left[-{\frac {\partial ^{2}}{\partial r^{2}}}+V(r)-k^{2}\right]G=-\delta (r-r').}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow>
<mo>[</mo>
<mrow>
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<msup>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<msup>
<mi>r</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mrow>
</mfrac>
</mrow>
<mo>+</mo>
<mi>V</mi>
<mo stretchy="false">(</mo>
<mi>r</mi>
<mo stretchy="false">)</mo>
<mo>−<!-- − --></mo>
<msup>
<mi>k</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mrow>
<mo>]</mo>
</mrow>
<mi>G</mi>
<mo>=</mo>
<mo>−<!-- − --></mo>
<mi>δ<!-- δ --></mi>
<mo stretchy="false">(</mo>
<mi>r</mi>
<mo>−<!-- − --></mo>
<msup>
<mi>r</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">)</mo>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \left[-{\frac {\partial ^{2}}{\partial r^{2}}}+V(r)-k^{2}\right]G=-\delta (r-r').}</annotation>
</semantics>
</math></span><img src="./2e070d5f1eada1f2d25a08538ef33a05c76a6593.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:37.359ex; height:6.343ex;" alt="{\displaystyle \left[-{\frac {\partial ^{2}}{\partial r^{2}}}+V(r)-k^{2}\right]G=-\delta (r-r').}" loading="lazy"></span></dd></dl>
<p>In fact,
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G^{+}(k;r,r')=-{\frac {\varphi (k,r\wedge r')f_{+}(k,r\vee r')}{\omega (k)}},}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>G</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>+</mo>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mi>k</mi>
<mo>;</mo>
<mi>r</mi>
<mo>,</mo>
<msup>
<mi>r</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>φ<!-- φ --></mi>
<mo stretchy="false">(</mo>
<mi>k</mi>
<mo>,</mo>
<mi>r</mi>
<mo>∧<!-- ∧ --></mo>
<msup>
<mi>r</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">)</mo>
<msub>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>+</mo>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>k</mi>
<mo>,</mo>
<mi>r</mi>
<mo>∨<!-- ∨ --></mo>
<msup>
<mi>r</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">)</mo>
</mrow>
<mrow>
<mi>ω<!-- ω --></mi>
<mo stretchy="false">(</mo>
<mi>k</mi>
<mo stretchy="false">)</mo>
</mrow>
</mfrac>
</mrow>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle G^{+}(k;r,r')=-{\frac {\varphi (k,r\wedge r')f_{+}(k,r\vee r')}{\omega (k)}},}</annotation>
</semantics>
</math></span><img src="./b1925394810d4ee71dd1f87e4c35e0302dbdcdbd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:40.606ex; height:6.509ex;" alt="{\displaystyle G^{+}(k;r,r')=-{\frac {\varphi (k,r\wedge r')f_{+}(k,r\vee r')}{\omega (k)}},}" loading="lazy"></span></dd></dl>
<p>where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r\wedge r'\equiv \min(r,r')}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>r</mi>
<mo>∧<!-- ∧ --></mo>
<msup>
<mi>r</mi>
<mo>′</mo>
</msup>
<mo>≡<!-- ≡ --></mo>
<mo movablelimits="true" form="prefix">min</mo>
<mo stretchy="false">(</mo>
<mi>r</mi>
<mo>,</mo>
<msup>
<mi>r</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle r\wedge r'\equiv \min(r,r')}</annotation>
</semantics>
</math></span><img src="./a431f9b2a3390be38593cb6a1815359ac0a9e930.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.964ex; height:3.009ex;" alt="{\displaystyle r\wedge r'\equiv \min(r,r')}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r\vee r'\equiv \max(r,r')}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>r</mi>
<mo>∨<!-- ∨ --></mo>
<msup>
<mi>r</mi>
<mo>′</mo>
</msup>
<mo>≡<!-- ≡ --></mo>
<mo movablelimits="true" form="prefix">max</mo>
<mo stretchy="false">(</mo>
<mi>r</mi>
<mo>,</mo>
<msup>
<mi>r</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle r\vee r'\equiv \max(r,r')}</annotation>
</semantics>
</math></span><img src="./7a8aaace2624fe785b9968632251fe4a3126f5c8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.414ex; height:3.009ex;" alt="{\displaystyle r\vee r'\equiv \max(r,r')}" loading="lazy"></span>.
</p><p>The analyticity of the Jost function in the particle momentum <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>k</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle k}</annotation>
</semantics>
</math></span><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"></span> allows to establish a relationship between
the scattering phase difference with infinite and zero momenta on one hand
and the number of bound states <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n_{b}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>n</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>b</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n_{b}}</annotation>
</semantics>
</math></span><img src="./bb3645f1f5418cbf5e44f9f5ca21e3353d3258f1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.332ex; height:2.009ex;" alt="{\displaystyle n_{b}}" loading="lazy"></span>, the number of <a href="Robert_Jaffe_(physicist)" title="Robert Jaffe (physicist)"> Jaffe </a>- Low primitives <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n_{p}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>n</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>p</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n_{p}}</annotation>
</semantics>
</math></span><img src="./ba23a1183bdfd6b7d242d6aa83a4a21cc300029d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.454ex; height:2.343ex;" alt="{\displaystyle n_{p}}" loading="lazy"></span>,
and the number of Castillejo -<a href="Richard_Dalitz" title="Richard Dalitz"> Daliz </a>-<a href="Freeman_Dyson" title="Freeman Dyson"> Dyson </a> poles <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n_{\text{CDD}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>n</mi>
<mrow class="MJX-TeXAtom-ORD">
<mtext>CDD</mtext>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n_{\text{CDD}}}</annotation>
</semantics>
</math></span><img src="./2703e26a9337ad6af931707bff84841674e360a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.325ex; height:2.009ex;" alt="{\displaystyle n_{\text{CDD}}}" loading="lazy"></span>
on the other (<a href="Levinson's_theorem" title="Levinson's theorem">Levinson's theorem</a>):
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta (+\infty )-\delta (0)=-\pi ({\frac {1}{2}}n_{0}+n_{b}+n_{p}-n_{\text{CDD}})}">
<semantics>
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<mo stretchy="false">)</mo>
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<mi>δ<!-- δ --></mi>
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<mo>+</mo>
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<mi>p</mi>
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<mo>−<!-- − --></mo>
<msub>
<mi>n</mi>
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<mtext>CDD</mtext>
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<mo stretchy="false">)</mo>
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<annotation encoding="application/x-tex">{\displaystyle \delta (+\infty )-\delta (0)=-\pi ({\frac {1}{2}}n_{0}+n_{b}+n_{p}-n_{\text{CDD}})}</annotation>
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</math></span><img src="./60b7778e40541a8a9aba8a2f4323f3ddee0d050a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:44.978ex; height:5.176ex;" alt="{\displaystyle \delta (+\infty )-\delta (0)=-\pi ({\frac {1}{2}}n_{0}+n_{b}+n_{p}-n_{\text{CDD}})}" loading="lazy"></span>.</dd></dl>
<p>Here <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta (k)}">
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<annotation encoding="application/x-tex">{\displaystyle \delta (k)}</annotation>
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</math></span><img src="./506df3167c39471b2f69864a6dd9c81c02b1f39f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.069ex; height:2.843ex;" alt="{\displaystyle \delta (k)}" loading="lazy"></span> is the scattering phase and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n_{0}}">
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<mrow class="MJX-TeXAtom-ORD">
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<annotation encoding="application/x-tex">{\displaystyle n_{0}}</annotation>
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</math></span><img src="./63584d203ecb012a7bcb90f422408bbfe4018956.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.449ex; height:2.009ex;" alt="{\displaystyle n_{0}}" loading="lazy"></span> = 0 or 1. The value <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n_{0}=1}">
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<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
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<annotation encoding="application/x-tex">{\displaystyle n_{0}=1}</annotation>
</semantics>
</math></span><img src="./e9161a208c40ff5385b51c66f82b313bf5dac192.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.71ex; height:2.509ex;" alt="{\displaystyle n_{0}=1}" loading="lazy"></span> corresponds to the exceptional case of a <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}">
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<mi>s</mi>
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<annotation encoding="application/x-tex">{\displaystyle s}</annotation>
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</math></span><img src="./01d131dfd7673938b947072a13a9744fe997e632.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}" loading="lazy"></span>-wave
scattering in the presence of a bound state with zero energy.
</p>
<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>
<ul><li><style data-mw-deduplicate="TemplateStyles:r1238218222">
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</style><cite id="CITEREFNewton1966" class="citation book cs1">Newton, Roger G. (1966). <i>Scattering Theory of Waves and Particles</i>. New York: McGraw-Hill. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1966stwp.book.....N">1966stwp.book.....N</a>. <a href="OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/362294">362294</a>.</cite></li>
<li><cite id="CITEREFYafaev1992" class="citation book cs1">Yafaev, D. R. (1992). <i>Mathematical Scattering Theory</i>. Providence: American Mathematical Society. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-8218-4558-6</bdi>.</cite></li></ul>
<p><br>
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