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```python
# AI Dev Tool: FastAPI Pydantic v2 Micro-SaaS with PostgreSQL Vector Search Extension
# ==============================================================================
# This is a production-ready developer code asset for building high-performance
# Micro-SaaS applications using FastAPI, Pydantic v2, and PostgreSQL Vector Search Extension.

from fastapi import FastAPI, HTTPException
from fastapi.responses import JSONResponse
from pydantic import BaseModel
from typing import List, Optional
import psycopg2
import psycopg2.extras
from psycopg2 import Error

# Initialize the FastAPI application
app = FastAPI(title="FastAPI Pydantic v2 Micro-SaaS",
              description="A high-performance Micro-SaaS application",
              version="1.0.0")

# Define the Pydantic model for the user
class User(BaseModel):
    """User model"""
    id: int
    name: str
    email: str

# Define the Pydantic model for the search query
class SearchQuery(BaseModel):
    """Search query model"""
    query: str

# Establish a connection to the PostgreSQL database
def connect_to_db():
    """Connect to the PostgreSQL database"""
    try:
        conn = psycopg2.connect(
            dbname="mydatabase

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