# Dirac delta distribution

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##### 1: 1.16 Distributions
The Dirac delta distribution is singular. …
##### 2: 2.6 Distributional Methods
2.6.17 ${\left\langle f,\phi\right\rangle}=\sum_{s=0}^{n-1}a_{s}\left\langle t^{-s-% \alpha},\phi\right\rangle-\sum_{s=1}^{n}c_{s}\left\langle{\delta}^{(s-1)},\phi% \right\rangle+\left\langle f_{n},\phi\right\rangle$
The Dirac delta distribution in (2.6.17) is given by
2.6.19 $\left\langle{\delta}^{(s)},\phi\right\rangle=(-1)^{s}\phi^{(s)}(0),$ $s=0,1,2,\dots$;
2.6.20 ${\left\langle f,\phi\right\rangle}=\sum_{s=0}^{n-1}a_{s}\left\langle t^{-s-1},% \phi\right\rangle-\sum_{s=1}^{n}d_{s}\left\langle{\delta}^{(s-1)},\phi\right% \rangle+\left\langle f_{n},\phi\right\rangle$
Furthermore, $K^{+}$ contains the distributions $H$, $\delta$, and $t^{\lambda}$, $t>0$, for any real (or complex) number $\lambda$, where $H$ is the distribution associated with the Heaviside function $H\left(t\right)$1.16(iv)), and $t^{\lambda}$ is the distribution defined by (2.6.12)–(2.6.14), depending on the value of $\lambda$. …
##### 3: 1.17 Integral and Series Representations of the Dirac Delta
In applications in physics and engineering, the Dirac delta distribution1.16(iii)) is historically and customarily replaced by the Dirac delta (or Dirac delta function) $\delta\left(x\right)$. …
##### 4: 14.30 Spherical and Spheroidal Harmonics
14.30.8 $\int_{0}^{2\pi}\!\!\int_{0}^{\pi}\overline{Y_{{l_{1}},{m_{1}}}\left(\theta,% \phi\right)}Y_{{l_{2}},{m_{2}}}\left(\theta,\phi\right)\sin\theta\mathrm{d}% \theta\mathrm{d}\phi=\delta_{l_{1},l_{2}}\delta_{m_{1},m_{2}}.$
###### Distributional Completeness
For a series representation of the product of two Dirac deltas in terms of products of spherical harmonics see §1.17(iii). …
##### 5: 1.18 Linear 2nd Order Differential Operators and Eigenfunction Expansions
of the Dirac delta function. Equation (1.18.19) is often called the completeness relation. Applying the representation (1.17.13), now symmetrized as in (1.17.14), as $\frac{1}{x}\delta\left(x-y\right)=\frac{1}{\sqrt{xy}}\delta\left(x-y\right)$ ,For $f(x)$ even in $x$ this yields the Fourier cosine transform pair (1.14.9) & (1.14.11), and for $f(x)$ odd the Fourier sine transform pair (1.14.10) & (1.14.12). These latter results also correspond to use of the $\delta\left(x-y\right)$ as defined in (1.17.12_1) and (1.17.12_2). Suppose that $X$ is the whole real line in one dimension, and that $q(x)$ , in (1.18.28) has (non-oscillatory) limits of $0$ at both $\pm\infty$ , and thus a continuous spectrum on $\boldsymbol{\sigma}\geq 0$ . What then is the condition on $q(x)$ to insure the existence of at least a single eigenvalue in the point spectrum? The discussions of §1.18(vi) imply that if $q(x)\equiv 0$ then there is only a continuous spectrum. Surprisingly, if $q(x)<0$ on any interval on the real line, even if positive elsewhere, as long as $\int_{X}q(x)\mathrm{d}x\leq 0$ , see Simon (1976, Theorem 2.5), then there will be at least one eigenfunction with a negative eigenvalue, with corresponding $L^{2}(X)$ eigenfunction. Thus, and this is a case where $q(x)$ is not continuous, if $q(x)=-\alpha\delta\left(x-a\right),\alpha>0$ , there will be an $L^{2}$ eigenfunction localized in the vicinity of $x=a$ , with a negative eigenvalue, thus disjoint from the continuous spectrum on $[0,\infty)$ . Similar results hold for two, but not higher, dimensional quantum systems. See Brownstein (2000) and Yang and de Llano (1989) for numerical examples, based on variational calculations, and Simon (1976) and Chadan et al. (2003) for rigorous mathematical discussion. See §18.39 for discussion of Schrödinger equations and operators. For fixed angular momentum $\ell$ the appropriate self-adjoint extension of the above operator may have both a discrete spectrum of negative eigenvalues $\lambda_{n},n=0,1,\dots,N-1$ , with corresponding $L^{2}([0,\infty),r^{2}\mathrm{d}r)$ eigenfunctions $\phi_{n}(r)$ , and also a continuous spectrum $\lambda\in[0,\infty)$ , with Dirac-delta normalized eigenfunctions $\phi_{\lambda}(r)$ , also with measure $r^{2}\mathrm{d}r$ . Unlike in the example in the paragraph above: in 3-dimensions a “dip below zero, or a potential well” in $V(r)$ does not always correspond to the existence of a discrete part of the spectrum. The well must be deep and broad enough to allow existence of such $L^{2}$ discrete states. The number, $N$ , of discrete states depends on the nature of $V(r)$ , as well as $\ell$ , and, again, $V(r)$ must vanish as $r\to\infty$ , corresponding to the traditionally assumed start of the energy continuum at $\lambda=0$ . In unusual cases $N=\infty$ , even for all $\ell$ , such as in the case of the Schrödinger Coulomb problem ( $V=-r^{-1}$ ) discussed in §18.39 and §33.14, where the point spectrum actually accumulates at the onset of the continuum at $\lambda=0$ , implying an essential singularity, as well as a branch point, in matrix elements of the resolvent, (1.18.67). See Bethe and Salpeter (1977, Ch. 1, (4.12)–(4.13)) for the resulting transform pair in this case.
##### 6: 20.13 Physical Applications
is also a solution of (20.13.2), and it approaches a Dirac delta1.17) at $t=0$. … In the singular limit $\Im\tau\rightarrow 0+$, the functions $\theta_{j}\left(z\middle|\tau\right)$, $j=1,2,3,4$, become integral kernels of Feynman path integrals (distribution-valued Green’s functions); see Schulman (1981, pp. 194–195). …
##### 7: Bibliography L
• D. A. Levine (1969) Algorithm 344: Student’s t-distribution [S14]. Comm. ACM 12 (1), pp. 37–38.
• Y. T. Li and R. Wong (2008) Integral and series representations of the Dirac delta function. Commun. Pure Appl. Anal. 7 (2), pp. 229–247.
• J. E. Littlewood (1914) Sur la distribution des nombres premiers. Comptes Rendus de l’Academie des Sciences, Paris 158, pp. 1869–1872 (French).